Chapter 13 Simple Harmonic Motion

Size: px
Start display at page:

Download "Chapter 13 Simple Harmonic Motion"

Transcription

1 We are to adit no ore causes of natural things than such as are both true and sufficient to explain their appearances. Isaac Newton 13.1 Introduction to Periodic Motion Periodic otion is any otion that repeats itself in equal intervals of tie. The uniforly rotating earth represents a periodic otion that repeats itself every 24 hours. The otion of the earth around the sun is periodic, repeating itself every 12 onths. A vibrating spring and a pendulu also exhibit periodic otion. The period of the otion is defined as the tie for the otion to repeat itself. A special type of periodic otion is siple haronic otion and we now proceed to investigate it Siple Haronic Motion An exaple of siple haronic otion is the vibration of a ass, attached to a spring of negligible ass, as the ass slides on a frictionless surface, as shown in figure We say that the ass, in the unstretched position, figure 13.1(a), is in its equilibriu position. If an applied force F A acts on the ass, the ass will be displaced to the right of its equilibriu position a distance x, figure 13.1(b). The distance that the spring stretches, obtained fro Hooke s law, is F A = kx The displaceent x is defined as the distance the body oves fro its equilibriu position. Because F A is a force that pulls the ass to the right, it is also the force that pulls the spring to the right. By Newton s third law there is an equal but opposite elastic force exerted by the spring on the ass pulling the ass toward the left. Since this force tends to restore the ass to its original position, we call it the restoring force F R. Because the restoring force is opposite to the applied force, it is given by F R = F A = kx (13.1) When the applied force F A is reoved, the elastic restoring force F R is then the only force acting on the ass, figure 13.1(c), and it tries to restore to its equilibriu position. We can then find the acceleration of the ass fro Newton s second law as Thus, a = F R = kx a = k x (13.2) 13-1

2 Figure 13.1 The vibrating spring. Equation 13.2 is the defining equation for siple haronic otion. Siple haronic otion is otion in which the acceleration of a body is directly proportional to its displaceent fro the equilibriu position but in the opposite direction. A vibrating syste that executes siple haronic otion is soeties called a haronic oscillator. Because the acceleration is directly proportional to the displaceent x in siple haronic otion, the acceleration of the syste is not constant but varies with x. At large displaceents, the acceleration is large, at sall displaceents the acceleration is sall. Describing the vibratory otion of the ass requires soe new techniques and we will do so in section However, let us first look at the otion fro a physical point of view. The ass in figure 13.2(a) is pulled a distance x = A to the right, and is then released. The axiu restoring force on acts to the left at this position because F Rax = kx ax = ka The axiu displaceent A is called the aplitude of the otion. At this position the ass experiences its axiu acceleration to the left. Fro equation 13-2

3 Figure 13.2 Detailed otion of the vibrating spring we obtain a = k A The ass continues to ove toward the left while the acceleration continuously decreases. At the equilibriu position, figure 13.2(b), x = 0 and hence, fro equation 13.2, the acceleration is also zero. However, because the ass has inertia it oves past the equilibriu position to negative values of x, thereby copressing the spring. The restoring force F R now points to the right, since for negative values of x, F R = k( x) = kx The force acting toward the right causes the ass to slow down, eventually coing to rest at x = A. At this point, figure 13.2(c), there is a axiu restoring force pointing toward the right F Rax = k( A) ax = ka 13-3

4 and hence a axiu acceleration a ax = k ( A) = k A also to the right. The ass oves to the right while the force F R and the acceleration a decreases with x until x is again equal to zero, figure 13.2(d). Then F R and a are also zero. Because of the inertia of the ass, it oves past the equilibriu position to positive values of x. The restoring force again acts toward the left, slowing down the ass. When the displaceent x is equal to A, figure 13.2(e), the ass oentarily coes to rest and then the otion repeats itself. One coplete otion fro x = A and back to x = A is called a cycle or an oscillation. The period T is the tie for one coplete oscillation, and the frequency f is the nuber of coplete oscillations or cycles ade in unit tie. The period and the frequency are reciprocal to each other, that is, f = 1 (13.3) T The unit for a period is the second, while the unit for frequency, called a hertz, is one cycle per second. The hertz is abbreviated, Hz. Also note that a cycle is a nuber not a diensional quantity and can be dropped fro the coputations whenever doing so is useful Analysis of Siple Haronic Motion -- The Reference Circle As pointed out in section 13.2, the acceleration of the ass in the vibrating spring syste is given by equation 13.2 as a = k x Since the acceleration a = d 2 x/dt 2, equation 13.2 can be written as d 2 x = k x (13.4) dt 2 Equation 13.4 is a second-order differential equation that copletely describes the siple haronic otion of the ass. Unfortunately, the solution of such differential equations is beyond the scope of this course. We also can not use the kineatic equations derived in chapter 2 because they were based on the assuption that the acceleration of the syste was a constant. As we can see fro 13-4

5 equation 13.2, this assuption is no longer valid because the acceleration varies with the displaceent x. Thus, a new set of equations ust be derived to describe siple haronic otion. It turns out that siple haronic otion is related to the unifor circular otion studied in chapter 4. An analysis of unifor circular otion gives us a set of equations to describe siple haronic otion. As an exaple, consider a point Q oving in unifor circular otion with an angular velocity ω, as shown in figure 13.3(a). At a particular instant of tie t, the angle θ that Q has turned through is θ = ωt (13.5) Figure 13.3 Siple haronic otion and the reference circle. The projection of point Q onto the x-axis gives the point P. As Q rotates in the circle, P oscillates back and forth along the x-axis, figure 13.3(b). That is, when Q is at position 1, P is at 1. As Q oves to position 2 on the circle, P oves to the left along the x-axis to position 2. As Q oves to position 3, P oves on the x-axis to position 3, which is of course the value of x = 0. As Q oves to position 4 on the circle, P oves along the negative x-axis to position 4. When Q arrives at position 5, P is also there. As Q oves to position 6 on the circle, P oves to position 6 on the x-axis. Then finally, as Q oves through positions 7, 8, and 1, P oves through 7, 8, and 1, respectively. The oscillatory otion of point P on the x-axis corresponds to 13-5

6 the siple haronic otion of a body oving under the influence of an elastic restoring force, as shown in figure The position of P on the x-axis and hence the position of the ass is described in ters of the point Q and the angle θ found in figure 13.3(a) as x = A cos θ (13.6) Here A is the aplitude of the vibratory otion and using the value of θ fro equation 13.5 we have x = A cos ωt (13.7) Equation 13.7 is the first kineatic equation for siple haronic otion; it gives the displaceent of the vibrating body at any instant of tie t. The angular velocity ω of point Q in the reference circle is related to the frequency of the siple haronic otion. Because the angular velocity was defined as ω = θ t then, for a coplete rotation of point Q, θ rotates through an angle of 2π rad. But this occurs in exactly the tie for P to execute one coplete vibration. We call this tie for one coplete vibration the period T. Hence, we can also write the angular velocity as ω = θ = 2π (13.8) t T Since the frequency f is the reciprocal of the period T (equation 13.3) we can write equation 13.8 as ω = 2πf (13.9) Thus, the angular velocity of the unifor circular otion in the reference circle is related to the frequency of the vibrating syste. Because of this relation between the angular velocity and the frequency of the syste, we usually call the angular velocity ω the angular frequency of the vibrating syste. We can substitute equation 13.9 into equation 13.7 to give another for for the first kineatic equation of siple haronic otion, naely x = A cos(2πft) (13.10) We can find the velocity of the ass attached to the end of the spring in figure 13.2 with the help of the reference circle in figure 13.3(c). The point Q oves with the tangential velocity V T. The x-coponent of this velocity is the velocity of the point P and hence the velocity of the ass. Fro figure 13.3(c) we can see that 13-6

7 v = V T sin θ (13.11) The inus sign indicates that the velocity of P is toward the left at this position. The linear velocity V T of the point Q is related to the angular velocity ω by equation 9.3 of chapter 9, that is v = rω For the reference circle, v = V T and r is the aplitude A. Hence, the tangential velocity V T is given by V T = ωa (13.12) Using equations 13.11, 13.12, and 13.5, the velocity of point P becoes v = ωa sin ωt (13.13) Equation is the second of the kineatic equations for siple haronic otion and it gives the speed of the vibrating ass at any tie t. We could also find the velocity equation by a direct differentiation of equation 13.7 as v = dx = d(a cos ωt) = A ( sin ωt) d(ωt) dt dt dt and v = ωa sin ωt The differentiation is a sipler derivation of the velocity of the ass but it does not show the physical relation between the reference circle and siple haronic otion. A third kineatic equation for siple haronic otion giving the speed of the vibrating body as a function of displaceent can be found fro equation by using the trigonoetric identity sin 2 θ + cos 2 θ = 1 or sin = ± 1 cos 2 Fro figure 13.3(a) or equation 13.6, we have Hence, cos θ = x A sin = ± 1 x2 A 2 (13.14) Substituting equation back into equation 13.13, we get 13-7

8 or v = ± A 1 x2 A 2 v = ± A 2 x 2 (13.15) Equation is the third of the kineatic equations for siple haronic otion and it gives the velocity of the oving body at any displaceent x. The ± sign in equation indicates the direction of the vibrating body. If the body is oving to the right, then the positive sign (+) is used. If the body is oving to the left, then the negative sign ( ) is used. Finally, we can find the acceleration of the vibrating body using the reference circle in figure 13.3(d). The point Q in unifor circular otion experiences a centripetal acceleration a c pointing toward the center of the circle in figure 13.3(d). The x-coponent of the centripetal acceleration is the acceleration of the vibrating body at the point P. That is, a = a c cos θ (13.16) The inus sign again indicates that the acceleration is toward the left. But recall fro chapter 4 that the agnitude of the centripetal acceleration is a c = v 2 (4.56) r where v represents the tangential speed of the rotating object, which in the present case is V T, and r is the radius of the circle, which in the present case is the radius of the reference circle A. Thus, 2 a c = V T A But we saw in equation that V T = ωa, therefore a c = ω 2 A The acceleration of the ass, equation 13.16, thus becoes a = ω 2 A cos ωt (13.17) Equation is the fourth of the kineatic equations for siple haronic otion. It gives the acceleration of the vibrating body at any tie t. Equation could also have been found by a direct differentiation of equation as 13-8

9 and a = dv = d( ωa sin ωt) = ωa d(sin ωt) = ωa cos ωt d(ωt) dt dt dt dt a = ω 2 A cos ωt The differentiation is a sipler derivation of the acceleration but it does not show the physical relation between the reference circle and siple haronic otion. We should also point out that since F = a by Newton s second law, the force acting on the ass, becoes F = ω 2 A cos ωt (13.18) which shows how the force acting on the ass varies with tie. Equations 13.7 and can be cobined into the siple equation If equation is copared with equation 13.2, a = ω 2 x (13.19) a = k x we see that the acceleration of the ass at P, equation 13.19, is directly proportional to the displaceent x and in the opposite direction. But this is the definition of siple haronic otion as stated in equation Hence, the projection of a point at Q, in unifor circular otion, onto the x-axis does indeed represent siple haronic otion. Thus, the kineatic equations developed to describe the otion of the point P, also describe the otion of a ass attached to a vibrating spring. An iportant relation between the characteristics of the spring and the vibratory otion can be easily deduced fro equations 13.2 and That is, because both equations represent the acceleration of the vibrating body they can be equated to each other, giving ω 2 = k or = k (13.20) The value of ω in the kineatic equations is expressed in ters of the force constant k of the spring and the ass attached to the spring. The physics of siple haronic otion is thus connected to the angular frequency ω of the vibration. In suary, the kineatic equations for siple haronic otion are x = A cos ωt (13.7) 13-9

10 where, fro equations 13.9 and 13.20, we have v = ωa sin ωt (13.13) v = ± A 2 x 2 (13.15) a = ω 2 A cos ωt (13.17) F = ω 2 A cos ωt (13.18) = 2 f = k A plot of the displaceent, velocity, and acceleration of the vibrating body as a function of tie are shown in figure We can see that the atheatical description follows the physical description in figure When x = A, the Figure 13.4 Displaceent, velocity, and acceleration in siple haronic otion. axiu displaceent, the velocity v is zero, while the acceleration is at its axiu value of ω 2 A. The inus sign indicates that the acceleration is toward the left. The force is at its axiu value of ω 2 A, where the inus sign shows 13-10

11 that the restoring force is pulling the ass back toward its equilibriu position. At the equilibriu position x = 0, a = 0, and F = 0, but v has its axiu velocity of ωa toward the left. As x goes to negative values, the force and the acceleration becoe positive, slowing down the otion to the left, and hence v starts to decrease. At x = A the velocity is zero and the force and acceleration take on their axiu values toward the right, tending to restore the ass to its equilibriu position. As x becoes less negative, the velocity to the right increases, until it picks up its axiu value of ωa at x = 0, the equilibriu position, where F and a are both zero. Because of this large velocity, the ass passes the equilibriu position in its otion toward the right. However, as soon as x becoes positive, the force and the acceleration becoe negative thereby slowing down the ass until its velocity becoes zero at the axiu displaceent A. One entire cycle has been copleted, and the otion starts over again. (We should ephasize here that in this vibratory otion there are two places where the velocity is instantaneously zero, x = A and x = A, even though the instantaneous acceleration is nonzero there.) Soeties the vibratory otion is so rapid that the actual displaceent, velocity, and acceleration at every instant of tie are not as iportant as the gross otion, which can be described in ters of the frequency or period of the otion. We can find the frequency of the vibrating ass in ters of the spring constant k and the vibrating ass by setting equation 13.9 equal to equation Thus, = 2 f = k Solving for the frequency f, we obtain f = 1 2 k (13.21) Equation gives the frequency of the vibration. Because the period of the vibrating otion is the reciprocal of the frequency, we get for the period T = 2 k (13.22) Equation gives the period of the siple haronic otion in ters of the ass in otion and the spring constant k. Notice that for a particular value of and k, the period of the otion reains a constant throughout the otion. Exaple 13.1 An exaple of siple haronic otion. A ass of kg is placed on a vertical spring and the spring stretches by 10.0 c. It is then pulled down an additional 5.00 c and then released. Find (a) the spring constant k, (b) the angular frequency ω, (c) the frequency f, (d) the period T, (e) the axiu velocity of the vibrating ass, 13-11

12 (f) the axiu acceleration of the ass, (g) the axiu restoring force, (h) the velocity of the ass at x = 2.00 c, and (i) the equation of the displaceent, velocity, and acceleration at any tie t. Solution Although the original analysis dealt with a ass on a horizontal frictionless surface, the results also apply to a ass attached to a spring that is allowed to vibrate in the vertical direction. The constant force of gravity on the kg ass displaces the equilibriu position to x = 10.0 c. When the additional force is applied to displace the ass another 5.00 c, the ass oscillates about the equilibriu position, located at the 10.0-c ark. Thus, the force of gravity only displaces the equilibriu position, but does not otherwise influence the result of the dynaic otion. a. The spring constant, found fro Hooke s law, is k = F A = g x x = (0.300 kg)(9.80 /s 2 ) = 29.4 N/ b. The angular frequency ω, found fro equation 13.20, is = k = 29.4 N/ kg = 9.90 rad/s c. The frequency of the otion, found fro equation 13.9, is f = ω 2π = 9.90 rad/s 2π rad = 1.58 cycles = 1.58 Hz s d. We could find the period fro equation but since we already know the frequency f, it is easier to copute T fro equation Thus, T = 1 = 1 = s f 1.58 cycles/s 13-12

13 e. The axiu velocity, found fro equation 13.13, is v ax = ωa = (9.90 rad/s)( ) = /s f. The axiu acceleration, found fro equation 13.17, is a ax = ω 2 A = (9.90 rad/s) 2 ( ) = 4.90 /s 2 g. The axiu restoring force, found fro Hooke s law, is F ax = kx ax = ka = (29.4 N/)( ) = 1.47 N h. The velocity of the ass at x = 2.00 c, found fro equation 13.15, is v = ± A 2 x 2 v = ± (9.90 rad/s) (5.00 % 10 2 ) 2 (2.00 % 10 2 ) 2 = ± /s where v is positive when oving to the right and negative when oving to the left. i. The equation of the displaceent at any instant of tie, found fro equation 13.7, is x = A cos ωt = ( ) cos(9.90 rad/s)t The equation of the velocity at any instant of tie, found fro equation 13.13, is v = ωa sin ωt = (9.90 rad/s)( )sin(9.90 rad/s)t = (0.495 /s)sin(9.90 rad/s)t The equation of the acceleration at any tie, found fro equation 13.17, is a = ω 2 A cos ωt = (9.90 rad/s) 2 ( )cos(9.90 rad/s)t = (4.90 /s 2 )cos(9.90 rad/s)t To go to this Interactive Exaple click on this sentence

14 13.4 Conservation of Energy and the Vibrating Spring The vibrating spring syste of figure 13.2 can also be described in ters of the law of conservation of energy. When the spring is stretched to its axiu displaceent A, work is done on the spring, and hence the spring contains potential energy. The ass attached to the spring also has that potential energy. The kinetic energy is equal to zero at this point because v = 0 at the axiu displaceent. The total energy of the syste is thus E tot = PE + KE = PE But recall fro chapter 7 that the potential energy of a spring was given by equation 7.25 as PE = 1 kx 2 2 Hence, the total energy is E tot = PE = 1 ka 2 (13.23) 2 When the spring is released, the ass oves to a saller displaceent x, and is oving at a speed v. At this arbitrary position x, the ass will have both potential energy and kinetic energy. The law of conservation of energy then yields E tot = PE + KE E tot = 1 kx v 2 (13.24) 2 2 But the total energy iparted to the ass is given by equation Hence, the law of conservation of energy gives E tot = E tot 1 ka 2 = 1 kx v 2 (13.25) We can also use equation to find the velocity of the oving body at any displaceent x. Thus, 1 v 2 = 1 ka 2 1 kx v 2 = k (A 2 x 2 ) v =! k (A2 x 2 ) (13.26) 13-14

15 We should note that this is the sae equation for the velocity as derived earlier (equation 13.15). It is inforative to replace the values of x and v fro their respective equations 13.7 and into equation Thus, or but since E tot = 1 k(a cos ωt) ( ωa sin ωt) E tot = 1 ka 2 cos 2 ωt + 1 ω 2 A 2 sin 2 ωt 2 2 ω 2 = k E tot = 1 ka 2 cos 2 ωt + 1 k A 2 sin 2 ωt 2 2 = 1 ka 2 cos 2 ωt + 1 ka 2 sin 2 ωt (13.27) 2 2 These ters are plotted in figure Figure 13.5 Conservation of energy and siple haronic otion. The total energy of the vibrating syste is a constant and this is shown as the horizontal line, E tot. At t = 0 the total energy of the syste is potential energy (v is zero, hence the kinetic energy is zero). As the tie increases the potential energy decreases and the kinetic energy increases, as shown. However, the total energy reains the sae. Fro equation and figure 13.5, we see that at x = 0 the potential energy is zero and hence all the energy is kinetic. This occurs when t = T/4. The axiu velocity of the ass occurs here and is easily found by equating the axiu kinetic energy to the total energy, that is, 1 v ax2 = 1 ka v ax = k A = A (13.28) 13-15

16 When the oscillating ass reaches x = A, the kinetic energy becoes zero since 1 ka 2 = 1 ka v v 2 = 1 ka 2 1 ka 2 = = KE = 0 As the oscillation continues there is a constant interchange of energy between potential energy and kinetic energy but the total energy of the syste reains a constant. Exaple 13.2 Conservation of energy applied to a spring. A horizontal spring has a spring constant of 29.4 N/. A ass of 300 g is attached to the spring and displaced 5.00 c. The ass is then released. Find (a) the total energy of the syste, (b) the axiu velocity of the syste, and (c) the potential energy and kinetic energy for x = 2.00 c. Solution a. The total energy of the syste is E tot = 1 ka 2 2 = 1 (29.4 N/)( ) 2 2 = J b. The axiu velocity occurs when x = 0 and the potential energy is zero. Therefore, using equation 13.28, v ax = k A v ax = 29.4 N/ 3.00 % 10 1 kg (5.00 % 10 2 ) = /s c. The potential energy at 2.00 c is PE = 1 kx 2 = 1 (29.4 N/)( ) = J The kinetic energy at 2.00 c is 13-16

17 KE = 1 v 2 = 1 k (A 2 x 2 ) 2 2 = 1 (29.4 N/)[( ) 2 ( ) 2 ] 2 = J Note that the su of the potential energy and the kinetic energy is equal to the sae value for the total energy found in part a. To go to this Interactive Exaple click on this sentence The Siple Pendulu Another exaple of periodic otion is a pendulu. A siple pendulu is a bob that is attached to a string and allowed to oscillate, as shown in figure 13.6(a). The Figure 13.6 The siple pendulu. bob oscillates because there is a restoring force, given by Restoring force = g sin θ (13.29) This restoring force is just the coponent of the weight of the bob that is perpendicular to the string, as shown in figure 13.6(b). If Newton s second law, F = a, is applied to the otion of the pendulu bob, we get g sin θ = a The tangential acceleration of the pendulu bob is thus 13-17

18 a = g sin θ (13.30) Note that although this pendulu otion is periodic, it is not, in general, siple haronic otion because the acceleration is not directly proportional to the displaceent of the pendulu bob fro its equilibriu position. However, if the angle θ of the siple pendulu is sall, then the sine of θ can be replaced by the angle θ itself, expressed in radians. (The discrepancy in using θ rather than the sin θ is less than 0.2% for angles less than 10 degrees.) That is, for sall angles The acceleration of the bob is then sin θ θ a = gθ (13.31) Fro figure 13.6 and the definition of an angle in radians (θ = arc length/radius), we have θ = s l where s is the actual path length followed by the bob. Thus a = g s (13.32) l The path length s is curved, but if the angle θ is sall, the arc length s is approxiately equal to the chord x, figure 13.6(c). Hence, a = g x (13.33) l which is an equation having the sae for as that of the equation for siple haronic otion. Therefore, if the angle of oscillation θ is sall, the pendulu will execute siple haronic otion. For siple haronic otion of a spring, the acceleration was found to be a = k x (13.2) We can now use the equations developed for the vibrating spring to describe the otion of the pendulu. We find an equivalent spring constant of the pendulu by setting equation 13.2 equal to equation That is k = g l 13-18

19 or k P = g (13.34) l Equation states that the otion of a pendulu can be described by the equations developed for the vibrating spring by using the equivalent spring constant of the pendulu k p. Thus, the period of otion of the pendulu, found fro equation 13.22, is T p = 2 k p = 2 g/l T p = 2 g l (13.35) The period of otion of the pendulu is independent of the ass of the bob but is directly proportional to the square root of the length of the string. If the angle θ is equal to 15 0 on either side of the central position, then the true period differs fro that given by equation by less than 0.5%. The pendulu can be used as a very siple device to easure the acceleration of gravity at a particular location. We easure the length l of the pendulu and then set the pendulu into otion. We easure the period by a clock and obtain the acceleration of gravity fro equation as g = 4π 2 l (13.36) T p 2 Exaple 13.3 The period of a pendulu. Find the period of a siple pendulu 1.50 long. The period, found fro equation 13.35, is Solution T p = 2 g l = /s 2 = 2.46 s To go to this Interactive Exaple click on this sentence

20 Exaple 13.4 The length of a pendulu. Find the length of a siple pendulu whose period is 1.00 s. Solution The length of the pendulu, found fro equation 13.35, is l = T p 2 g 4π 2 = (1.00 s) 2 (9.80 /s 2 ) 4π 2 = To go to this Interactive Exaple click on this sentence. Exaple 13.5 The pendulu and the acceleration of gravity. A pendulu 1.50 long is observed to have a period of 2.47 s at a certain location. Find the acceleration of gravity there. Solution The acceleration of gravity, found fro equation 13.36, is g = 4π 2 l T p 2 = 4π 2 (1.50 ) (2.47 s) 2 = 9.71 /s 2 To go to this Interactive Exaple click on this sentence. We can also use a pendulu to easure an acceleration. If a pendulu is placed on board a rocket ship in interstellar space and the rocket ship is accelerated at 9.80 /s 2, the pendulu oscillates with the sae period as it would at rest on the surface of the earth. An enclosed person or thing on the rocket ship could not distinguish between the acceleration of the rocket ship at 9.80 /s 2 and the 13-20

21 acceleration of gravity of 9.80 /s 2 on the earth. (This is an exaple of Einstein s principle of equivalence in general relativity.) An oscillating pendulu of easured length l can be placed in an elevator and the period T easured. We can use equation to easure the resultant acceleration experienced by the pendulu in the elevator Springs in Parallel and in Series Soeties ore than one spring is used in a vibrating syste. The otion of the syste will depend on the way the springs are connected. As an exaple, suppose there are three assless springs with spring constants k 1, k 2, and k 3. These springs can be connected in parallel, as shown in figure 13.7(a), or in series, as in figure 13.7(b). The period of otion of either configuration can be found by using an equivalent spring constant k E. Figure 13.7 Springs in parallel and in series. Springs in Parallel If the total force pulling the ass a distance x to the right is F tot, this force will distribute itself aong the three springs such that there will be a force F 1 on spring 1, a force F 2 on spring 2, and a force F 3 on spring 3. If the displaceent of each spring is equal to x, then the springs are said to be in parallel. Then we can write the total force as F tot = F 1 + F 2 + F 3 (13.37) However, since we assued that each spring was displaced the sae distance x, Hooke s law for each spring is F 1 = k 1 x F 2 = k 2 x (13.38) F 3 = k 3 x Substituting equation into equation gives F tot = k 1 x + k 2 x + k 3 x 13-21

22 = (k 1 + k 2 + k 3 )x We now define an equivalent spring constant k E for springs connected in parallel as Hooke s law for the cobination of springs is given by k E = k 1 + k 2 + k 3 (13.39) F tot = k E x (13.40) The springs in parallel will execute a siple haronic otion whose period, found fro equation 13.22, is T = 2 = 2 (13.41) k E k 1 + k 2 + k 3 Springs in Series If the sae springs are connected in series, as in figure 13.7(b), the total force F tot displaces the ass a distance x to the right. But in this configuration, each spring stretches a different aount. Thus, the total displaceent x is the su of the displaceents of each spring, that is, The displaceent of each spring, found fro Hooke s law, is x = x 1 + x 2 + x 3 (13.42) x 1 = F 1 k 1 x 2 = F 2 k 2 x 3 = F 3 (13.43) k 3 Substituting these values of the displaceent into equation 13.42, yields x = F 1 + F 2 + F 3 (13.44) k 1 k 2 k 3 But because the springs are in series the total applied force is transitted equally fro spring to spring. Hence, F tot = F 1 = F 2 = F 3 (13.45) Substituting equation into equation 13.44, gives 13-22

23 and x = F tot + F tot + F tot k 1 k 2 k 3 x = 1 k k k 3 F tot (13.46) We now define the equivalent spring constant for springs connected in series as Thus, the total displaceent, equation 13.46, becoes 1 = (13.47) k E k 1 k 2 k 3 and Hooke s law becoes x = F tot (13.48) k E F tot = k E x (13.49) where k E is given by equation Hence, the cobination of springs in series executes siple haronic otion and the period of that otion, given by equation 13.22, is T = 2 k E = 2 1 k k k 3 (13.50) Exaple 13.6 Springs in parallel. Three springs with force constants k 1 = 10.0 N/, k 2 = 12.5 N/, and k 3 = 15.0 N/ are connected in parallel to a ass of kg. The ass is then pulled to the right and released. Find the period of the otion. Solution The period of the otion, found fro equation 13.41, is T = 2 T = 2 k 1 + k 2 + k kg 10.0 N/ N/ N/ = s To go to this Interactive Exaple click on this sentence

24 Exaple 13.7 Springs in series. The sae three springs as in exaple 13.6 are now connected in series. Find the period of the otion. The period, found fro equation 13.50, is Solution T = 2 1 k k k 3 = 2 (0.500 kg) N/ N/ N/ = 2.21 s To go to this Interactive Exaple click on this sentence. The Language of Physics Periodic otion Motion that repeats itself in equal intervals of tie (p. ). Displaceent The distance a vibrating body oves fro its equilibriu position (p. ). Siple haronic otion Periodic otion in which the acceleration of a body is directly proportional to its displaceent fro the equilibriu position but in the opposite direction. Because the acceleration is directly proportional to the displaceent, the acceleration of the body is not constant. The kineatic equations developed in chapter 3 are no longer valid to describe this type of otion (p. ). Aplitude The axiu displaceent of the vibrating body (p. ). Cycle One coplete oscillation or vibratory otion (p. ). Period The tie for the vibrating body to coplete one cycle (p. )

25 Frequency The nuber of coplete cycles or oscillations in unit tie. The frequency is the reciprocal of the period (p. ). Reference circle A body executing unifor circular otion does so in a circle. The projection of the position of the rotating body onto the x- or y-axis is equivalent to siple haronic otion along that axis. Thus, vibratory otion is related to otion in a circle, the reference circle (p. ). Angular velocity The angular velocity of the unifor circular otion in the reference circle is related to the frequency of the vibrating syste. Hence, the angular velocity is called the angular frequency of the vibrating syste (p. ). Potential energy of a spring The energy that a body possesses by virtue of its configuration. A copressed spring has potential energy because it has the ability to do work as it expands to its equilibriu configuration. A stretched spring can also do work as it returns to its equilibriu configuration (p. ). Siple pendulu A bob that is attached to a string and allowed to oscillate to and fro under the action of gravity. If the angle of the pendulu is sall the pendulu will oscillate in siple haronic otion (p. ). Suary of Iportant Equations Restoring force in a spring F R = kx (13.1) Defining relation for siple haronic otion a = k x (13.2) Frequency f = 1 (13.3) T Displaceent in siple haronic otion x = A cos ωt (13.7) Angular frequency ω = 2πf (13.9) Velocity as a function of tie in siple haronic otion v = ωa sin ωt (13.13) 13-25

26 Velocity as a function of displaceent v = ± A 2 x 2 (13.15) Acceleration as a function of tie a = ω 2 A cos ωt (13.17) Angular frequency of a spring = k (13.20) Frequency in siple haronic otion f = 1 k (13.21) 2 Period in siple haronic otion T = 2 (13.22) k Potential energy of a spring PE = 1 kx 2 (7.25) 2 Conservation of energy for a vibrating spring 1 ka 2 = 1 kx v 2 (13.25) Period of otion of a siple pendulu T p = 2 g l (13.35) Equivalent spring constant for springs in parallel k E = k 1 + k 2 + k 3 (13.39) Period of otion for springs in parallel T = 2 (13.41) k 1 + k 2 + k 3 Equivalent spring constant for springs in series 1 = (13.47) k E k 1 k 2 k 3 Period of otion for springs in series T = (13.50) k k 1 2 k 3 Questions for Chapter Can the periodic otion of the earth be considered to be an exaple of siple haronic otion? 2. Can the kineatic equations derived in chapter 2 be used to describe siple haronic otion? 3. In the siple haronic otion of a ass attached to a spring, the velocity of the ass is equal to zero when the acceleration has its axiu value. How is this possible? Can you think of other exaples in which a body has zero velocity with a nonzero acceleration? 4. What is the characteristic of the restoring force that akes siple haronic otion possible? 13-26

27 5. Discuss the significance of the reference circle in the analysis of siple haronic otion. 6. How can a ass that is undergoing a one-diensional translational siple haronic otion have anything to do with an angular velocity or an angular frequency, which is a characteristic of two or ore diensions? 7. How is the angular frequency related to the physical characteristics of the spring and the vibrating ass in siple haronic otion? *8. In the entire derivation of the equations for siple haronic otion we have assued that the springs are assless and friction can be neglected. Discuss these assuptions. Describe qualitatively what you would expect to happen to the otion if the springs are not sall enough to be considered assless. *9. Describe how a geological survey for iron ight be undertaken on the oon using a siple pendulu. *10. How could a siple pendulu be used to ake an acceleroeter? *11. Discuss the assuption that the displaceent of each spring is the sae when the springs are in parallel. Under what conditions is this assuption valid and when would it be invalid? Probles for Chapter Siple Haronic Motion and 13.3 Analysis of Siple Haronic Motion 1. A ass of kg is attached to a spring of spring constant 30.0 N/. If the ass executes siple haronic otion, what will be its frequency? 2. A 30.0-g ass is attached to a vertical spring and it stretches 10.0 c. It is then stretched an additional 5.00 c and released. Find its period of otion and its frequency. 3. A kg ass on a spring executes siple haronic otion at a frequency f. What ass is necessary for the syste to vibrate at a frequency of 2f? 4. A siple haronic oscillator has a frequency of 2.00 Hz and an aplitude of 10.0 c. What is its axiu acceleration? What is its acceleration at t = 0.25 s? 5. A ball attached to a string travels in unifor circular otion in a horizontal circle of 50.0 c radius in 1.00 s. Sunlight shining on the ball throws its shadow on a wall. Find the velocity of the shadow at (a) the end of its path and (b) the center of its path. 6. A 50.0-g ass is attached to a spring of force constant 10.0 N/. The spring is stretched 20.0 c and then released. Find the displaceent, velocity, and acceleration of the ass at s. 7. A 25.0-g ass is attached to a vertical spring and it stretches 15.0 c. It is then stretched an additional 10.0 c and then released. What is the axiu velocity of the ass? What is its axiu acceleration? 13-27

28 8. The displaceent of a body in siple haronic otion is given by x = (0.15 )cos[(5.00 rad/s)t]. Find (a) the aplitude of the otion, (b) the angular frequency, (c) the frequency, (d) the period, and (e) the displaceent at 3.00 s. 9. A 500-g ass is hung fro a coiled spring and it stretches 10.0 c. It is then stretched an additional 15.0 c and released. Find (a) the frequency of vibration, (b) the period, and (c) the velocity and acceleration at a displaceent of 10.0 c. 10. A ass of kg is placed on a vertical spring and the spring stretches by 15.0 c. It is then pulled down an additional 10.0 c and then released. Find (a) the spring constant, (b) the angular frequency, (c) the frequency, (d) the period, (e) the axiu velocity of the ass, (f) the axiu acceleration of the ass, (g) the axiu restoring force, and (h) the equation of the displaceent, velocity, and acceleration at any tie t Conservation of Energy and the Vibrating Spring 11. A siple haronic oscillator has a spring constant of 5.00 N/. If the aplitude of the otion is 15.0 c, find the total energy of the oscillator. 12. A body is executing siple haronic otion. At what displaceent is the potential energy equal to the kinetic energy? 13. A 20.0-g ass is attached to a horizontal spring on a sooth table. The spring constant is 3.00 N/. The spring is then stretched 15.0 c and then released. What is the total energy of the otion? What is the potential and kinetic energy when x = 5.00 c? 14. A body is executing siple haronic otion. At what displaceent is the speed v equal to one-half the axiu speed? 13.6 The Siple Pendulu 15. Find the period and frequency of a siple pendulu 0.75 long. 16. If a pendulu has a length L and a period T, what will be the period when (a) L is doubled and (b) L is halved? 17. Find the frequency of a child s swing whose ropes have a length of What is the period of a pendulu on the oon where g = (1/6)g e? 19. What is the period of a pendulu long on a spaceship (a) accelerating at 4.90 /s 2 and (b) oving at constant velocity? 20. What is the period of a pendulu in free-fall? 21. A pendulu has a period of s at the equator at sea level. The pendulu is carried to another place on the earth and the period is now found to be s. Find the acceleration due to gravity at this location Springs in Parallel and in Series *22. Springs with spring constants of 5.00 N/ and 10.0 N/ are connected in parallel to a 5.00-kg ass. Find (a) the equivalent spring constant and (b) the period of the otion

29 *23. Springs with spring constants 5.00 N/ and 10.0 N/ are connected in series to a 5.00-kg ass. Find (a) the equivalent spring constant and (b) the period of the otion. Additional Probles 24. A 500-g ass is attached to a vertical spring of spring constant 30.0 N/. How far should the spring be stretched in order to give the ass an upward acceleration of 3.00 /s 2? 25. A ball is caused to ove in a horizontal circle of 40.0-c radius in unifor circular otion at a speed of 25.0 c/s. Its projection on the wall oves in siple haronic otion. Find the velocity and acceleration of the shadow of the ball at (a) the end of its otion, (b) the center of its otion, and (c) halfway between the center and the end of the otion. *26. The otion of the piston in the engine of an autoobile is approxiately siple haronic. If the stroke of the piston (twice the aplitude) is equal to 20.3 c and the engine turns at 1800 rp, find (a) the acceleration at x = A and (b) the speed of the piston at the idpoint of the stroke. *27. A 535-g ass is dropped fro a height of 25.0 c above an uncopressed spring of k = 20.0 N/. By how uch will the spring be copressed? 28. A siple pendulu is used to operate an electrical device. When the pendulu bob sweeps through the idpoint of its swing, it causes an electrical spark to be given off. Find the length of the pendulu that will give a spark rate of 30.0 sparks per inute. *29. The general solution for the period of a siple pendulu, without aking the assuption of sall angles of swing, is given by T = 2 l g 1 + ( 1 2 )2 sin ( 1 2 )2 ( 3 4 )2 sin Find the period of a pendulu for θ = , , and and copare with the period obtained with the sall angle approxiation. Deterine the percentage error in each case by using the sall angle approxiation. 30. A pendulu clock on the earth has a period of 1.00 s. Will this clock run slow or fast, and by how uch if taken to (a) Mars, (b) Moon, and (c) Venus? *31. A pendulu bob, 355 g, is raised to a height of 12.5 c before it is released. At the botto of its path it akes a perfectly elastic collision with a 500-g ass that is connected to a horizontal spring of spring constant 15.8 N/, that is at rest on a sooth surface. How far will the spring be copressed? 13-29

30 Diagra for proble 31. Diagra for proble 32. *32. A 500-g block is in siple haronic otion as shown in the diagra. A ass is added to the top of the block when the block is at its axiu extension. How uch ass should be added to change the frequency by 25%? *33. A pendulu clock keeps correct tie at a location at sea level where the acceleration of gravity is equal to 9.80 /s 2. The clock is then taken up to the top of a ountain and the clock loses 3.00 s per day. How high is the ountain? *34. Three people, who together weigh 1880 N, get into a car and the car is observed to ove 5.08 c closer to the ground. What is the spring constant of the car springs? *35. In the accopanying diagra, the ass is pulled down a distance of 9.50 c fro its equilibriu position and is then released. The ass then executes siple haronic otion. Find (a) the total potential energy of the ass with respect to the ground when the ass is located at positions 1, 2, and 3; (b) the total energy of the ass at positions 1, 2, and 3; and (c) the speed of the ass at position 2. Assue = 55.6 g, k = 25.0 N/, h 0 = 50.0 c. Diagra for proble 35. *36. A 20.0-g ball rests on top of a vertical spring gun whose spring constant is 20 N/. The spring is copressed 10.0 c and the gun is then fired. Find how high the ball rises in its vertical trajectory. *37. A toy spring gun is used to fire a ball as a projectile. Find the value of the spring constant, such that when the spring is copressed 10.0 c, and the gun 13-30

31 is fired at an angle of , the range of the projectile will be The ass of the ball is 25.2 g. *38. In the siple pendulu shown in the diagra, find the tension in the string when the height of the pendulu is (a) h, (b) h/2, and (c) h = 0. The ass = 500 g, the initial height h = 15.0 c, and the length of the pendulu l = Diagra for proble 38. *39. A ass is attached to a horizontal spring. The ass is given an initial aplitude of 10.0 c on a rough surface and is then released to oscillate in siple haronic otion. If 10.0% of the energy is lost per cycle due to the friction of the ass oving over the rough surface, find the axiu displaceent of the ass after 1, 2, 4, 6, and 8 coplete oscillations. *40. Find the axiu aplitude of vibration after 2 periods for a 85.0-g ass executing siple haronic otion on a rough horizontal surface of µ k = The spring constant is 24.0 N/ and the initial aplitude is 20.0 c. 41. A 240-g ass slides down a circular chute without friction and collides with a horizontal spring, as shown in the diagra. If the original position of the ass is 25.0 c above the table top and the spring has a spring constant of 18 N/, find the axiu distance that the spring will be copressed. Diagra for proble 41. Diagra for proble 42. *42. A 235-g block slides down a frictionless inclined plane, 1.25 long, that akes an angle of with the horizontal. At the botto of the plane the block slides along a rough horizontal surface 1.50 long. The coefficient of kinetic friction between the block and the rough horizontal surface is The block then collides 13-31

32 with a horizontal spring of k = 20.0 N/. Find the axiu copression of the spring. *43. A 335-g disk that is free to rotate about its axis is connected to a spring that is stretched 35.0 c. The spring constant is 10.0 N/. When the disk is released it rolls without slipping as it oves toward the equilibriu position. Find the speed of the disk when the displaceent of the spring is equal to 17.5 c. Diagra for proble 43. Diagra for proble 44. *44. A 25.0-g ball oving at a velocity of 200 c/s to the right akes an inelastic collision with a 200-g block that is initially at rest on a frictionless surface. There is a hole in the large block for the sall ball to fit into. If k = 10 N/, find the axiu copression of the spring. *45. Show that the period of siple haronic otion for the ass shown is equivalent to the period for two springs in parallel. Diagra for proble 45. Diagra for proble 46. *46. A nail is placed in the wall at a distance of l/2 fro the top, as shown in the diagra. A pendulu of length 85.0 c is released fro position 1. (a) Find the tie it takes for the pendulu bob to reach position 2. When the bob of the pendulu reaches position 2, the string hits the nail. (b) Find the tie it takes for the pendulu bob to reach position 3. *47. A spring is attached to the top of an Atwood s achine as shown. The spring is stretched to A = 10 c before being released. Find the velocity of 2 when x = A/2. Assue 1 = 28.0 g, 2 = 43.0 g, and k = 10.0 N/

33 Diagra for proble 47. Diagra for proble 48. *48. A 280-g block is connected to a spring on a rough inclined plane that akes an angle of with the horizontal. The block is pulled down the plane a distance A = 20.0 c, and is then released. The spring constant is 40.0 N/ and the coefficient of kinetic friction is Find the speed of the block when the displaceent x = A/ The rotational analog of siple haronic otion, is angular siple haronic otion, wherein a body rotates periodically clockwise and then counterclockwise. Hooke s law for rotational otion is given by τ = C θ where τ is the torque acting on the body, θ is the angular displaceent, and C is a constant, like the spring constant. Use Newton s second law for rotational otion to show α = C θ I Use the analogy between the linear result, a = ω 2 x, to show that the frequency of vibration of an object executing angular siple haronic otion is given by f = 1 2 C I 13-33

34 Interactive Tutorials 50. Siple Pendulu. Calculate the period T of a siple pendulu located on a planet having a gravitational acceleration of g = 9.80 /s 2, if its length l = 1.00 is increased fro 1 to 10 in steps of Plot the results as the period T versus the length l. 51. Siple Haronic Motion. The displaceent x of a body undergoing siple haronic otion is given by the forula x = A cos ωt, where A is the aplitude of the vibration, ω is the angular frequency in rad/s, and t is the tie in seconds. Plot the displaceent x as a function of t for an aplitude A = and an angular frequency ω = 5.00 rad/s as t increases fro 0 to 2 s in 0.10 s increents. 52. The Vibrating Spring. A ass = kg is attached to a spring on a sooth horizontal table. An applied force F A = 4.00 N is used to stretch the spring a distance x 0 = (a) Find the spring constant k of the spring. The ass is returned to its equilibriu position and then stretched to a value A = 0.15 and then released. The ass then executes siple haronic otion. Find (b) the angular frequency ω, (c) the frequency f, (d) the period T, (e) the axiu velocity v ax of the vibrating ass, (f) the axiu acceleration a ax of the vibrating ass, (g) the axiu restoring force F Rax, and (h) the velocity of the ass at the displaceent x = (i) Plot the displaceent x, velocity v, acceleration a, and the restoring force F R at any tie t. 53. Conservation of Energy and the Vibrating Horizontal Spring. A ass = kg is attached to a horizontal spring. The ass is then pulled a distance x = A = fro its equilibriu position and when released the ass executes siple haronic otion. Find (a) the total energy E tot of the ass when it is at its axiu displaceent A fro its equilibriu position. When the ass is at the displaceent x = find, (b) its potential energy PE, (c) its kinetic energy KE, and (d) its speed v. (e) Plot the total energy, potential energy, and kinetic energy of the ass as a function of the displaceent x. The spring constant k = 35.5 N/. 54. Conservation of Energy and the Vibrating Vertical Spring. A ass = kg is attached to a vertical spring. The ass is at a height h 0 = 1.50 fro the floor. The ass is then pulled down a distance A = fro its equilibriu position and when released executes siple haronic otion. Find (a) the total energy of the ass when it is at its axiu displaceent A below its equilibriu position, (b) the gravitational potential energy when it is at the displaceent x = 0.120, (c) the elastic potential energy when it is at the sae displaceent x, (d) the kinetic energy at the displaceent x, and (e) the speed of the ass when it is at the displaceent x. The spring constant k = 35.5 N/. To go to these Interactive Tutorials click on this sentence. To go to another chapter, return to the table of contents by clicking on this sentence

Lesson 44: Acceleration, Velocity, and Period in SHM

Lesson 44: Acceleration, Velocity, and Period in SHM Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain

More information

Physics 211: Lab Oscillations. Simple Harmonic Motion.

Physics 211: Lab Oscillations. Simple Harmonic Motion. Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.

More information

Work, Energy, Conservation of Energy

Work, Energy, Conservation of Energy This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and non-conservative forces, with soe

More information

Lecture L9 - Linear Impulse and Momentum. Collisions

Lecture L9 - Linear Impulse and Momentum. Collisions J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,

More information

Answer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 7 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Hoework 7 David McIntyre 453 Mar 5, 004 This print-out should have 4 questions. Multiple-choice questions ay continue on the next colun or page find all choices before aking your selection.

More information

Answer: Same magnitude total momentum in both situations.

Answer: Same magnitude total momentum in both situations. Page 1 of 9 CTP-1. In which situation is the agnitude of the total oentu the largest? A) Situation I has larger total oentu B) Situation II C) Sae agnitude total oentu in both situations. I: v 2 (rest)

More information

Physics 41 HW Set 1 Chapter 15

Physics 41 HW Set 1 Chapter 15 Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,

More information

PHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true?

PHY231 Section 2, Form A March 22, 2012. 1. Which one of the following statements concerning kinetic energy is true? 1. Which one of the following statements concerning kinetic energy is true? A) Kinetic energy can be measured in watts. B) Kinetic energy is always equal to the potential energy. C) Kinetic energy is always

More information

PHY231 Section 1, Form B March 22, 2012

PHY231 Section 1, Form B March 22, 2012 1. A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is 0.20. What is the maximum speed with which the car can safely negotiate

More information

Version 001 test 1 review tubman (IBII201516) 1

Version 001 test 1 review tubman (IBII201516) 1 Version 001 test 1 review tuban (IBII01516) 1 This print-out should have 44 questions. Multiple-choice questions ay continue on the next colun or page find all choices before answering. Crossbow Experient

More information

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

More information

LAWS OF MOTION PROBLEM AND THEIR SOLUTION

LAWS OF MOTION PROBLEM AND THEIR SOLUTION http://www.rpauryascienceblog.co/ LWS OF OIO PROBLE D HEIR SOLUIO. What is the axiu value of the force F such that the F block shown in the arrangeent, does not ove? 60 = =3kg 3. particle of ass 3 kg oves

More information

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N

C B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a

More information

Practice Test SHM with Answers

Practice Test SHM with Answers Practice Test SHM with Answers MPC 1) If we double the frequency of a system undergoing simple harmonic motion, which of the following statements about that system are true? (There could be more than one

More information

AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?

AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The

More information

Homework 8. problems: 10.40, 10.73, 11.55, 12.43

Homework 8. problems: 10.40, 10.73, 11.55, 12.43 Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks

More information

9. The kinetic energy of the moving object is (1) 5 J (3) 15 J (2) 10 J (4) 50 J

9. The kinetic energy of the moving object is (1) 5 J (3) 15 J (2) 10 J (4) 50 J 1. If the kinetic energy of an object is 16 joules when its speed is 4.0 meters per second, then the mass of the objects is (1) 0.5 kg (3) 8.0 kg (2) 2.0 kg (4) 19.6 kg Base your answers to questions 9

More information

F=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu

F=ma From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.edu Chapter 4 F=a Fro Probles and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, orin@physics.harvard.edu 4.1 Introduction Newton s laws In the preceding two chapters, we dealt

More information

AP Physics C. Oscillations/SHM Review Packet

AP Physics C. Oscillations/SHM Review Packet AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

More information

Chapter 6 Work and Energy

Chapter 6 Work and Energy Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system

More information

VELOCITY, ACCELERATION, FORCE

VELOCITY, ACCELERATION, FORCE VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how

More information

Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

More information

B) 286 m C) 325 m D) 367 m Answer: B

B) 286 m C) 325 m D) 367 m Answer: B Practice Midterm 1 1) When a parachutist jumps from an airplane, he eventually reaches a constant speed, called the terminal velocity. This means that A) the acceleration is equal to g. B) the force of

More information

Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel

Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel Physics 125 Practice Exam #3 Chapters 6-7 Professor Siegel Name: Lab Day: 1. A concrete block is pulled 7.0 m across a frictionless surface by means of a rope. The tension in the rope is 40 N; and the

More information

Simple Harmonic Motion MC Review KEY

Simple Harmonic Motion MC Review KEY Siple Haronic Motion MC Review EY. A block attache to an ieal sprin uneroes siple haronic otion. The acceleration of the block has its axiu anitue at the point where: a. the spee is the axiu. b. the potential

More information

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

More information

226 Chapter 15: OSCILLATIONS

226 Chapter 15: OSCILLATIONS Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion

More information

PHYS 211 FINAL FALL 2004 Form A

PHYS 211 FINAL FALL 2004 Form A 1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

More information

Physics 1120: Simple Harmonic Motion Solutions

Physics 1120: Simple Harmonic Motion Solutions Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

More information

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc. Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

More information

circular motion & gravitation physics 111N

circular motion & gravitation physics 111N circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would

More information

Phys101 Lectures 14, 15, 16 Momentum and Collisions

Phys101 Lectures 14, 15, 16 Momentum and Collisions Phs0 Lectures 4, 5, 6 Moentu and ollisions Ke points: Moentu and ipulse ondition for conservation of oentu and wh How to solve collision probles entre of ass Ref: 9-,,3,4,5,6,7,8,9. Page Moentu is a vector:

More information

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc. Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

More information

Problem Set 5 Work and Kinetic Energy Solutions

Problem Set 5 Work and Kinetic Energy Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physics Physics 8.1 Fall 1 Problem Set 5 Work and Kinetic Energy Solutions Problem 1: Work Done by Forces a) Two people push in opposite directions on

More information

PHY121 #8 Midterm I 3.06.2013

PHY121 #8 Midterm I 3.06.2013 PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

More information

Tennessee State University

Tennessee State University Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an F-grade. Other instructions will be given in the Hall. MULTIPLE CHOICE.

More information

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013 PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

More information

Lecture L26-3D Rigid Body Dynamics: The Inertia Tensor

Lecture L26-3D Rigid Body Dynamics: The Inertia Tensor J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall

More information

CHAPTER 6 WORK AND ENERGY

CHAPTER 6 WORK AND ENERGY CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS. REASONING AND SOLUTION The work done by F in moving the box through a displacement s is W = ( F cos 0 ) s= Fs. The work done by F is W = ( F cos θ). s From

More information

PHYSICS 151 Notes for Online Lecture 2.2

PHYSICS 151 Notes for Online Lecture 2.2 PHYSICS 151 otes for Online Lecture. A free-bod diagra is a wa to represent all of the forces that act on a bod. A free-bod diagra akes solving ewton s second law for a given situation easier, because

More information

AP Physics C Fall Final Web Review

AP Physics C Fall Final Web Review Name: Class: _ Date: _ AP Physics C Fall Final Web Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. On a position versus time graph, the slope of

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

More information

Simple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines

Simple Harmonic Motion(SHM) Period and Frequency. Period and Frequency. Cosines and Sines Simple Harmonic Motion(SHM) Vibration (oscillation) Equilibrium position position of the natural length of a spring Amplitude maximum displacement Period and Frequency Period (T) Time for one complete

More information

AP Physics - Chapter 8 Practice Test

AP Physics - Chapter 8 Practice Test AP Physics - Chapter 8 Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A single conservative force F x = (6.0x 12) N (x is in m) acts on

More information

F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26

F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26 Physics 23 Exam 2 Spring 2010 Dr. Alward Page 1 1. A 250-N force is directed horizontally as shown to push a 29-kg box up an inclined plane at a constant speed. Determine the magnitude of the normal force,

More information

Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

More information

KE =? v o. Page 1 of 12

KE =? v o. Page 1 of 12 Page 1 of 12 CTEnergy-1. A mass m is at the end of light (massless) rod of length R, the other end of which has a frictionless pivot so the rod can swing in a vertical plane. The rod is initially horizontal

More information

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

PHYS 101-4M, Fall 2005 Exam #3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. PHYS 101-4M, Fall 2005 Exam #3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A bicycle wheel rotates uniformly through 2.0 revolutions in

More information

The Virtual Spring Mass System

The Virtual Spring Mass System The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a

More information

and that of the outgoing water is mv f

and that of the outgoing water is mv f Week 6 hoework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign ersions of these probles, arious details hae been changed, so that the answers will coe out differently. The ethod to find the solution is

More information

Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion

Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion Physics: Principles and Applications, 6e Giancoli Chapter 4 Dynamics: Newton's Laws of Motion Conceptual Questions 1) Which of Newton's laws best explains why motorists should buckle-up? A) the first law

More information

The Mathematics of Pumping Water

The Mathematics of Pumping Water The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role

More information

From Last Time Newton s laws. Question. Acceleration of the moon. Velocity of the moon. How has the velocity changed?

From Last Time Newton s laws. Question. Acceleration of the moon. Velocity of the moon. How has the velocity changed? Fro Last Tie Newton s laws Law of inertia F=a ( or a=f/ ) Action and reaction Forces are equal and opposite, but response to force (accel.) depends on ass (a=f/). e.g. Gravitational force on apple fro

More information

Sample Questions for the AP Physics 1 Exam

Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiple-choice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each

More information

Chapter 7: Momentum and Impulse

Chapter 7: Momentum and Impulse Chapter 7: Momentum and Impulse 1. When a baseball bat hits the ball, the impulse delivered to the ball is increased by A. follow through on the swing. B. rapidly stopping the bat after impact. C. letting

More information

Review Assessment: Lec 02 Quiz

Review Assessment: Lec 02 Quiz COURSES > PHYSICS GUEST SITE > CONTROL PANEL > 1ST SEM. QUIZZES > REVIEW ASSESSMENT: LEC 02 QUIZ Review Assessment: Lec 02 Quiz Name: Status : Score: Instructions: Lec 02 Quiz Completed 20 out of 100 points

More information

Mechanics 1: Conservation of Energy and Momentum

Mechanics 1: Conservation of Energy and Momentum Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

More information

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations

Spring Simple Harmonic Oscillator. Spring constant. Potential Energy stored in a Spring. Understanding oscillations. Understanding oscillations Spring Simple Harmonic Oscillator Simple Harmonic Oscillations and Resonance We have an object attached to a spring. The object is on a horizontal frictionless surface. We move the object so the spring

More information

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

More information

Curso2012-2013 Física Básica Experimental I Cuestiones Tema IV. Trabajo y energía.

Curso2012-2013 Física Básica Experimental I Cuestiones Tema IV. Trabajo y energía. 1. A body of mass m slides a distance d along a horizontal surface. How much work is done by gravity? A) mgd B) zero C) mgd D) One cannot tell from the given information. E) None of these is correct. 2.

More information

Physics 1A Lecture 10C

Physics 1A Lecture 10C Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

More information

2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position.

2. The acceleration of a simple harmonic oscillator is zero whenever the oscillating object is at the equilibrium position. CHAPTER : Vibrations and Waes Answers to Questions The acceleration o a siple haronic oscillator is zero wheneer the oscillating object is at the equilibriu position 5 The iu speed is gien by = A k Various

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com

Copyright 2011 Casa Software Ltd. www.casaxps.com Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations

More information

Chapter 6 Circular Motion

Chapter 6 Circular Motion Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

More information

8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential

8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential 8. Potential Energy and Conservation of Energy Potential Energy: When an object has potential to have work done on it, it is said to have potential energy, e.g. a ball in your hand has more potential energy

More information

Chapter 11 Equilibrium

Chapter 11 Equilibrium 11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

More information

Serway_ISM_V1 1 Chapter 4

Serway_ISM_V1 1 Chapter 4 Serway_ISM_V1 1 Chapter 4 ANSWERS TO MULTIPLE CHOICE QUESTIONS 1. Newton s second law gives the net force acting on the crate as This gives the kinetic friction force as, so choice (a) is correct. 2. As

More information

PHYS 117- Exam I. Multiple Choice Identify the letter of the choice that best completes the statement or answers the question.

PHYS 117- Exam I. Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. PHYS 117- Exam I Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Car A travels from milepost 343 to milepost 349 in 5 minutes. Car B travels

More information

Unit 3 Work and Energy Suggested Time: 25 Hours

Unit 3 Work and Energy Suggested Time: 25 Hours Unit 3 Work and Energy Suggested Time: 25 Hours PHYSICS 2204 CURRICULUM GUIDE 55 DYNAMICS Work and Energy Introduction When two or more objects are considered at once, a system is involved. To make sense

More information

Physics 201 Homework 8

Physics 201 Homework 8 Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

More information

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6

Lecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6 Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.

More information

LAB 6 - GRAVITATIONAL AND PASSIVE FORCES

LAB 6 - GRAVITATIONAL AND PASSIVE FORCES L06-1 Name Date Partners LAB 6 - GRAVITATIONAL AND PASSIVE FORCES OBJECTIVES And thus Nature will be very conformable to herself and very simple, performing all the great Motions of the heavenly Bodies

More information

Newton s Laws. Physics 1425 lecture 6. Michael Fowler, UVa.

Newton s Laws. Physics 1425 lecture 6. Michael Fowler, UVa. Newton s Laws Physics 1425 lecture 6 Michael Fowler, UVa. Newton Extended Galileo s Picture of Galileo said: Motion to Include Forces Natural horizontal motion is at constant velocity unless a force acts:

More information

Ph\sics 2210 Fall 2012 - Novcmbcr 21 David Ailion

Ph\sics 2210 Fall 2012 - Novcmbcr 21 David Ailion Ph\sics 2210 Fall 2012 - Novcmbcr 21 David Ailion Unid: Discussion T A: Bryant Justin Will Yuan 1 Place answers in box provided for each question. Specify units for each answer. Circle correct answer(s)

More information

Prelab Exercises: Hooke's Law and the Behavior of Springs

Prelab Exercises: Hooke's Law and the Behavior of Springs 59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically

More information

3600 s 1 h. 24 h 1 day. 1 day

3600 s 1 h. 24 h 1 day. 1 day Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

More information

HOOKE S LAW AND SIMPLE HARMONIC MOTION

HOOKE S LAW AND SIMPLE HARMONIC MOTION HOOKE S LAW AND SIMPLE HARMONIC MOTION Alexander Sapozhnikov, Brooklyn College CUNY, New York, alexs@brooklyn.cuny.edu Objectives Study Hooke s Law and measure the spring constant. Study Simple Harmonic

More information

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad. Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

More information

Problem Set V Solutions

Problem Set V Solutions Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3

More information

A Gas Law And Absolute Zero Lab 11

A Gas Law And Absolute Zero Lab 11 HB 04-06-05 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution

More information

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

More information

Work, Energy and Power Practice Test 1

Work, Energy and Power Practice Test 1 Name: ate: 1. How much work is required to lift a 2-kilogram mass to a height of 10 meters?. 5 joules. 20 joules. 100 joules. 200 joules 5. ar and car of equal mass travel up a hill. ar moves up the hill

More information

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Rotational Motion: 11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

More information

A Gas Law And Absolute Zero

A Gas Law And Absolute Zero A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very

More information

AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s

AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s AP Physics Circular Motion Practice Test B,B,B,A,D,D,C,B,D,B,E,E,E, 14. 6.6m/s, 0.4 N, 1.5 m, 6.3m/s, 15. 12.9 m/s, 22.9 m/s Answer the multiple choice questions (2 Points Each) on this sheet with capital

More information

Chapter 14 Oscillations

Chapter 14 Oscillations Chapter 4 Oscillations Conceptual Probles 3 n object attached to a spring exhibits siple haronic otion with an aplitude o 4. c. When the object is. c ro the equilibriu position, what percentage o its total

More information

Chapter 4: Newton s Laws: Explaining Motion

Chapter 4: Newton s Laws: Explaining Motion Chapter 4: Newton s Laws: Explaining Motion 1. All except one of the following require the application of a net force. Which one is the exception? A. to change an object from a state of rest to a state

More information

LAB 6: GRAVITATIONAL AND PASSIVE FORCES

LAB 6: GRAVITATIONAL AND PASSIVE FORCES 55 Name Date Partners LAB 6: GRAVITATIONAL AND PASSIVE FORCES And thus Nature will be very conformable to herself and very simple, performing all the great Motions of the heavenly Bodies by the attraction

More information

Chapter 4. Forces and Newton s Laws of Motion. continued

Chapter 4. Forces and Newton s Laws of Motion. continued Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting

More information

5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity.

5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity. 5. Forces and Motion-I 1 Force is an interaction that causes the acceleration of a body. A vector quantity. Newton's First Law: Consider a body on which no net force acts. If the body is at rest, it will

More information

Problem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani

Problem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Vector A has length 4 units and directed to the north. Vector B has length 9 units and is directed

More information

Chapter 7 WORK, ENERGY, AND Power Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power

Chapter 7 WORK, ENERGY, AND Power Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power Chapter 7 WORK, ENERGY, AND Power Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power Examples of work. (a) The work done by the force F on this

More information

Vectors & Newton's Laws I

Vectors & Newton's Laws I Physics 6 Vectors & Newton's Laws I Introduction In this laboratory you will eplore a few aspects of Newton s Laws ug a force table in Part I and in Part II, force sensors and DataStudio. By establishing

More information

The Velocities of Gas Molecules

The Velocities of Gas Molecules he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes

More information

Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

More information

Oscillations. Vern Lindberg. June 10, 2010

Oscillations. Vern Lindberg. June 10, 2010 Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1

More information

Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

More information

BHS Freshman Physics Review. Chapter 2 Linear Motion Physics is the oldest science (astronomy) and the foundation for every other science.

BHS Freshman Physics Review. Chapter 2 Linear Motion Physics is the oldest science (astronomy) and the foundation for every other science. BHS Freshman Physics Review Chapter 2 Linear Motion Physics is the oldest science (astronomy) and the foundation for every other science. Galileo (1564-1642): 1 st true scientist and 1 st person to use

More information

Exam Three Momentum Concept Questions

Exam Three Momentum Concept Questions Exam Three Momentum Concept Questions Isolated Systems 4. A car accelerates from rest. In doing so the absolute value of the car's momentum changes by a certain amount and that of the Earth changes by:

More information