Physics B AP Review Packet: Mechanics Name:


 Derick Sherman Poole
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1 Name: Position Location of a particle in space. (x) or (x,y) or (x,y,z) Distance The total length of the path traveled by an object. Does not depend upon direction. Displacement The change in position of an object. Depends only on the initial and final positions, not on path. Includes direction. Represented by Δx. Problem: Distance versus Displacement 1. A hiker hikes 5 miles due north and then all the way back to the starting point. a) How far does the hiker hike? Show your work. Simple, 5 m up and 5 back is = 50 m total b) What is the hiker s displacement? Show your work. Zero, returns to start Acceleration A change in velocity. Acceleration can be speeding up, slowing down, or turning. The SI unit for acceleration is m/s. If the sign of the velocity and the sign of the acceleration is the same, the object speeds up. If the sign of the velocity and the sign of the acceleration are different, the object slows down. Uniformly Accelerated Motion a ave = v/ t Problem: Acceleration (1993) 1. In which of the following situations would an object be accelerated? I. It moves in a straight line at constant speed. II. It moves with uniform circular motion. III. It travels as a projectile in a gravitational field with negligible air resistance. (A) I only (B) III only (C) I and II only (D) II and III only (E) I, II, and III Explain your answer: Change direction = accelerate, projectile accelerates at 9.8 Average speed v ave = d / t Average Velocity v ave = d/ t Problem: Average Velocity (1988) 6. A truck traveled 400 meters north in 80 seconds, and then it traveled 300 meters east in 70 seconds. The magnitude of the average velocity of the truck was most nearly (A) 1. m/s (B) 3.3 m/s (C) 4.6 m/s (D) 6.6 m/s (E) 9.3 m/s Kinematic Equations v = v o + at x = x o + v o t + 1/ at v = v o + a( x) Problem: Kinematic Equations (1984) 65. A body moving in the positive x direction passes the origin at time t = 0. Between t = 0 and t = 1 second, the body has a constant speed of 4 meters per second. At t = 1 second, the body is given a constant acceleration of 6 meters per second squared in the negative x direction. The position x of the body at t = 11 seconds is (A) +99 m (B) +36 m (C) 36 m (D) 75 m (E) 99 m D= (V 1 T 1 ) + (V I T +.5 a t ) plug in, V1=4, Vi=4, T1=1, T=10, a=  6 1
2 Problem: Sketch the Kinematic Graphs Stationary particle d vs t v vs t a vs t Particle moving with constant velocity (A) Between O and 1 s (B) 1 s (C) Between 1 and s (D) s (E) Between and 3 s The area is the distance moved. Its ALL negative in the first 1 second, so have to make an equal positive area to bring back to 0 in the next section. seconds would be too much area d vs t v vs t a vs t Particle moving with constant nonzero acceleration d vs t v vs t a vs t Problem: Kinematic Graphs (1988) 1. The displacement x of an object moving along the xaxis is shown above as a function of time t. The acceleration of this object must be (A) zero (B) constant but not zero (C) increasing (D) decreasing (E) equal to g Explain your answer: Slope = velocity, slope is constant so velocity is constant so that means a=0 Problem: Kinematic Graphs (1984) 3. The graph shows the velocity versus time for an object moving in a straight line. At what time after time = 0 does the abject again pass through its initial position? Free Fall Occurs when an object falls unimpeded. Gravity accelerates the object toward the earth. g = 9.8 m/s downward. a = g if up is positive. acceleration is down when ball is thrown up EVERYWHERE in the balls flight. Problem: Free Fall (1993) 5. An object is released from rest on a planet that has no atmosphere. The object falls freely for 3.0 meters in the first second. What is the magnitude of the acceleration due to gravity on the planet? (A) l.5 m/s (B) 3.0 m/s (C) 6.0 m/s (D) 10.0 m/s (E) 1.0 m/s D=vi t +.5 a t T = 1, D = 3m, Vi=0, solve for a Projectile Motion An example of dimensional motion. Something is fired, thrown, shot, or hurled near the earth s surface. Horizontal velocity is constant. Vertical velocity is accelerated. Air resistance is ignored. Trajectory of Projectile The trajectory of a projectile is the path it follows. It is defined by a parabola. The RANGE of the projectile is how far it travels horizontally. The MAXIMUM HEIGHT of the projectile occurs halfway through its range, provided the projectile is fired over level ground. Acceleration points down at 9.8 m/s for the entire trajectory. Velocity is tangent to the path for the entire trajectory. The vertical velocity changes while the horizontal velocity remains constant.
3 Problem: Draw Graphs for D Projectiles Problem: Graphs of Projectile Motion (1988) xcomponent of motion d vs t v vs t a vs t ycomponent of motion d vs t v vs t a vs t Problems: Projectile Motion (1993) 63. A projectile is fired with initial velocity v 0 at an angle θ with the horizontal and follows the trajectory shown above. Which of the following pairs of graphs best represents the vertical components of the velocity and acceleration, v and a, respectively, of the projectile as functions of time t? Explain your reasoning: Veritcal component is influenced by gravity so gets less on the way up and is + sin ce going up but then move down so is negative and gets faster. Accel is always How do the speeds of the ball at the three points compare? (A) v P < v Q < v R (B) v R < v Q < v P (C) v Q < v R < v P (D) v Q < v P = v R (E) v P = v R < v Q Explain your choice: Speeds at equal points are the same (fact). At top you have the lowest speed 65. Which of the following diagrams best shows the direction of the acceleration of the ball at point P? Explain your choice: Gravity only acts in y direction D Motion problems Resolve vectors into components. Work as onedimensional problems. Horizontal Component of Velocity Not accelerated Not influence by gravity Follows equation: x = V o,x t Problem: Horizontal Component (1988) 9. A diver initially moving horizontally with speed v dives off the edge of a vertical cliff and lands in the water a distance d from the base of the cliff. How far from the base of the cliff would the diver have landed if the diver initially had been moving horizontally with speed v? (A) d (B) d (C) d (D) 4d (E) It cannot be determined unless the height of the cliff is known. Show your work or explain your reasoning: In x direction. Dx = vx * t. T is based on y direction which is unchanged. Vx is doubled so Dx will be doubles also 3
4 Vertical Component of Velocity Accelerated by gravity (9.8 m/s down) V y = V o,y  gt y = y o + V o,y t  1 / gt V y = V o,y  g(y y o ) Problem: Vertical Component (1988) 5. A kilogram block rests at the edge of a platform that is 10 meters above level ground. The block is launched horizontally from the edge of the platform with an initial speed of 3 meters per second. Air resistance is negligible. The time it will take for the block to reach the ground is most nearly (A) 0.3 s (B) 1.0 s (C) 1.4 s (D).0 s (E) 3.0 s Y direction motion. Viy = 0. Dy=Viy T +.5 a t Find t Problem: Vertical Component (1998) Force A force is a push or pull on an object. Forces cause an object to accelerate To speed up To slow down To change direction Unit: Newton (SI system) Newton s First Law The Law of Inertia. A body in motion stays in motion at constant velocity and a body at rest stays at rest unless acted upon by an external force. Problem: Newton s 1 st Law (1998) 7. Three forces act on an object. If the object is in translational equilibrium, which of the following must be true? I. The vector sum of the three forces must equal zero. II. The magnitudes of the three forces must be equal III. All three forces must be parallel (A) I only (B) II only (C) I and III only (D) II and III only (E) I, II, and III Explain your reasoning Equilibrium means Fnet=0. That defines # I. II and III are wrong based on the following example possibly making Fnet = 0 Problem: Newton s 1 st Law (1998) A rock of mass m is thrown horizontally off a building from a height h, as shown above. The speed of the rock as it leaves the thrower's hand at the edge of the building is υ How much time does it take the rock to travel from the edge of the building to the ground? (A) hυ 0 (B) h υ 0 (C) hυ 0 g (D) h g (E) h g Y direction motion. Viy = 0. Dy=Viy T +.5 a t Rearrange for t 44. The sum of the forces on the object is zero in which of the cases? (A) II only (B) III only (C) I and II only (D) I and III only (E) I, II, and III Explain your reasoning Fnet = 0 means a = 0. d vst t graph accelerates when curved, so only III is not possible 4
5 Problem: Newton s 1 st Law (1984) 5. A hall of mass m is suspended from two strings of unequal length as shown above. The tensions T 1 and T in the strings must satisfy which of the following relations? (A) T 1 = T (B) T 1 >T (C) T 1 <T (D) T 1 +T =mg (E) T 1 T =mg X components must be equal so ball does not move. Since T is at a higher angle it would have a larger y component for the same x component as T1, making the resultant of T bigger. Problem: Newton s 1 st Law (1988) 58. When an object of weight W is suspended from the center of a massless string as shown above, the tension at any point in the string is (A) W cosθ (B) W cosθ (C) W cosθ W (D) cosθ W (E) cosθ A little big of math involved. Both Y components are identical with equal angle so = MG/. Then using cos we have cos()=a/h. Sub in the MG/ and you get the answer General Procedure for Solving Second Law Problems Step 1: Draw the problem Step : Free Body Diagram Step 3: Set up equations ΣF = ma ΣF x = ma x ΣF y = ma y Step 4: Substitute Make a list of givens from the word problem. Substitute in what you know. Step 5: Solve Problem: Second Law (1984) 11. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension in the string between the blocks is (A) F (B) F (C) 3 F (D) 1 F 1 (E) 3 F Use the whole FBD to get F = (3m) a, so a = F/3m. Then do FBD of only the 1 kg mass, and get F T = ma. Sub in a and get F T = m ( F / 3m) = 1/3 F Problem: Second Law (1993). A ball falls straight down through the air under the influence of gravity. There is a retarding force F on the ball with magnitude given by F = bv, where v is the speed of the ball and b is a positive constant. The magnitude of the acceleration a of the ball at any time is equal to which of the following? (A) g (B) g (C) g b bv m + bv m Ball is moving down so make down +. FBD has mg down and force BV up.. Fnet = ma MGBV = m a a = ( mg bv ) / m Newton s Second Law ΣF = ma (D) g b (E) bv m 5
6 Problem: Second Law (1993) (A) W  Fcos θ (B) W  Fsin θ (C) W (D) W + Fsin θ (E) W + Fcos θ Show your work Fy = Fsin() upwards, Fn upwards, mg down. Fnet(y) gives Fn = MG Fsin(). 45. A block of mass 3m can move without friction on a horizontal table. This block is attached to another block of mass m by a cord that passes over a frictionless pulley, as shown above. If the masses of the cord and the pulley are negligible, what is the magnitude of the acceleration of the descending block? (A) Zero (B) g/4 (C) g/3 (D) g/3 (E) g Use the whole FBD, F = ma mg = (4m) a so a = g/4 Newton s Third Law For every action there exists an equal and opposite reaction. If A exerts a force F on B, then B exerts a force of F on A. Weight W = mg Normal Force Force that prevents objects from penetrating each other Reaction to other forces Commonly a reaction to gravity Problem: Normal Force Flat (1988) Problem: Normal Force Ramp (1993) A plane 5 meters in length is inclined at an angle of 37, as shown above. A block of weight 0 newtons is placed at the top of the plane and allowed to slide down. 6. The magnitude of the normal force exerted on the block by the plane is most nearly (A) l0 N (B) 1N (C) l6 N (D) 0 N (E) 33 N Show your work Some fancy math helps solve without calculator. Fn = y component of weight. = mg cos (). Using cos() = a/h and the triangle provided, we have cos () = 4/5. Now sub this in. MG cos() = (0) (4/5) = 16 N Problem: Elevators and Normal Force (PAB). A 50kg middle school student stands on a scale in an elevator that is moving downward, but slowing with an acceleration of magnitude.0 m/s. What does the scale read (in N)? a) 300 b) 400 c) 500 d) 600 e) 700 Show your work 4. A block of weight W is pulled along a horizontal surface at constant speed v by a force F. which acts at an angle of θ with the horizontal, as shown above. The normal force exerted on the block by the surface has magnitude Elevator method. Fscale points up, MG points down. Move down so down is +. Fnet = ma. Sub in gives MG Fsc = Ma (plug in, use a =  ) solve and get 600 Friction The force that opposes a sliding motion. Static friction exists before sliding occurs Kinetic friction exists after sliding occurs In general Kinetic friction <= Static friction 6
7 Calculating Static Friction f s μ s N Static friction increases as the force trying to push an object increases up to a point! Calculating Kinetic Friction f k = μ k N Problems: Friction on Flat Surface (1984) Explain your Reasoning: FBD s duh 7. The magnitude of the frictional force along the plane is most nearly (A).5 N (B) 5N (C) 6 N (D) 10 N (E) 16 N Fnet = ma. Mg sin () f K = m a. plug in and solve 61. A push broom of mass m is pushed across a rough horizontal floor by a force of magnitude T directed at angle θ as shown above. The coefficient of friction between the broom and the floor is µ. The frictional force on the broom has magnitude (A) μ(mg +Tsinθ) (B) μ(mg Tsinθ) (C) μ(mg +Tcosθ) (D) μ(mg Tcosθ) (E) μmg Show your work Fnet(y) gives Fn = mg + Tsin(). Then f = u Fn and your done. Problems: Friction on Ramp(1984) Questions 67 A kilogram block slides down a 30 incline as shown above with an acceleration of meters per second squared. 6. Which of the following diagrams best represents the gravitational force W, the frictional force f, and the normal force N that act on the block? Uniform Circular Motion An object moves at uniform speed in a circle of constant radius. Acceleration in Uniform Circular Motion Turns object; doesn t speed it up or slow it down. Acceleration points toward center of the circle. Called centripetal acceleration. Centripetal Acceleration a c = v /r Force in Uniform Circular Motion Any force or responsible for uniform circular motion is referred to as a centripetal force. Centripetal force can arise from one force, or from a combination of sources. F c = ΣF = ma c F c = ΣF = m v / r Centripetal forces always arise from other forces. Since speed of object remains constant, kinetic energy remains constant, and work is zero. Friction, tension, normal force, gravity and the magnetic force are common forces that can act centripetally to cause uniform circular motion. Problem: Centripetal Force (1993) 46. A car initially travels north and then turns to the left along a circular curve. This causes a package on the seat of the car to slide toward the right side of the car. Which of the following is true of the net force on the package while it is sliding? (A) The force is directed away from the center of the circle. (B) The force is directed north. (C) There is not enough force directed north to keep the package from sliding. (D) There is not enough force tangential to the car's path to keep the package from sliding. (E) There is not enough force directed toward the center of the circle to keep the package from sliding. Fc points towards center and is friction in this case. The lack of friction towards center is the lack of centripetal force needed to make the box go in the circle with the car 7
8 Problem: Centripetal Force (1988). The horizontal turntable shown below rotates at a constant rate. As viewed from above, a coin on the turntable moves counterclockwise in a circle as shown. Which of the following vectors best represents the direction of the frictional force exerted on the coin by the turntable when the coin is in the position shown? Friction acts as the centripetal force allowing the coin to go in the circle so it must point to center SIDE BAR Time to learn a little something new. Two topics were skipped over, they are not on the midterm but as long as we are reviewing, might as well introduce these two very minor areas of study. Technically, they are on the AP exam but very rare and usually a single multiple choice question. Problem: Angular Momentum (1998) 6. A kg object moves in a circle of radius 4 m at a constant speed of 3 m/s. A net force of 4.5 N acts on the object. What is the angular momentum of the object with respect to an axis perpendicular to the circle and through its center? (A) N m 9 kg (B) 1 m s (C) kg m 135. s (D) N m 18 kg (E) kg m 4 s Based on the new concept discussed in the side bar, Angular momentum (L) = mvr simply plug in and solve Problem: Angular Momentum (1984) I.) ANGULAR MOMENTUM The tendency of a spinning body to remain spinning. It is similar o to Torque. L: angular momentum (kg m /s) Angular momentum is found by the equation: L = p x r L = (mv) x r Where momentum (mv) is perpendicular to the distance r Angular momentum, like linear momentum is conserved and when objects move in a circle, the angular momentum stays the same unless outside force acts. For an elliptical orbit, as the velocity changes, the radius changes as well to make the quantity (L=mvr) a constant II. Gravitaional Potential Energy for space systems U g = GM e m/r (close to earth s surface) U g has been defined to be zero when an object is infinitely far from the earth, and it gets increasingly negative as an object approaches the earth. 14. An asteroid moves in an elliptic orbit with the Sun at one focus as shown above. Which of the following quantities increases as the asteroid moves from point P in its orbit to point Q? (A) Speed (B) Angular momentum (C) Total energy (D) Kinetic energy (E) Potential energy Angular momentum is conserved so that s not it, Closer to planet = faster so speed is wrong. Energy is also conserved so that s wrong. Going from P to Q, you slow down so lose K which mean U must be gained 8
9 Universal Law of Gravity F g = Gm 1 m /r The negative sign simply means the force is attractive. Most orbit problems can be solved by setting the gravitational force equal to the centripetal force. Gm 1 m / r = m 1 v / r (a good starting point for most problems!) Orbit the above equation simplifies to v = GM/r for velocity in an orbit which also = πr/t if period is known Acceleration of gravity g is found by equating mg=gmm/r Problem: Orbit (1988) 6l. A satellite of mass M moves in a circular orbit of radius R at a constant speed v. Which of the following must be true? I. The net force on the satellite is equal to mv /R and is directed toward the center of the orbit. II. The net work done on the satellite by gravity in one revolution is zero. III. The angular momentum of the satellite is a constant. (A) I only (B) III only (C) I and II only (D) II and III only (E) I, II, and III Angular momentum is conserved so III is good, There is a circle so the force would be centripetal so I is good, and work is always zero when in a circle since the force is always perpendicular to motion. Problem: Orbit (1998) 40. What is the kinetic energy of a satellite of mass m that orbits the Earth, of mass M, in a circular orbit of radius R? (A) Zero (B) 1 GMm R (D) 1 GMm (E) GMm R R (C) 1 4 GMm R Velocity in orbit is sqrt(gm/r). Then KE = 1/ m v then plug in orbit velocity and you get B Problem: Law of Gravity and Weight (1998) 39. An object has a weight W when it is on the surface of a planet of radius R. What will be the gravitational force on the object after it has been moved to a distance of 4R from the center of the planet? (A) 16W Show your work ; (B) 4W Based on Fg = G M M / r (C) W so 4 x R (D) 1 gives 1/16 the force. 4 W Problem: Law of Gravity and Accel. (1993) 48. The planet Mars has mass 6.4 x 10 3 kilograms and radius 3.4 x 10 6 meters. The acceleration of an object in freefall near the surface of Mars is most nearly (A) zero (B) l.0 m/s (C) 1.9 m/s (D) 3.7 m/s (E) 9.8 m/s This problem was given before calculators were banned on multiple choice so you simply plug into g = G M / R Alternatively if you are really smart you can estimate the answer by knowing mass of earth 6x10 4 and radius of earth 6.4x10 6 m we can see that mars has about 1/10 the mass and 1/ the radius, so based on the formula it should have 4/10 the value of g on earth. Periodic Motion Repeats itself over a fixed and reproducible period of time. Mechanical devices that do this are known as oscillators. Simple Harmonic Motion (SHM) Periodic motion described by a sine or cosine function. Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs). All SHO s experience a restoring force. Restoring Force: F = kx (basic form of restoring force) Restoring force is greatest at maximum displacement and zero at equilibrium Equilibrium The midpoint of the oscillation of a simple harmonic oscillator. Position of minimum potential energy and maximum kinetic energy. Amplitude (A) How far the oscillating mass is from equilibrium at its maximum displacement. Period (T) The length of time it takes for one cycle of periodic motion to complete itself. Frequency (f): How fast the oscillation is occurring. Frequency is inversely related to period. f = 1/T The units of frequency is the Herz (Hz) where 1 Hz = 1 s 1. (E) 1 16 W 9
10 Problem: Simple Harmonic Motion (1993) 43. A particle oscillates up and down in simple harmonic motion. Its height y as a function of time t is shown in the diagram above. At what time t does the particle achieve its maximum positive acceleration? (A) 1s (B) s (C) 3s (D) 4s (E) None of the above, because the acceleration is constant When the mass on the spring is below equibrium at a negative displacement the force points upwards and is + and is at a maximum so gives maximum + acceleration there at max stretch. Springs A common Simple Harmonic Oscillator. F s = kx (Hooke s Law) Problem: Hooke s Law (1993) 3. An ideal spring obeys Hooke's law, F =  kx. A mass of 0.50 kilogram hung vertically from this spring stretches the spring meter. The value or the force constant for the spring is most nearly (A) 0.33 N/m (B) 0.66 N/m (C) 6.6 N/m (D) 33 N/m (E) 66 N/m Simply plug it in. Use fractions. 5/10 and 75/1000 to ease calculation Period of a spring T = π m/k Potential Energy of a Spring U s = ½ k x Pendulum The pendulum can be thought of as an oscillator. The displacement needs to be small for it to work properly. Pendulum Forces: Gravity and tension Period of a pendulum T = π l/g Potential Energy of a Pendulum U g = mgh Problem: Period of Pendulum (1988) 8. The length of a simple pendulum with a period on Earth of one second is most nearly (A) 0.1 m (B) 0.5 m (C) 0.50 m (D) 1.0 m (E) 10.0 m T= pi * rad (L/g) rearrange for L, use g = 10 and use pi = 3. Get estimated value closest to 0.5 Problem: Period of Spring and of Pendulum (1988) 44. An object swings on the end of a cord as a simple pendulum with period T. Another object oscillates up and down on the end of a vertical spring, also with period T. If the masses of both objects are doubled, what are the new values for the periods? Pendulum Spring (A) T T (B) T T (C) T T (D) T T (E) T T Actually quite simple, mass does not change the pendulum period so it must be B. Based on spring formula, you get rad T as change for spring system. Work The bridge between force and energy. Work is a scalar. W = F Δx cos θ SI Unit: Joule (N m) Counterintuitive Results There is no work if there is no displacement. Forces perpendicular to displacement don t work. By doing positive work on an object, a force or collection of forces increases its mechanical energy in some way. The two forms of mechanical energy are called potential and kinetic energy. 10
11 Problem: Work (B1988) 6. A horizontal force F is used to pull a 5kilogran block across a floor at a constant speed of 3 meters per second. The frictional force between the block and the floor is 10 newtons. The work done by the force F in 1 minute is most nearly (A) 0 J (B) 30 J (C) 600 J (D) 1,350 J (E) 1,800 J W = F d. Constant speed of F pull = friction = 10 N, v = 3 m/s, t = 60 sec, so d=vt, d = 180 m. Now W = F d = 10 (180) = 1800 J Problem: Power (B1998). A student weighing 700 N climbs at constant speed to the top of an 8 m vertical rope in 10 s. The average power expended by the student to overcome gravity is most nearly (A) 1.1 W (B) 87.5 W (C) 560 W (D) 875 W (E) 5,600 W Kinetic Energy Energy due to motion K = ½ m v P = F d / t simply plug in. Force to lift = weight Problem: Kinetic Energy (B1988) 3. Which of the following quantities is a scalar that is always positive or zero? (A) Power (B) Work (C) Kinetic energy (D) Linear momentum (F) Angular momentum State your reasoning: K = 1/ m v there is no way this could be negative. The WorkEnergy Theorem W net = ΔK Net work is used in this theorem. This is work due to ALL FORCES acting upon object. When net work is positive, the kinetic energy of the object will increase (it will speed up). When net work is negative, the kinetic energy of the object will decrease (it will slow down). When there is no net work, the kinetic energy is unchanged (constant speed). Work and graphs The area under the curve of a graph of force vs displacement gives the work done by the force in performing the displacement. Springs: stretching Springs:compressing Problem: Power (B1998) 9. A child pushes horizontally on a box of mass m which moves with constant speed υ across a horizontal floor. The coefficient of friction between the box and the floor is μ. At what rate does the child do work on the box? (A) μmgυ (B) mgυ (C) υ/μmg (D) μmg/υ (E) μmυ Rate of work = W / t = Power. Power = F v constant speed of F = friction = u Fn P = u Fn v = u mg v Power The rate of which work is done. When we run upstairs, t is small so P is big. When we walk upstairs, t is large so P is small. P = W/t work/time P = F V (force )(velocity) SI unit for Power is the Watt. 1 Watt = 1 Joule/s 11
12 Force Types Conservative forces: Work in moving an object is path independent. Work in moving an object along a closed path is zero. Work done against conservative forces increases potential energy; work done by them decreases it. Ex: gravity, springs Nonconservative forces: Work is path dependent. Work along a closed path is NOT zero. Work may be related to a change in total energy (including thermal energy). Ex: friction, drag Potential energy Energy an object possesses by virtue of its position or configuration. Examples: Gravitational Potential Energy Spring Potential Energy Potential energy is related to work done by CONSERVATIVE FORCES only. ΔU g = W g (gravity) ΔU g = W s (spring) Spring potential energy Springs also can possess potential energy (U s ). U s = ½ kx U s is zero when a spring is in its preferred, or equilibrium, position where the spring is neither compressed or extended. Problem: Conservation of Energy and Springs(B1988) A block oscillates without friction on the end of a spring as shown above. The minimum and maximum lengths of the spring as it oscillates are, respectively, x min and x max The graphs below can represent quantities associated with the oscillation as functions of the length x of the spring. Gravitational potential energy close to earth s surface. W g = mgh (close to earth s surface) ΔU = W g = mgh Note: we calculate changes in potential energy only using this method. We assign the potential energy to be zero at some certain point, usually the surface of the earth. Problem: Work due to gravity (B1993) 11. Which graph can represent the total mechanical energy of the blockspring system as a function of x? (A) A (B) B (C) C (D) D (E) E A plane 5 meters in length is inclined at an angle of 37, as shown above. A block of weight 0 newtons is placed at the top of the plane and allowed to slide down. 63. The work done on the block by the gravitational force during the 5meter slide down the plane is most nearly (A) 0 J (B) 60 J (C) 80 (D) 100 J (E) l30 J Work of gravity is work of Fgx. Fgx = mg sin (). Using the triangle provided and that sin() = o/h, we get sin() = 3/5. Then work = W=Fgx * D = mgsin() d = (0)*(3/5)*(5) Energy is conserved so ME is always the same 1. Which graph can represent the kinetic energy of the block as a function of x? (A) A (B) B (C) C (D) D (E) E State your reasoning: At the max displacement the block stops so there cant be any K there. C or D are the only ones left. Since K and Usp are both squared relationships, as displacement changes it should not be linear, so it D. 1
13 Law of Conservation of Mechanical Energy U + K = Constant ΔU + ΔK = 0 ΔU = ΔK Note: Sometimes conservation of energy problems can be best worked with potential energy alone, such as in the problem below. Conservation of Energy and Dissipative Forces. Dissipative forces cause loss of mechanical energy by producing heat. W nc = ΔU + ΔK Problem: Conservation of Energy (B1998) height to which the block rises on the other side of the track? (A) It is equal to h/π. (B) It is equal to h/4. (C) It is equal to h/. (D) It is equal to h. (E) It is between zero and h; the exact height depends on how much energy is lost to friction. Work of friction equals the change in the energy which controls how high it goes. We don t know anything about speeds or forces or distance so there is no way of knowing how much the friction work is. Problem: Conservation of Energy (B1998) 38. A block of mass 3.0 kg is hung from a spring, causing it to stretch 1 cm at equilibrium, as shown above. The 3.0 kg block is then replaced by a 4.0 kg block, and the new block is released from the position shown above, at which the spring is unstretched. How far will the 4.0 kg block fall before its direction is reversed? (A) 9 cm (B) 18 cm (C) 4 cm (D) 3 cm (E) 48 cm First use 3 kg mass situation. F = 3 N, /\x = 1/100 so F=kx gives us k = 3000/1. Then looking at 4 kg mass we do energy conservation. Ugrav at start = Usp at end. And h= /\x so mgh = ½ kx mgx = ½ k x mg = k x / so x = mg /k = (4)(10) / (3000/1) = 960 / 3000 = 30/1000 = 0.3 done. A rock of mass m is thrown horizontally off a building from a height h, as shown above. The speed of the rock as it leaves the thrower's hand at the edge of the building is υ How much time does it take the rock to travel from the edge of the building to the ground? (F) hυ 0 h υ 0 (G) hυ 0 g h g Problem: Conservation of Energy (B1993) (H) h g 4. The figure above shows a rough semicircular track whose ends are at a vertical height h. A block placed at point P at one end of the track is released from rest and slides past the bottom of the track. Which of the following is true of the Simple projectile problem using Y direction stuff. Viy is zero since it goes horizontal. 13
14 60. What is the kinetic energy of the rock just before it hits the ground? Problem: Conservation of Energy and Pendulums (B1988) (A) mgh (B) 1 mυ 0 (C) 1 mυ 0 mgh (D) 1 mυ 0 + mgh (E) mgh 1 m υ 0 ME top = ME bottom. At top its moving with height do is both K and U, and bottom its just K, so the K = the energy at the top which is K and U = choice D Problem: Conservation of Energy (B1993) 47. A block of mass m slides on a horizontal frictionless table with an initial speed v 0. It then compresses a spring of force constant k and is brought to rest. How much is the spring compressed from its natural length? (A) v 0 g (B) mg k (C) m k v 0 (D) (E) m k v 0 k m v 0 A ball swings freely back and forth in an arc from point I to point IV, as shown above. Point II is the lowest point in the path, III is located 0.5 meter above II, and IV is l meter above II. Air resistance is negligible. 51. If the potential energy is zero at point II, where will the kinetic and potential energies of the ball be equal? (A) At point II (B) At some point between II and III (C) At point III (D) At some point between III and IV (E) At point IV State your reasoning: U = mgh, so comparing 1m to 0.5 m it s the half way point for energy, so if half the energy is potential the other half is kinetic 5. The speed of the ball at point II is most nearly (A) 3.0 m/s (B) 4.5 m/s (C) 9.8 m/s (D) l4 m/s (E) 0 m/s Pt II is all K, and Pt I is all U, so U1 = K so mgh I = ½ mv m cancels, plug in, V = rad ( (10) (1) ) = rad (0). 4.5 is best estimate ME = ME, start with K then become Usp. Plug in formulas and rearrange. 14
15 Momentum How hard it is to stop a moving object. Related to both mass and velocity. For one particle p = mv For a system of multiple particles P = Σp i = Σm i v i Units: N s or kg m/s Momentum is a vector! Problem: Momentum (1998) Problem: Impulse (1998) 57. A ball of mass 0.4 kg is initially at rest on the ground. It is kicked and leaves the kicker's foot with a speed of 5.0 m/s in a direction 60 above the horizontal. The magnitude of the impulse imparted by the ball to the foot is most nearly (A) 1N s (B) 3 N s (C) N s (D) 3 N s (E) 4N s 43. The magnitude of the momentum of the object is increasing in which of the cases? (A) II only (B) III only (C) I and II only (D) I and III only (E) I, II, and III Momentum increases when velocity increases. Slope is velocity and only in III is velocity increasing. Impulse (J) The product of an external force and time, which results in a change in momentum J = F t J = ΔP Units: N s or kg m/s Problem: Impulse (1984) 56. Two planets have the same size, but different masses, and no atmospheres. Which of the following would be the same for objects with equal mass on the surfaces of the two planets? I. The rate at which each would fall freely II. The amount of mass each would balance on an equalarm balance III. The amount of momentum each would acquire when given a certain impulse (A) I only (B) III only (C) I and II only (D) II and III only (E) I, II, and III More massive planet has a larger value of g. I cant be same. For balancing, the weight of the mass balanced would be more but as long as masses are equal it will balance and total mass is equivalent in both places. Impulse is Ft and equals chance in momentum, this is not affected by gravity so also is same The angle is irrelevant. Impulse (Ft) equals momentum change so simply plug in Ft Law of Conservation of Momentum If the resultant external force on a system is zero, then the momentum of the system will remain constant. The sum of the momentums before a collision is equal to the sum of the momentums after a collision. ΣP b = ΣP a Problem: Conservation of Momentum (1998) 4. An open cart on a level surface is rolling without frictional loss through a vertical downpour of rain, as shown above. As the cart rolls, an appreciable amount of rainwater accumulates in the cart. The speed of the cart will (A) increase because of conservation of momentum (B) increase because of conservation of mechanical energy (C) decrease because of conservation of momentum (D) decrease because of conservation of mechanical energy (E) remain the same because the raindrops are falling perpendicular to the direction of the cart's motion Momentum must be conserved. Increasing the mass required v to decrease to have the same momentum 15
16 Collisions Follow Newton s Third Law which tells us that the force exerted by body A on body B in a collision is equal and opposite to the force exerted on body B by body A. During a collision, external forces are ignored. The time frame of the collision is very short. The forces are impulsive forces (high force, short duration). Collision Types Elastic: P is conserved, K is conserved Inelastic: P is conserved, K is NOT conserved Perfectly Inelastic means the bodies stick together Problem: Collisions (1993) 10. Which of the following is true when an object of mass m moving on a horizontal frictionless surface hits and sticks to an object of mass M > m, which is initially at rest on the surface? (A) The collision is elastic. (B) All of the initial kinetic energy of the lessmassive object is lost. (C) The momentum of the objects that are stuck together has a smaller magnitude than the initial momentum of the lessmassive object. (D) The speed of the objects that are stuck together will be less than the initial speed of the lessmassive object. (E) The direction of motion of the objects that are stuck together depends on whether the hit is a headon collision. Based on collision P before = P after. m v = (M+m) v clearly to conserve momentum the speed must drop Problem: Collision (1998) 3. A railroad car of mass m is moving at speed υ when it collides with a second railroad car of mass M which is at rest. The two cars lock together instantaneously and move along the track. What is the speed of the cars immediately after the collision? (A) υ (B) mυ M (C) Mυ m (D) (m + M) υ m (E) mv m+m Collision P before = P after. m v = (M+m) v solve for v Problem: Momentum Change (1988) Problem: Collisions (1993) 11. Two objects having the same mass travel toward each other on a flat surface, each with a speed of 10 meter per second relative to the surface. The objects collide headon and are reported to rebound after the collision, each with a speed of 0 meters per second relative to the surface. Which of the following assessments of this report is most accurate? (A) Momentum was not conserved, therefore the report is false. (B) If potential energy was released to the objects during the collision, the report could be true. (C) If the objects had different masses, the report could be true. (D) If the surface was inclined, the report could be true. (E) If there was no friction between the objects and the surface, the report could be true. Before the collision, each has momentum and energy based on velocity of 10 m/s. After, there is more velocity so a gain of energy. The only way to gain energy is to have some sort of input of energy into the system 7. A tennis ball of mass m rebounds from a racquet with the same speed v as it had initially, as shown above. The magnitude of the momentum change of the ball is (A) 0 (B) mv (C) mv (D) mv sin θ (E) mv cos θ D collision must be analized in X and Y directions. In y direction there is no change, the racket only hits in x direction. Also the velocity in y diriection is unchanged. So the momentum change occurs in the X direction. Before we have MVx = mv cos() then after we have  mv cos (). Change in momentum = pf pi =  mv cos()  mv cos () done. 16
17 Problem: Collision (1998) Problem: Explosion (1998) 41. Two objects of mass 0. kg and 0.1 kg, respectively, move parallel to the xaxis, as shown above. The 0. kg object overtakes and collides with the 0. 1 kg object. Immediately after the collision, the ycomponent of the velocity of the 0. kg object is 1 m/s upward. What is the ycomponent of the velocity of the 0.1 kg object immediately after the collision? (A) m/s downward (B) 0.5 m/s downward (C) 0 m/s (D) 0.5 m/s upward (E) m/s upward A D collision, we look in y direction.. Before there is no y momentum so there can be any after either. Since the 0. kg object goes up and has momentum (0.)*(1) the 0.1 kg mass must go down at m/s to have an equal negative momentum to cancel out and give zero Explosion Mathematically, handled just like an ordinary perfectly inelastic collision. Momentum is conserved, kinetic energy is not. 67. A stationary object explodes, breaking into three pieces of masses m, m, and 3m. The two pieces of mass m move off at right angles to each other with the same magnitude of momentum mv, as shown in the diagram above. What are the magnitude and direction of the velocity of the piece having mass 3m? Before explosion, momentum is zero, so it must be zero after also. In the x direction there is mv and in the y direction there is mv. So mass 3m must have momentum mv in both x and v direction. For 3m to have momentum = mv, the velocity needs to be V/3. It moves at V/3 in the x and y direction now use Pythagorean theorem to solve = v = sqrt (a + b ) v = sqrt ( (v/3) + (v/3) ) = sqrt ( v / 9) 17
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