LABORATORY 9. Simple Harmonic Motion

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1 LABORATORY 9 Simple Harmonic Motion Purpose In this experiment we will investigate two examples of simple harmonic motion: the mass-spring system and the simple pendulum. For the mass-spring system we will measure the period of oscillation and compare that with the value predicted by theory. We will use the pendulum to calculate a value for g, the acceleration of gravity. Equipment Motion sensor & connecting cable Science Workshop interface & DataStudio software Table clamp Support rod Pendulum clamp Spring Mass hanger Masses: g and 1 50-g Pendulum bob String Meter stick Principles Simple Harmonic Motion Simple harmonic motion is motion that repeats itself in a regular fashion. That is, the motion is cyclic. If we know how the motion takes place during one cycle, we know the motion for all time (for an ideal, frictionless system), since succeeding cycles are the same as the first. A key piece of information about a system in simple harmonic motion is its period: the time for the system to go through one cycle. We will see below that Newton s Laws of Motion make a definite prediction for the period of any such system. For the mass-spring system, the period turns out to depend on the mass of the system and the stiffness of the spring (also called the force constant of the spring). For the pendulum, the relevant parameters are the length of the pendulum and the acceleration of gravity. Linear Restoring Forces Simple harmonic motion is caused by a linear restoring force. A linear restoring 11

2 force is one that is proportional to the displacement of the system (i.e., the graph of the force with displacement is a straight line), and that is directed opposite to the displacement (i.e., the force resists the displacement). The farther the system moves from equilibrium, the stronger the force pushes or pulls it back, with the result that the system oscillates around equilibrium. Since the mass develops a velocity during this process, it will overshoot the equilibrium position and compress the spring. The spring will resist this compression, again with a force proportional to the amount of compression, so that the mass will be slowed to a stop and then accelerated toward equilibrium again. It will again overshoot, and the process will repeat itself. For instance, if we attach a mass to a horizontal spring on a frictionless surface and stretch it out, the spring will pull back on the mass with a force that is proportional to the amount of stretch. (See Diagram 1, where the unstretched position is indicated by x = 0.). x = 0 F m If we then release the mass, it will be accelerated towards the equilibrium position, x = 0, where the force of the spring and the displacement of the mass are both zero. Diagram 1 We can graph the force on the mass as a function of displacement from equilibrium as follows: Force v. Displacement Force (F) Displacement (x) 1

3 From the graph, we see that a linear restoring force has the following general form: (1) F = kx where x is the displacement from equilibrium and k is the magnitude of the slope of the graph. For the mass-spring system, k is the stiffness of the spring, but other parameters play the role of k for other systems. Angular frequency If the force on a system is as above, then by Newton s nd Law, F = ma, we have which means that kx = ma a = giving us the acceleration of the system at any displacement x. k m It is convenient to define a positive constant ω as: () ω k m so that the acceleration can be written x a = ω x. ω is called the angular frequency of the system. It measures how quickly the system goes through its cycle. The units of ω are radians per second, or sec -1, also known as hertz. Equation of Motion Now let s find a general equation of motion for a system accelerated by a linear restoring force in terms of its angular frequency. Recall that acceleration is the rate of change of velocity with time: it is the slope of the velocity-time graph at any point in time. In turn, the velocity is the rate of change of position (or displacement): it is the slope of the position-time graph at any point in time. This means that we can express acceleration as: v x x a = = ( ) = where we have expressed things in terms of average acceleration and average velocity for simplicity. (The reader should understand that to be exact we really should write the above in terms of the limit as 0, but we ll omit this for simplicity; it won t change our argument.) Putting this into our expression for acceleration, we have x (3) = ω x so that the only variables are x and t (ω is a constant parameter). Now x is a function of time, not just a number. To solve the above equation, we need to find a function that has the 13

4 property that the equation specifies: that the rate of change of the rate of change of the function is the same as the function itself multiplied by ω. It is shown in Calculus that both the sine function and the cosine function have this property, and that a solution to the above differential equation is either (4a) x( t) = x0 sin( ωt) or (4b) x( t) = x0 cos( ωt) In our experiment, we will use the motion detector to confirm that both the displacement and the acceleration of the system are sinusoidal (described by a sine or cosine) functions of time. We will see that the velocity is sinusoidal as well. In our case, the solution using the cosine function is appropriate, since the systems we will look at will start at time zero with a non-zero displacement. (That is, x ( t = 0) = x0, where x 0 is the initial displacement of the system. Since the system will periodically return to x 0, and since it is the largest displacement of the system in absolute value, x 0 is also called the amplitude of the motion.) Period of the Motion Now let s find the period T of the motion. At time zero, the system is located at x 0 ; at time T the system has returned to x 0. Since the equation of motion (3b) tells us that x ( T ) = x0 cos( ω T ) = x0, we see that we must have: cos( ω T ) = 1 This will be true when or (5) ωt = π T π = ω which is the general formula relating the period of an oscillating system to its angular frequency. Now let s apply the above general analysis to the mass-spring system and to the pendulum. The Mass-Spring System When a mass is attached to a horizontal spring on a frictionless surface and the mass is displaced from equilibrium (as in Diagram 1), the force that the spring experts on the mass is given by Hooke s Law: (6) = kx F s where x is the displacement from equilibrium and k is the spring constant. F s has the general form described above, so the angular frequency of the massspring system is: ω = and the period is: T = π k m m k The above applies to a massless spring. Our real spring has some mass m s, and this should be included in the mass of 14

5 the oscillating system. Theory shows that the effective mass of the system is that of the attached mass m plus onethird of the mass of the spring, so that the period formula becomes: (7) meff T = π with k 1 meff = m + m s. 3 The Spring Constant We will need to determine the spring constant k experimentally. If we hang the spring vertically, gravity will stretch out the spring because of the spring s own weight. The position where the lower end of the spring comes to rest without oscillating is the equilibrium position for the vertical spring. If we hang some additional mass on the spring, the spring will stretch out more, until the restoring force of the spring balances the weight W of the mass. At the new equilibrium position we have: F s = kx = W where x is the elongation of the spring because of the weight of the hanging mass. In the above, F s is directed upward while x and W are directed downward. Thus (8) W = kx The graph of this equation will be a straight line, with the spring constant k as the slope. We can hang several masses from the spring and measure the resulting elongation x for each. Graphing W as a function of x and taking the slope will give us the spring constant. Once we have k, we can time the oscillation of the system for several masses to determine the period experimentally. We will compare our experimental values for the period with theoretical values given by equations (7). The Pendulum A simple pendulum is another oscillating system which, for small displacements, is acted upon by a linear restoring force. A suspended weight, called the pendulum bob, travels in a circular arc about the vertical direction. The forces on the bob are its own weight, directed vertically, and the tension in the string, directed toward the center of the arc (see Diagram ). The situation is most naturally expressed in angular coordinates. The displacement from equilibrium, x, is the distance along the arc from the vertical. In radian measure, an angle θ is defined s as θ, where s is the arc length and r r is the radius of the arc. In our notation, x θ =, where L is the length of the L pendulum. Thus x = Lθ and the acceleration along the arc can be written as x θ a = = L and Newton s nd Law becomes θ F = ml 15

6 for the acceleration tangent to the arc. L -θ θ Equilibrium position (θ=0) -Wsinθ Wcosθ Diagram Now the force tangent to the arc is the tangential component of the bob s weight: F = -Wsinθ. This is a restoring force, but it is not linear because of the term sinθ. However, for small angles (θ < 10 degrees) we can use the approximation sin θ θ (this approximation can be rigorously justified*). In this case, the restoring force becomes F = -Wθ = -mgθ, which is linear in the angular displacement θ. Plugging this into the angular form of Newton s nd Law we have: which is the angular form of equation (3) with ω = g/l. Thus, for the simple pendulum, ω = g L so that the period T will be (9) T π = = π ω L g and the equation of motion will be or θ mgθ = ml θ g = θ L (10) θ ( t) = θ 0 cos( ωt) (Compare with equation (4b); all we have done is to replace the translational coordinate x with the angular coordinate θ and determined what ω must be in this case.) 16

7 We can use equation (9) to get an experimental value for g. If we solve this equation for g, we get L g = 4π T We can time the period for several values of the length L and graph the square of the period (T ) as a function of L. The slope of this graph will be the ratio T /L, so that we can calculate g from: (11) 4π g = m where m is the slope of the graph. * It is shown in Calculus that the sine function can be expressed as 3 θ sinθ = θ 3! 5 θ + 5! 7 θ 7! + where θ is expressed in radian measure. For small angles, θ < 1 and all terms on the right are negligible except the first. Procedures: Spring 1. Graph the spring s motion using the motion detector. Set up the spring for vertical oscillation. Clamp an upright rod at the end of a lab table. Attach a pendulum clamp (or a horizontal bar) to the upright rod. Hang the spring from the pendulum clamp with the narrow end down. Hang a mass hanger and 400 grams from the end of the spring. Arrange the system so it can vibrate up and down freely. Open DataStudio and set up the motion detector. The yellow plug goes into digital channel one and the black plug goes into digital channel two. Set the motion sensor on wide beam. Open the file for this lab ( SHM.ds ). Three blank graphs will be arranged on the screen, one above the other. These are graphs for the position, velocity and acceleration of the hanging mass as a function of time. Place the motion detector directly underneath the mass hanger. Turn the screen on the motion detector to point to the bottom of the mass hanger. There should be at least 14 cm between the screen and the bottom of the hanger. Set the mass-spring system in motion, then click the start button on DataStudio. The activity is set up to start timing after a onesecond delay and to stop timing after 5 seconds. For each of the graph windows, click on scale-to-fit. Also click on align x-axes. Make sure you are getting good sine wave traces in all 3 windows. If not, adjust the position of the motion sensor. Also make sure that the hanger is not getting too close to the sensor screen. Compare the position, velocity and acceleration graphs. In particular, note that the velocity is a maximum when the position is at the equilibrium (elongation 17

8 x = 0) point, and that acceleration is positive when elongation is negative and vice versa.. Determine the spring constant k. Remove the motion sensor. Tape a -meter stick next to the vertically suspended spring so you can measure the position of its end. The meter stick should be resting on the floor with the high end down. Measure the position of the bottom of the mass hanger when the spring is at rest. Place 50, 150, 50, 350 and 450 grams on the mass hanger in succession and measure the new position of equilibrium for each. When hanging the masses, let the spring stretch out slowly so that it does not oscillate. Calculate the elongation of the spring for each mass, relative to the position for mass hanger alone. Calculate the weight of each hanging mass and tabulate your data in the table below. Graph the elongation (x) as a function of the weight suspended from the spring. Draw the bestfit line through your data points. Determine the spring constant k from the slope of the graph. 3. Find the period of the mass-spring system and compare with theory. Weigh the spring and record the value. Now hang 150, 50, 350 and 450 grams from the spring in succession. (The 50 grams is the mass of the hanger.) Calculate the effective mass, m eff in each case: m eff = m + m spring /3. For each mass, displace the bottom of the hanger 5 centimeters from equilibrium and let the system oscillate. Start your timer as you release the system and count 0 cycles. In the data table, record the time for 0 cycles. Calculate the period by dividing this time by 0: T exp = T 0 /0. Calculate the period using equation (7). Take percent difference for each case. 4. Use the pendulum to find g Get a piece of string about 1. m long. Run the string through the center hole of a pendulum bob and tie it. Wind the other end of the string around one of the clamps on the pendulum clamp and tighten the screw. It is not necessary to cut the string: to change the length of the string, simple wind or unwind the string. Time the period of oscillation for pendulum lengths of about L = 0, 40, 60, 80 and 100 cm. (It is not necessary to use exactly these lengths: just measure and record the lengths you actually use.) The length of the pendulum is the distance from the bottom of the pendulum clamp to the middle of the ball. 18

9 Let each pendulum oscillate 0 times and record the time for 0 swings. Calculate the period, T exp for each length by dividing the total time by 0. Square T exp and record the value in the column labeled T. Graph T as a function of the length of the pendulum, L. Find the slope of your graph. Use equation (11) to calculate an experimental value for g. Compare your value with the accepted value of 9.80 m/sec. Calculate the theoretical period for each of the lengths from equation (9) and record the values in the table. Find the percent difference between the experimental and theoretical values of the period. 19

10 130

11 Lab 9: Simple Harmonic Motion Name: Date: Class: Data & Analysis Spring Constant Data Equilibrium position: (with no mass on hanger) Suspended mass Suspended weight Position of bottom of hanger Elongation (x) Slope of graph: Spring Constant k: 131

12 Lab 9: Simple Harmonic Motion Name: Date: Class: Data & Analysis Spring Period Data Mass of spring: Amplitude (x 0 ): Suspended Mass Effective Mass (m eff ) Time for 0 vibrations Measured Period Calculated Percent difference (T exp ) T theory 13

13 Lab 9: Simple Harmonic Motion Name: Date: Class: Data & Analysis Pendulum Data Length of Pendulum Time for 0 vibrations Experimental Period (T exp ) Square of Period (T ) Theoretical Period (T theory) % error Slope of graph: Calculated value of g: Percent difference from accepted value 133

14 Lab 9: Simple Harmonic Motion Name: Date: Class: Data & Analysis Questions 1. Look at the DataStudio plot of position, velocity and acceleration. How do the maxima and minima of the curves on these plots relate?. How could you calculate the period and the angular frequency of the spring from the DataStudio plots? 3. Why is only part of the mass of the spring included in the effective mass of the system? 4. In the pendulum experiment, what is the advantage of timing 0 swings rather than timing one swing 0 times and taking an average? How would the possible measurement error compare between the two methods? 134

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