M07. Simple Harmonic Motion

 Introduction

In L123, you determined the spring constant of a spring by adding different weights to the platform to see how that changed the equilibrium position. Review the introduction to that exercise now. Then continue below.

Hooke's Law relates a change in the tension force ΔT exerted by the spring to the corresponding displacement Δx from equilibrium as follows: ΔT = -kΔx. Recall that k is defined to be positive, and the negative sign indicates that the change in tension force is opposite the displacement.

In L123, you were dealing with the spring in equilibrium. You changed the tension in the spring by changing the amount of weight placed on the platform. Since you took measurements with the spring in equilibrium, you were able to say that ΔT = ΔW. What will be new in the present exercise is that the spring and the mass attached to it will be in continual oscillation. The acceleration will generally not be 0. Nevertheless, Hooke's Law still applies. The use of Hooke's Law to describe the motion of a spring-object system is simply one application of this general statement:

F_res = -kx,

where the subscript res on the force indicates that the force is a restoring force. That is, the force tends to restore the object to its equilibrium position. Note that when using the equation in the form given above, F_res and x are vectors, and the position of x = 0 is the equilibrium position. 

 Investigation

Enter your answers to the following in WebAssign M07.

1. Begin by opening this animation. When you play the animation, you'll see a ball oscillating horizontally about the origin on a frictionless table. The origin is in the center, the direction of +x is to the right, and the grid spacing is 0.02 m. The oscillation is the result of a Hooke's Law force applied by a spring to the ball. The heights of the vertical bars shown below the table are proportional to the values of kinetic and elastic potential energy of the ball-spring system. The mass and spring constant are given as well as two other quantities--amplitude and phase--that you'll investigate.

Unless indicated otherwise, leave the mass at 0.5 kg and the spring constant at 10 N/m.. When reading positions of the block, use the geometric center.

You'll examine the kinematics of the situation in this first problem.

1. Without making any changes to the inputs, run the animation. Note the starting position and the direction of the velocity as the animation begins.

2. Change the amplitude to 0.05 m, reset, and play the animation. Note the starting position. Repeat with a value of -0.05 m for the amplitude.

3. You should have seen in step 2 that while you can enter a negative value for the amplitude, that makes no difference. This is because the applet always takes the absolute value of the number in the amplitude box, since amplitude is by definition a magnitude. In order to get the motion to start at a negative value of position, you have to change the phase. For what range of phases is the initial position of the block less than or equal to 0?  (Give phase as a number between 0 and 360 degrees.)

4. For what range of phases are both the initial position and initial velocity ≤ 0?

5. For what range of phases is the initial position ≥ 0 and the initial velocity ≤ 0?

6. Change the amplitude to 0.09 m and the phase to 0°.  Run the animation. What is the first time to the nearest 0.01 s at which the velocity has its largest positive value? After what fraction of a cycle from t = 0 (when x has its largest positive value) does this occur?

7. For the same situation as step 6, what is the first time to the nearest 0.01 s at which the acceleration has its largest positive value? After what fraction of a cycle from t = 0 does this occur?

8. Click Show Graph to display a position vs. time graph of the motion. Click xVel and Apply to display a graph of velocity vs. time. Note that both the graphs of position vs. time and velocity vs. time are sinusoidal in form with the same period. However, their peaks (highest positive value) occur at different times. What fraction of a cycle after a peak of the position graph occurs does the velocity peak? You should see that this result is consistent with what you found in part f.

9. Click xAccel and Apply to display a graph of acceleration vs. time. Change the vertical scale attributes to PVA max = 2 and PVA min = -2 and click Apply in order to display the entire curve. Once again, the shape is sinusoidal, the period is the same as for the position vs. time graph, and the peaks occur at different times. What fraction of a cycle after a peak of the position graph occurs does the acceleration peak? You should see that this result is consistent with what you found in part g.

2. Continue with the dynamics.

1. Which way does the spring (tension) force point when the displacement is positive? negative?

2. Consider the situation where the block has its maximum positive acceleration. Draw a force diagram for the block in this position.

3. Write the horizontal net force equation corresponding to the force diagram in terms of k and x.

4. Explain why your equation in the previous part gives the correct sign for F_net. (Consider what the sign of the position is.)

5. Use your net force equation together with Hooke's Law to obtain a formula for the maximum value of the acceleration of the block in terms of the mass, m, the amplitude, A, and the spring constant, k. You can verify your formula by checking that it gives the same numerical result as the applet.

3. Now you'll examine the energy of the system of block and spring.

1. What is the value of the maximum kinetic energy of the block to the nearest 0.0001 J, and what is the block's position when this occurs?

2. What is the value of the total mechanical energy of the system at any time to the nearest 0.0001 J?

3. Do a conservation of energy problem to obtain a formula for the maximum value of the velocity of the block in terms of the mass, m, the amplitude, A, and the spring constant, k.

4. Verify that your formula gives the same numerical result as the applet.

4. Lastly you'll examine the angular velocity and the period.

   1. Calculate the angular velocity of the block to the nearest 0.01 s^-1.

   2. Double the mass of the block. What effect does that have on the period?

   3. Keeping the mass at 1.0 kg, how must you change the spring constant to restore the period to the value it had before the mass was doubled?

   4. How is the period of the motion related to the ratio m/k?

   5. Is the relationship that you gave in the last item a linear relationship? Use an argument based on units analysis to explain your answer.

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