## Module 13 Class Activities

## Simple Harmonic Motion

### I. Qualitative analysis of motion

A block is connected to a spring, one end of which is attached to a wall. (Neglect the mass of the spring, and assume the surface is frictionless.)

The block is moved 0.5 m to the right of equilibrium and released from rest at instant 1. The strobe diagram at right shows the subsequent motion of the block (i.e., the block is shown at equal time intervals).

A. For each instant, draw a vector that represents the instantaneous velocity of the block at that instant. Explain how you decided to draw your vectors.

B. Use your velocity vectors from part A to determine graphically the direction of the average acceleration from instant 2 to instant 3. Discuss your reasoning with your partners.

Modify your approach as necessary to determine:

- the direction of the average acceleration from instant 4 to instant 5, and from instant 5 to instant 6
- the direction of the instantaneous acceleration at instants 2, 4, and 5

Are your results above consistent with your knowledge of forces and Newton’s laws? If so, explain why. If not, resolve the inconsistencies.

*Consult an LA before proceeding.*

### II. Differential equation of motion

Consider again the situation depicted in section I, in which a block of mass *m* attached to an (ideal) spring of force constant *k* undergoes simple harmonic motion on a level, frictionless surface.

A. Using Newton’s second law in one dimension, , write down the differential equation that governs the motion of the block.

The net force exerted on the block may be called a restoring force. Justify this term on the basis of your differential equation above.

B. Show by direct substitution that the following functions are solutions to the differential equation you wrote down in part A. As part of your answer, specify the conditions (if any) that must be met by the parameters A, ω, and φ_{o} in order for each function to be a valid solution.

*Consult with an LA before proceeding.*

### III. Period and Frequency of Simple Harmonic Motion

A spring is used to connect a block to a wall. (Neglect the mass of the spring, and assume the surface is frictionless.)

A student moves the block 0.5 m to the right of its equilibrium position and at a certain instant (instant 1) releases it from rest. The subsequent motion of the block is shown. (The diagrams show the position of the block at time intervals 0.1 s apart.)

A. What is the period of motion of the block? Explain how you can tell.

B. Suppose that the student repeated the experiment shown here, except with one change to the setup.

For each change listed below, how (if at all) would that change affect the period of motion? Be as specific as possible in your answers. Explain your reasoning.

- The block is released 0.3 m to the right of equilibrium.
- The spring is replaced with a stiffer spring.
- The block is replaced with another block with four times the mass as the original one.

*Consult an LA before proceeding.*

### IV. Putting it all together

A 500-g block is placed on a level, frictionless surface and attached to an ideal spring. At t = 0 the block moves through the equilibrium position with speed vo in the –x direction, as shown below. At t = π sec, the block reaches its maximum displacement of 40 cm to the left of equilibrium.

A. Determine the value of each of the following quantities:

- period
- spring constant

B. Using x(t) = A cos(ωt + φ_{o}) as the solution to the differential equation of motion:

- Determine the form of the function v(t) that represents the velocity of the block.
- Evaluate all constant parameters (A, ω, and φ
_{o}) so as to completely describe both the position and velocity of the block as functions of time.

C. Consider the following statement about the situation described above.

“It takes the first π seconds for the block to travel 40 cm, so the initial speed vo can be found by dividing 40 cm by π seconds.”

Do you agree or disagree with this statement? If so, explain why you agree. If not, explain why you disagree and calculate the initial speed v_{o} of the block.

*Consult an LA before proceeding.*