Representing and Analyzing SHM (College Board AP® Physics 1: Algebra-Based): Exam Questions

A small block of mass is attached to an ideal horizontal spring of spring constant and is set into simple harmonic motion. The system oscillates on a frictionless surface. The block is displaced by a distance and released from rest. While the block is oscillating, it has a maximum speed . The motion can be described by the equation:

where is the frequency of oscillation.

Figure 1

i) On the axes provided in Figure 1, sketch graphs of position as a function of time and velocity as a function of time for two complete cycles of motion.

ii) Derive an expression for the magnitude of the maximum acceleration of the block. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii) Derive an expression for the magnitude of the maximum velocity of the block. Express your answer in terms of , , and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 1

A cart on a horizontal surface is attached to a spring. The other end of the spring is attached to a wall. The cart is initially held at rest, as shown in Figure 1. When the cart is released, the system consisting of the cart and spring oscillates between the positions and .

Figure 2

On the axes in Figure 2, draw a graph of the kinetic energy of the cart-spring system as a function of the system's potential energy.

• Frictional forces are negligible.

• The initial potential energy of the system at is 4 J.

Figure 1

A block of mass is placed on a frictionless table and is attached to one end of a horizontal spring of spring constant . The other end of the spring is attached to a fixed wall. The block is set into oscillatory motion by stretching the spring and releasing the block from rest at time . A motion detector is used to record the position of the block as it oscillates. Figure 1 shows the graph of the velocity of the block as a function of time . The positive direction for all quantities is to the right.

Using Figure 1, determine an expression for the velocity of the block in terms of and numerical constants as appropriate.

Figure 1

A block of mass is attached to an ideal spring, whose other end is fixed to a wall. The block is displaced a distance to the left of the spring’s equilibrium position, as shown in Figure 1. The block is then released from rest and oscillates with negligible friction along the horizontal surface. While the block is oscillating, it has a maximum speed .

Figure 2 shows the position of the block as a function of time. Figure 3 shows the force exerted by the spring on the block as a function of time.

Figure 2

Figure 3

On the axes provided in Figure 4 sketch a graph of the velocity of the block as a function of time.

Figure 4

Figure 5

A student sketches the free-body diagram in Figure 5, and makes the following claim:

“The free-body diagram shows the forces exerted on the block at time ”.

Justify why the student’s sketch (Figure 5) and claim are or are not consistent with the graph of velocity as a function of time you sketched in part a).