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

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

$x=A\cos \left(2\mathrm{\pi }ft\right)$

where $f$ is the frequency of oscillation.

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

Figure 1

ii) Derive an expression for the magnitude of the maximum acceleration of the block. Express your answer in terms of $A$, $k$, $M$ 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 $v_{max}$ of the block. Express your answer in terms of $A$, $k$, $M$ 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 $x=+L$ and $x=-L$.

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 $x=+L$ is 4 J.

Figure 1

A block of mass $m=0.30\mathrm{kg}$ is placed on a frictionless table and is attached to one end of a horizontal spring of spring constant $k$. 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 $t=0\mathrm{s}$. A motion detector is used to record the position of the block as it oscillates. Figure 1 shows the graph of the velocity $v$ of the block as a function of time $t$. The positive direction for all quantities is to the right.

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

Figure 1

A block of mass $m$ is attached to an ideal spring, whose other end is fixed to a wall. The block is displaced a distance $x_0$ 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 $v_{max}$ .

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 $t=1.5t_0$”.

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).

Figure 1

Two ideal springs, 1 and 2, of spring constant $k_1$ and $k_2$ respectively, are connected end to end. A block of mass $m$ is attached to the end of Spring 2 and the other end of Spring 1 is fixed to a wall. The block is displaced to the left of the spring's equilibrium position, $x=0$, and held stationary at position $x=-A$, as shown in Figure 1. The block is then released.

At time $t=t_0$, the block's position is $x=\frac{1}{2}A$and it is travelling to the right.

Figure 2

The energy bar chart in Figure 2 represents the spring potential energy $U_s$ of the block-spring system, and the kinetic energy $K$ of the block at time $t=t_0$, and position $x=\frac{1}{2}A$.

Figure 3

A student sketches a graph of the block's position as a function of time, as shown in Figure 3.

Indicate whether the student's graph in Figure 3 is consistent with the energy bar chart in Figure 2. Justify your answer using the position equation for simple harmonic motion.

A block of mass $m=0.30\mathrm{kg}$ is placed on a frictionless table and is attached to one end of a horizontal spring of spring constant $k$. 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 $t=0\mathrm{s}$. A motion detector is used to record the position of the block as it oscillates. The resulting graph of velocity $v$ versus time $t$ is shown below in Figure 1. The positive direction for all quantities is to the right.

Figure 1

The block and spring arrangement is now placed on a rough surface, as shown below. The block is displaced so that the spring is compressed a distance and released from rest.

Sketch a graph of velocity as a function of time in this case on the axes in Figure 2. Assume that there is a negligible change in the period and that the positive direction is still to the right. Specifying values on the velocity axis is not necessary.

Figure 2