Energy of Simple Harmonic Oscillators (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.

A student places a second identical block onto the first block when it is at maximum displacement. The new system continues oscillating with frequency $f_2$. Air resistance is negligible.

Indicate whether $f_2$ is greater than, less than, or equal to $f_1$ by writing one of the following:

• $f_2>f_1$

• $f_2=f_1$

• $f_2<f_1$

Justify your reasoning. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

Figure 1

A pendulum is displaced to angle $\theta$ before being released. Upon being released, the pendulum has angular acceleration ${\alpha }_0$. The string has length $R$ and the bob at the end has mass $m$.

In Experiment 1, the pendulum is displaced to $\theta$ before being released. In Experiment 2, the pendulum is displaced to $x\theta$, where $x>1$ and $x\theta <\frac{\mathrm{\pi }}{2}rad$.

Indicate whether the maximum speed $v_1$ of the pendulum in Experiment 1 is greater than, less than, or equal to the maximum speed $v_2$ of the pendulum in Experiment 2 by writing one of the following:

• $v_1>v_2$

• $v_1=v_2$

• $v_1<v_2$

Justify your reasoning. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

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 $v_{max}$ .

The energy bar charts in Figure 2 represent the spring potential energy $U_s$ of the block-spring system, and the kinetic energy $K$ of the block, as the block passes through positions $x=-x_0$, $x=0$ and $x=+x_0$ while the block oscillates. The bar chart at $x=-x_0$ is complete. Draw shaded rectangles to complete the energy bar charts in Figure 2 for positions $x=0$ and $x=+x_0$.

• Positive energy values are above the zero-energy line (“0”), and negative energy values are below the zero-energy line.

• Shaded regions should start at the dashed line representing zero energy.

• Represent any energy that is equal to zero with a distinct line on the zero-energy line.

• The relative height of each shaded region should reflect the magnitude of the respective energy consistent with the scale shown.

Figure 2

Figure 3 shows the position of the block as a function of time. Figure 4 shows the force exerted by the spring on the block as a function of time.

Figure 3

Figure 4

i) Using figures 3 and 4, determine an expression for the spring constant of the spring.

ii) Starting with the equation for the period of a mass on a spring, derive an expression for the mass of the block. Express your answer in terms of $t_0$, $x_0$, $F_{max}$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 1

A spring of unknown spring constant $k_0$ is attached to a ceiling. A lightweight hanger is attached to the lower end of the spring, and a motion detector is placed on the floor facing upward directly under the hanger, as shown in the figure above. The bottom of the hanger is 1.00 m above the motion detector.

An object of mass $m$ is then placed on the hanger and allowed to come to rest at the equilibrium position, such that the bottom of the hanger is distance $d$ below its initial position. The spring is then stretched downward from equilibrium and released at time $t$ = 0 s. The motion detector records the height of the bottom of the hanger as a function of time. The output from the motion detector is shown in Figure 2.

Figure 2

At time 0.75 s, the system has total kinetic energy $K_0$ and total potential energy $U_0$. At time 1.13 s, the system has kinetic energy $K$ and potential energy $U$.

i) Indicate whether $K_0$ is greater than, less than, or equal to $K$ by writing one of the following:

• $K_0>K$

• $K_0=K$

• $K_0<K$

Justify your claim using features from Figure 2. In your justification, include qualitative reasoning beyond mathematical derivations or expressions.

ii) The experiment is repeated. The mass is displaced from equilibrium by the same amount but a new spring is used with spring constant $4k_0$.

Predict how Figure 2 will change. Justify your claim.

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 $t=0$.

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

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 $t=t_0$. Draw shaded rectangles to complete the energy bar charts in Figure 2 for the block-spring system at time $t=t_0$.

• Positive energy values are above the zero-energy line (“0”), and negative energy values are below the zero-energy line.

• Shaded regions should start at the dashed line representing zero energy.

• Represent any energy that is equal to zero with a distinct line on the zero-energy line.

• The relative height of each shaded region should reflect the magnitude of the respective energy consistent with the scale shown.

Figure 2

Figure 1

Block P of mass $m$ is on a horizontal, frictionless surface and is attached to a spring with spring constant $k$. The block is oscillating with period $T_P$ and amplitude $A_P$ about the spring's equilibrium position $x_0$. A second block Q of mass $2m$ is then dropped from rest and lands on block P at the instant it passes through the equilibrium position, as shown above. Block Q immediately sticks to the top of block P, and the two-block system oscillates with period $T_{PQ}$ and amplitude $A_{PQ}$·

Indicate whether the amplitude $A_{PQ}$ is greater than, lesser than, or equal to $A_P$. Justify your claim using qualitative energy reasoning beyond referencing equations.