AP®PhysicsCollege BoardPhysics 1: Algebra-BasedExam QuestionsUnit 7: OscillationsEnergy of Simple Harmonic Oscillators

Energy of Simple Harmonic Oscillators (College Board AP® Physics 1: Algebra-Based): Exam Questions

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Energy of Simple Harmonic Oscillators

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1a
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3 marks

State the three main types of energy in a vertically oscillating ideal spring-block-Earth system.

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1b
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2 marks

When the vertical ideal spring-block-Earth system is at equilibrium, identify whether the amount of elastic potential energy in the system is either maximum, zero, or sub-maximal but non-zero. Justify your answer.

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2a
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1 mark

Describe how the total energy in a system undergoing simple harmonic motion changes over time.

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2b
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1 mark

Describe how the amplitude of a system in simple harmonic motion changes over time when nonconservative resistive forces act on the system.

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2c
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1 mark

Describe how the maximum velocity of a system in simple harmonic motion changes over time when nonconservative resistive forces act on the system.

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2d
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1 mark

A mass-spring system is oscillating horizontally. At maximum extension, the spring has elastic potential energy U subscript m a x end subscript. At equilibrium, the spring has kinetic energy K subscript m a x end subscript.

Write an expression for the total energy of the system.

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3a
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1 mark

A pendulum is displaced by a small angle and released at time t space equals space 0. It then oscillates with a frequency of 2.0 Hz.

Calculate the period of this oscillation.

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3b
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1 mark

At t space equals space 1.0 space straight s, the pendulum-Earth system has a potential energy of U space equals space 0.25 space straight J.

Determine the system's kinetic energy at t space equals space 1.25 space straight s.

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3c
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1 mark

At an unknown time, the Earth-pendulum system has a kinetic energy of K space equals space 0.10 space straight J.

Determine the system's potential energy at this time.

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1
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3 marks

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_{m a x}. The motion can be described by the equation:

x space equals space A cos open parentheses 2 straight pi f t close parentheses

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.

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2
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3 marks
A pendulum of length R with a mass m at an angle θ to the vertical

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 less than straight pi over 2 r a d.

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.

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1a
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3 marks
Diagram showing a spring-mass system with a block labelled "m" displaced to the left from its original position by a hand, past point -x₀.

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 subscript 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 subscript m a x end subscript .

The energy bar charts in Figure 2 represent the spring potential energy U subscript s of the block-spring system, and the kinetic energy K of the block, as the block passes through positions x equals negative x subscript 0, x equals 0 and x equals plus x subscript 0 while the block oscillates. The bar chart at x equals negative x subscript 0 is complete. Draw shaded rectangles to complete the energy bar charts in Figure 2 for positions x equals 0 and x equals plus x subscript 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.

Three energy bar charts, labelled x = -x_0, x = 0 and x = +x_0 respectively. Each has a central x axis labelled "0". The y axis extends 4 dashed lines above this and 4 dashed lines below. Each bar chart has two spaces for a bar labelled U_s and K. The first bar chart, x = -x_0, has the U_s bar already filled. This is a height of 3 dashed lines above the "0" line.

Figure 2

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1b
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4 marks

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.

Graph of a sinusoidal wave showing position versus time, with peaks at +x₀ and troughs at -x₀. Time is marked at intervals t₀ to 4t₀.

Figure 3

Positive cosine displacement-time graph, starting at +F_max at time 0. At t_0, force is -F_max, at 2t_0 force is +F_max and at 3t_0 force is -F-max again.

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 subscript 0, x subscript 0, F subscript m a x end subscript, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

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2
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4 marks
Diagram showing a spring with a mass hanging from it, positioned 1.00 metre above a motion detector, indicating measurement setup.

Figure 1

A spring of unknown spring constant k subscript 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.

A graph of height (cm) as a function of time (s). A negative cosine graph oscillates between 55 cm and 65 cm around an equilibrium position at a height of 60 cm. The period of the graph is 1.25 s. At 0.75 s, height is 64 cm. At 1.13 s, height is 56 cm.

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 subscript 0 is greater than, less than, or equal to K by writing one of the following:

  • K subscript 0 space greater than space K

  • K subscript 0 space equals space K

  • K subscript 0 space less than space 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 4 k_{0}.

Predict how Figure 2 will change. Justify your claim.

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3
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2 marks
Spring 1 has constant k_1. It is in series with spring 2, with constant k_2. This is connected to a block with mass m. The block is to the left of equilibrium at x = -A, such that the springs are compressed.

Figure 1

Two ideal springs, 1 and 2, of spring constant k subscript 1 and k subscript 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 space equals space 0, and held stationary at position x space equals space minus A, as shown in Figure 1. The block is then released at time t space equals space 0.

At time t space equals space t subscript 0, the block's position is x space equals space 1 half A and it is travelling to the right.

The energy bar chart in Figure 2 represents the spring potential energy U subscript s of the block-spring system and the kinetic energy K of the block at time t space equals space t subscript 0. Draw shaded rectangles to complete the energy bar charts in Figure 2 for the block-spring system at time t space equals space t subscript 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.

Graph with a vertical arrow marked "E tot" and a horizontal line intersected by a vertical line at "0". Labels include "U s" and "K". Dotted horizontal lines.

Figure 2

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4
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4 marks
A spring is attached to block P of mass m at one end and to a wall at the other end. Block P is at position x_0 and is moving left. Bock Q of mass 2m is being dropped onto block P.

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 subscript P and amplitude A subscript P about the spring's equilibrium position x subscript 0. A second block Q of mass 2 m 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 subscript P Q end subscript and amplitude A subscript P Q end subscript·

Indicate whether the amplitude A subscript P Q end subscript is greater than, lesser than, or equal to A subscript P. Justify your claim using qualitative energy reasoning beyond referencing equations.

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