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AP Physics C: Mechanics Practice Quiz: Potential Energy

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 11 questions to check your progress.

Question 1 of 11

A particle moves in one dimension, and its potential energy as a function of position is U(x). The conservative force acting on the particle is given by F_x = -dU/dx. The force on the particle is directed in the positive x-direction when:

All Questions (11)

A particle moves in one dimension, and its potential energy as a function of position is U(x). The conservative force acting on the particle is given by F_x = -dU/dx. The force on the particle is directed in the positive x-direction when:

A) the potential energy U(x) is positive.

B) the slope of the U(x) vs. x graph is positive.

C) the slope of the U(x) vs. x graph is negative.

D) the potential energy U(x) is at a minimum.

Correct Answer: C

According to the relationship F_x = -dU/dx, for the force F_x to be positive, the derivative dU/dx (the slope of the potential energy graph) must be negative. Therefore, the force is in the positive x-direction where the potential energy is decreasing with position.

An ideal spring with spring constant k is stretched a distance Δx from its equilibrium position, storing an amount of potential energy U_s. If the spring is instead stretched a distance of 2Δx, what is the new elastic potential energy stored in the spring?

A) 1/2 U_s

B) U_s

C) 2 U_s

D) 4 U_s

Correct Answer: D

Elastic potential energy is given by the equation U_s = (1/2)k(Δx)². Since the energy is proportional to the square of the displacement, doubling the displacement (from Δx to 2Δx) results in a factor of 2² = 4 times the original energy.

The gravitational potential energy of a system of two masses, m₁ and m₂, separated by a distance r is given by U_g = -G(m₁m₂)/r. What is the physical significance of the negative sign in this equation?

A) It indicates that the gravitational force is always repulsive.

B) It signifies that the potential energy increases as the masses are brought closer together.

C) It establishes a convention where the potential energy is zero at infinite separation and the system is in a bound state.

D) It is a mathematical artifact with no physical meaning and can be ignored in calculations.

Correct Answer: C

The negative sign indicates that the gravitational force is attractive, creating a bound system. By convention, the potential energy is defined as zero when the masses are infinitely far apart (r → ∞). To separate the masses from a finite distance r to infinity, positive work must be done on the system, increasing its potential energy from a negative value to zero.

The change in potential energy of a system is related to the work done by a conservative force F via the equation ΔU = -∫[a, b] F⋅dr. This relationship implies that:

A) the potential energy of the system increases when the conservative force does positive work.

B) the potential energy of the system decreases when the conservative force does positive work.

C) the potential energy is always equal to the negative of the work done by the conservative force.

D) the work done by a conservative force is always positive.

Correct Answer: B

The equation can be written as ΔU = -W_c, where W_c is the work done by the conservative force. If the conservative force does positive work (W_c > 0), the change in potential energy (ΔU) must be negative, meaning the system's potential energy decreases.

The potential energy U(x) of a particle is shown in the graph. At which labeled point is the conservative force acting on the particle equal to zero?

A) Point A (positive slope)

B) Point B (negative slope)

C) Point C (local minimum)

D) Point D (different negative slope)

Correct Answer: C

The conservative force is the negative of the slope of the potential energy curve: F_x = -dU/dx. The force is zero where the slope (dU/dx) is zero. This occurs at local minima and maxima of the potential energy function. Point C is at a local minimum, where the tangent to the curve is horizontal and the slope is zero.

How does the dependence of potential energy on separation distance differ between an ideal spring and a system of two masses interacting gravitationally?

A) Spring potential energy is proportional to displacement (Δx), while gravitational potential energy is proportional to distance (r).

B) Spring potential energy is proportional to the square of displacement (Δx²), while gravitational potential energy is inversely proportional to distance (1/r).

C) Spring potential energy is inversely proportional to displacement (1/Δx), while gravitational potential energy is proportional to the square of distance (r²).

D) Both potential energies are proportional to the square of the separation distance.

Correct Answer: B

The elastic potential energy of a spring is U_s = (1/2)k(Δx)², which is proportional to (Δx)². The gravitational potential energy is U_g = -G(m₁m₂)/r, which is proportional to 1/r (or inversely proportional to r).

A satellite of mass m orbits a planet of mass M at a distance r from the planet's center. The gravitational potential energy of the satellite-planet system is U_g. If the satellite is moved to a new orbit with a radius of 3r, what is the new gravitational potential energy of the system in terms of U_g?

A) 1/9 U_g

B) 1/3 U_g

C) 3 U_g

D) 9 U_g

Correct Answer: B

Gravitational potential energy is given by U_g = -G(m₁m₂)/r. The energy is inversely proportional to the distance r. If the distance is tripled (from r to 3r), the new potential energy will be 1/3 of the original value: U_new = -G(m₁m₂)/(3r) = (1/3) * [-G(m₁m₂)/r] = (1/3)U_g.

The relationship F_x = -dU(x)/dx is a specific, one-dimensional case of the more general relationship ΔU = -∫ F⋅dr. The existence of a potential energy function U for a force F implies that the force must be:

A) a non-conservative force, such as friction.

B) a constant force, independent of position.

C) a conservative force, for which the work done is path-independent.

D) an applied force from an external agent.

Correct Answer: C

The concept of potential energy is only defined for conservative forces. Both equations provided relate a system's potential energy to a force, and therefore that force must be conservative. A key property of conservative forces is that the work they do in moving an object between two points is independent of the path taken.

Spring A has a spring constant k. Spring B has a spring constant of 2k. If both springs are compressed by the same distance Δx, how does the potential energy stored in Spring B, U_B, compare to the potential energy stored in Spring A, U_A?

A) U_B = 1/2 U_A

B) U_B = U_A

C) U_B = 2 U_A

D) U_B = 4 U_A

Correct Answer: C

The elastic potential energy is U_s = (1/2)k(Δx)². Since the displacement Δx is the same for both springs, the potential energy is directly proportional to the spring constant k. Because Spring B has twice the spring constant of Spring A, it will store twice the potential energy: U_B = (1/2)(2k)(Δx)² = 2 * [(1/2)k(Δx)²] = 2U_A.

The potential energy U of a system varies with position x as shown in the graph. The conservative force F is related to the potential energy by F_x = -dU/dx. At which of the labeled points is the magnitude of the force F the greatest?

A) Point A

B) Point B

C) Point C

D) Point D

Correct Answer: B

The magnitude of the force is equal to the magnitude of the slope of the U vs. x graph (|F_x| = |-dU/dx| = |dU/dx|). The force is greatest where the graph is steepest (i.e., the slope has the largest absolute value). Point B is located on the steepest part of the curve shown.

Consider a system of two spherical masses, m₁ and m₂. According to the equation U_g = -G(m₁m₂)/r, what happens to the gravitational potential energy of the system as the two masses are brought closer together (i.e., as r decreases)?

A) The potential energy becomes more positive and approaches infinity.

B) The potential energy becomes more negative (decreases).

C) The potential energy remains constant because mass is conserved.

D) The potential energy becomes less negative and approaches zero.

Correct Answer: B

The potential energy is given by U_g = -G(m₁m₂)/r. As the separation distance r decreases, the denominator gets smaller, which makes the magnitude of the fraction |G(m₁m₂)/r| larger. Since the value is negative, a larger magnitude means the potential energy becomes more negative. A more negative energy is a decrease in potential energy.