AP Physics C Mechanics Quiz: Defining Simple Harmonic Motion Shm
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Defining Simple Harmonic Motion ShmQuestion 1 of 20

A mass attached to a spring oscillates horizontally on a frictionless surface. At which position(s) is the acceleration of the mass equal to zero?

Only at the points of maximum displacement from equilibrium where the mass momentarily stops before changing direction
Only at the equilibrium position where the net force on the mass is zero and no restoring force acts
At both the equilibrium position and at the points of maximum displacement where the velocity is instantaneously zero
At no point during the oscillation since the mass is always subject to the gravitational force acting downward
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AP Physics C Mechanics Quiz

AP Physics C Mechanics Quiz: Defining Simple Harmonic Motion Shm

Practice Defining Simple Harmonic Motion Shm in AP Physics C Mechanics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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Question 1

A mass attached to a spring oscillates horizontally on a frictionless surface. At which position(s) is the acceleration of the mass equal to zero?

  1. Only at the points of maximum displacement from equilibrium where the mass momentarily stops before changing direction
  2. Only at the equilibrium position where the net force on the mass is zero and no restoring force acts (correct answer)
  3. At both the equilibrium position and at the points of maximum displacement where the velocity is instantaneously zero
  4. At no point during the oscillation since the mass is always subject to the gravitational force acting downward
Explanation: In SHM, acceleration is proportional to displacement: a=−kmxa = -\frac{k}{m}x. Acceleration equals zero only when displacement x=0x = 0, which occurs at the equilibrium position where the net restoring force is zero. Choice A is incorrect because acceleration is maximum at points of maximum displacement. Choice C incorrectly includes maximum displacement points. Choice D is incorrect because we consider only forces affecting the oscillatory motion (gravity is balanced by normal force).

Question 2

Which of the following mathematical relationships correctly describes the acceleration of an object undergoing simple harmonic motion?

  1. The acceleration is directly proportional to the velocity and directed in the same direction as the velocity vector
  2. The acceleration is directly proportional to the displacement and directed in the same direction as the displacement vector
  3. The acceleration is directly proportional to the displacement and directed opposite to the displacement vector from equilibrium (correct answer)
  4. The acceleration is inversely proportional to the displacement and directed opposite to the displacement vector from equilibrium
Explanation: For SHM, the defining relationship is a=−ω2xa = -\omega^2 x, where acceleration is directly proportional to displacement but directed opposite to it (hence the negative sign). This creates the restoring force characteristic of SHM. Choice A incorrectly relates acceleration to velocity. Choice B has the wrong direction (same as displacement). Choice D incorrectly describes an inverse relationship.

Question 3

A physical system undergoes oscillatory motion described by x(t)=Ae−γtcos⁡(ωt)x(t) = A e^{-\gamma t} \cos(\omega t). Is this simple harmonic motion?

  1. Yes, because the motion contains a cosine function which is the mathematical signature of simple harmonic motion in all cases
  2. Yes, because the angular frequency ω\omega is constant, indicating that the restoring force follows Hooke's law exactly
  3. No, because the exponentially decreasing amplitude indicates the presence of damping forces that violate the SHM force condition (correct answer)
  4. No, because the phase of the oscillation changes with time due to the exponential factor affecting the timing
Explanation: The exponential decay indicates damping, meaning additional forces beyond the ideal restoring force act on the system. True SHM requires only the restoring force F=−kxF = -kx; damping forces make this motion approximately harmonic but not truly simple harmonic. Choice A incorrectly assumes any cosine represents SHM. Choice B ignores the amplitude decay. Choice D incorrectly interprets the exponential factor's effect on phase.

Question 4

Which of the following systems would NOT exhibit simple harmonic motion, even for small displacements?

  1. A mass attached to a spring oscillating vertically under the influence of gravity with small amplitude oscillations
  2. A ball rolling back and forth in a parabolic bowl under the influence of gravitational force near the bottom
  3. A charged particle oscillating between two oppositely charged parallel plates with small displacement amplitudes from center (correct answer)
  4. A mass sliding on a frictionless surface attached to a rubber band that obeys Hooke's law perfectly
Explanation: Between parallel plates, the electric field is uniform, creating a constant force on the charged particle regardless of position. This violates the SHM requirement that force be proportional to displacement. Choice A exhibits SHM about a shifted equilibrium position. Choice B creates a restoring force proportional to displacement (F∝xF \propto x in a parabolic potential). Choice D follows Hooke's law directly, producing perfect SHM.

Question 5

For a mass-spring system undergoing simple harmonic motion, which statement correctly describes the equilibrium position?

  1. The equilibrium position is where the spring force exactly balances all other forces, resulting in zero net force on the mass (correct answer)
  2. The equilibrium position is the point of maximum potential energy where the mass has zero kinetic energy and zero velocity
  3. The equilibrium position is the center point between the two extreme positions where the mass has maximum acceleration
  4. The equilibrium position is where the mass would naturally come to rest if all external driving forces were removed completely
Explanation: The equilibrium position is defined as the location where the net force on the system is zero, meaning all forces (including the spring force) are balanced. At this position, there is no acceleration. Choice B incorrectly describes the turning points where potential energy is maximum. Choice C is wrong because acceleration is zero (not maximum) at equilibrium. Choice D describes where the mass would stop due to damping, not the equilibrium position.

Question 6

Simple harmonic motion (SHM) occurs when a system experiences which type of restoring force?

  1. A force that is directly proportional to the displacement from equilibrium and directed toward the equilibrium position (correct answer)
  2. A force that is inversely proportional to the displacement from equilibrium and directed away from the equilibrium position
  3. A force that is directly proportional to the square of the displacement and directed toward the equilibrium position
  4. A force that is constant in magnitude and directed toward the equilibrium position regardless of displacement magnitude
Explanation: Simple harmonic motion is defined by a restoring force that follows Hooke's law: F=−kxF = -kx, where the force is directly proportional to displacement and directed opposite to the displacement (toward equilibrium). Choice B describes an inverse relationship and wrong direction. Choice C involves a quadratic relationship which would not produce SHM. Choice D describes a constant force, which would produce constant acceleration, not SHM.

Question 7

A particle undergoes periodic motion. Which condition is necessary and sufficient to classify this motion as simple harmonic motion?

  1. The motion must be sinusoidal in nature with a constant frequency and amplitude that remains unchanged over time
  2. The restoring force must be proportional to displacement with the proportionality constant being positive and velocity-independent
  3. The net force must be proportional to displacement from equilibrium and directed toward the equilibrium position at all times (correct answer)
  4. The motion must exhibit equal time intervals for each complete cycle and maintain constant total mechanical energy
Explanation: The defining characteristic of SHM is F=−kxF = -kx where the net restoring force is proportional to displacement and directed toward equilibrium. This single condition is both necessary and sufficient. Choice A describes properties that result from SHM but aren't the fundamental definition. Choice B incorrectly states the proportionality constant should be positive (it should be negative for restoring force). Choice D describes consequences of SHM rather than the defining condition.

Question 8

A pendulum swings back and forth with small angular displacements. What makes this motion an example of simple harmonic motion?

  1. The gravitational force component tangent to the arc is approximately proportional to the angular displacement for small angles (correct answer)
  2. The tension in the string provides a centripetal force that varies sinusoidally with time throughout the oscillation cycle
  3. The total mechanical energy remains constant and is equally divided between kinetic and potential energy at all times
  4. The period of oscillation is independent of the amplitude and depends only on the length and gravitational acceleration
Explanation: For small angles, sin⁡θ≈θ\sin\theta \approx \theta, so the restoring torque τ=−mglsin⁡θ≈−mglθ\tau = -mgl\sin\theta \approx -mgl\theta is proportional to angular displacement, satisfying the SHM condition. Choice B incorrectly focuses on tension rather than the restoring force. Choice C describes energy conservation but not the defining characteristic of SHM. Choice D describes a property of SHM but not what makes it SHM.

Question 9

Which mathematical condition must be satisfied for an oscillating system to be classified as undergoing simple harmonic motion?

  1. The second derivative of position with respect to time must be proportional to the negative of the position coordinate (correct answer)
  2. The first derivative of position with respect to time must be proportional to the negative of the position coordinate
  3. The second derivative of position must be proportional to the positive value of the position coordinate at all times
  4. The ratio of kinetic energy to potential energy must remain constant throughout the oscillation cycle for all amplitudes
Explanation: The differential equation for SHM is d2xdt2=−ω2x\frac{d^2x}{dt^2} = -\omega^2 x, where the second derivative of position (acceleration) is proportional to the negative of position. This is the fundamental mathematical definition of SHM. Choice B describes the first derivative (velocity), which has a different relationship. Choice C has the wrong sign. Choice D describes an energy relationship that doesn't define SHM.

Question 10

A particle oscillates such that its position varies as x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi). What can be concluded about the motion?

  1. The motion is definitely simple harmonic since any cosine function represents simple harmonic motion regardless of parameters
  2. The motion is simple harmonic because the acceleration is proportional to negative displacement when differentiated twice (correct answer)
  3. The motion cannot be simple harmonic because the phase constant ϕ\phi indicates external driving forces are present
  4. The motion is simple harmonic only if both the angular frequency ω\omega and amplitude AA remain constant
Explanation: Taking the second derivative: d2xdt2=−ω2Acos⁡(ωt+ϕ)=−ω2x\frac{d^2x}{dt^2} = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x, which satisfies the SHM condition a=−ω2xa = -\omega^2 x. Choice A is incorrect because not all cosine functions represent physical SHM. Choice C incorrectly interprets the phase constant. Choice D states conditions already implicit in the given function.

Question 11

Which characteristic distinguishes simple harmonic motion from other types of periodic motion?

  1. Simple harmonic motion uses sinusoidal functions while other periodic motions use different mathematical functions
  2. Simple harmonic motion requires linear restoring force while other motions may have nonlinear restoring forces (correct answer)
  3. Simple harmonic motion maintains constant amplitude while other periodic motions experience amplitude decay
  4. Simple harmonic motion has amplitude-independent period while other motions show amplitude-dependent periods
Explanation: The defining characteristic of SHM is the linear restoring force (F=−kxF = -kx). Other periodic motions may have nonlinear restoring forces. Choice A confuses mathematical description with physical definition. Choice C incorrectly assumes SHM has no damping. Choice D describes a consequence rather than the fundamental distinguishing characteristic.

Question 12

Consider two oscillating systems: System X has restoring force FX=−kxF_X = -kx and System Y has restoring force FY=−kx3F_Y = -kx^3 where k>0k > 0 in both cases. A physics student argues that both systems exhibit simple harmonic motion because both have restoring forces that oppose displacement. Which analysis correctly evaluates this argument?

  1. The argument is correct because both restoring forces are directed toward equilibrium and both systems will oscillate periodically about x=0x = 0
  2. The argument is incorrect because only System X satisfies a∝−xa \propto -x; System Y has a∝−x3a \propto -x^3, violating the SHM definition (correct answer)
  3. The argument is correct because both systems conserve energy and exhibit periodic motion with well-defined equilibrium positions at the origin
  4. The argument is incorrect because System Y's period depends on amplitude, while SHM requires the period to be independent of amplitude
Explanation: Simple harmonic motion specifically requires acceleration proportional to displacement: a=−ω2xa = -\omega^2 x. System X: F=ma=−kxF = ma = -kx, so a=−kmxa = -\frac{k}{m}x (SHM). System Y: F=ma=−kx3F = ma = -kx^3, so a=−kmx3a = -\frac{k}{m}x^3 (not SHM). The defining characteristic of SHM is linear restoring force, not just any restoring force. Choice A confuses restoring force with SHM definition. Choice C lists properties that both motions share but misses the key distinction. Choice D mentions a consequence rather than the fundamental definition.

Question 13

A mass on a spring oscillates vertically. When the mass is at position y=+0.10y = +0.10 m above equilibrium, its acceleration is a=−4.0a = -4.0 m/s² (downward). When the mass is at position y=−0.05y = -0.05 m below equilibrium, its acceleration is a=+2.0a = +2.0 m/s² (upward). What can be concluded about this motion?

  1. This is simple harmonic motion with spring constant k=40k = 40 N/m, assuming the mass equals 1.0 kg based on the given data
  2. This is not simple harmonic motion because the ratio of acceleration to displacement differs between the two positions given
  3. This is simple harmonic motion with angular frequency ω=210\omega = 2\sqrt{10} rad/s because acceleration is proportional to displacement from equilibrium (correct answer)
  4. This is not simple harmonic motion because the acceleration values suggest the equilibrium position is not where y=0y = 0
Explanation: When analyzing oscillatory motion, you need to determine whether the acceleration is proportional to displacement from equilibrium. For simple harmonic motion, the relationship is a=−ω2ya = -\omega^2 y, where the acceleration is always directed toward equilibrium. Let's check if this relationship holds by calculating the ratio a/ya/y at both positions. At y=+0.10y = +0.10 m, a=−4.0a = -4.0 m/s², so a/y=−4.0/0.10=−40a/y = -4.0/0.10 = -40 s⁻². At y=−0.05y = -0.05 m, a=+2.0a = +2.0 m/s², so a/y=+2.0/(−0.05)=−40a/y = +2.0/(-0.05) = -40 s⁻². Since this ratio is constant, we have −ω2=−40-\omega^2 = -40, giving us ω=40=210\omega = \sqrt{40} = 2\sqrt{10} rad/s. Choice C correctly identifies this as simple harmonic motion with the right angular frequency. The acceleration is indeed proportional to displacement, confirming SHM. Choice A makes an unjustified assumption about the mass being 1.0 kg, which isn't given in the problem. Even if true, the spring constant calculation would require knowing the mass explicitly. Choice B incorrectly claims the acceleration-to-displacement ratio differs between positions. As shown above, this ratio is actually constant at both locations. Choice D misinterprets the data by suggesting the equilibrium position isn't at y=0y = 0. The fact that acceleration points toward y=0y = 0 in both cases (downward when above, upward when below) confirms that y=0y = 0 is indeed equilibrium. Study tip: Always check if a/ya/y is constant when determining whether motion is simple harmonic. If the ratio is constant and negative, you have SHM with ω=∣a/y∣\omega = \sqrt{|a/y|}.

Question 14

A particle moves along a straight line with position given by x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi). The particle's acceleration as a function of position can be written as a=−kxa = -kx where kk is a positive constant. Which statement correctly identifies the relationship between kk and the other parameters, and explains why this motion constitutes simple harmonic motion?

  1. k=ω2k = \omega^2; this is SHM because the acceleration is proportional to displacement with a negative constant of proportionality (correct answer)
  2. k=ωk = \omega; this is SHM because the restoring force varies linearly with displacement and the motion is periodic
  3. k=Aω2k = A\omega^2; this is SHM because the maximum acceleration occurs at maximum displacement from equilibrium
  4. k=ω2/Ak = \omega^2/A; this is SHM because the period is independent of amplitude and depends only on system parameters
Explanation: Taking the second derivative of x(t)=Acos⁡(ωt+ϕ)x(t) = A\cos(\omega t + \phi) gives a=−Aω2cos⁡(ωt+ϕ)=−ω2xa = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x. Comparing with a=−kxa = -kx, we get k=ω2k = \omega^2. Simple harmonic motion is defined as motion where acceleration is proportional to displacement with a negative constant of proportionality. Choice B is wrong because k≠ωk \neq \omega. Choice C incorrectly includes amplitude in the constant. Choice D has incorrect units and reasoning.

Question 15

A particle's motion is described by the differential equation d2xdt2+9x=12cos⁡(3t)\frac{d^2x}{dt^2} + 9x = 12\cos(3t). To determine if any component of the motion exhibits simple harmonic motion, which analysis is most appropriate?

  1. The homogeneous equation d2xdt2+9x=0\frac{d^2x}{dt^2} + 9x = 0 represents SHM, while the driving term prevents the total motion from being simple harmonic (correct answer)
  2. The complete motion exhibits SHM because the acceleration depends linearly on position with additional periodic forcing at the natural frequency
  3. No component exhibits SHM because the presence of the driving term 12cos⁡(3t)12\cos(3t) means acceleration is not proportional to displacement alone
  4. The motion exhibits SHM because the driving frequency matches the natural frequency ω0=3\omega_0 = 3 rad/s, creating resonant oscillation
Explanation: The homogeneous equation d2xdt2+9x=0\frac{d^2x}{dt^2} + 9x = 0 has solutions of the form xhom=Acos⁡(3t+ϕ)x_{hom} = A\cos(3t + \phi), which represents SHM with ω=3\omega = 3 rad/s. However, the complete solution includes both homogeneous and particular solutions. The driving term creates forced oscillation, so the total motion is not simple harmonic motion (acceleration is not simply proportional to displacement). Choice B incorrectly calls forced motion SHM. Choice C misses that the homogeneous part is SHM. Choice D confuses resonance with SHM definition.

Question 16

Two objects undergo motion described by the following equations: Object 1: x1(t)=3cos⁡(4t)x_1(t) = 3\cos(4t) and Object 2: x2(t)=2sin⁡(4t+π/3)+5x_2(t) = 2\sin(4t + \pi/3) + 5. Which statement correctly characterizes these motions?

  1. Both objects exhibit simple harmonic motion with the same frequency but different equilibrium positions and amplitudes
  2. Only Object 1 exhibits simple harmonic motion because Object 2 has a non-zero average position over one complete cycle
  3. Only Object 1 exhibits simple harmonic motion because Object 2's equation contains both sine and cosine components when expanded
  4. Both objects exhibit simple harmonic motion with identical periods, but Object 2 oscillates about a displaced equilibrium position (correct answer)
Explanation: Simple harmonic motion requires a∝−xa \propto -x relative to equilibrium position. For Object 1: equilibrium at x=0x = 0, so a1=−16⋅3cos⁡(4t)=−16x1a_1 = -16 \cdot 3\cos(4t) = -16x_1. For Object 2: equilibrium at x=5x = 5, displacement from equilibrium is (x2−5)=2sin⁡(4t+π/3)(x_2 - 5) = 2\sin(4t + \pi/3), so a2=−16⋅2sin⁡(4t+π/3)=−16(x2−5)a_2 = -16 \cdot 2\sin(4t + \pi/3) = -16(x_2 - 5). Both exhibit SHM with period T=π/2T = \pi/2. Choice A is wrong about frequency vs period terminology. Choice B incorrectly suggests non-zero average position disqualifies SHM. Choice C misunderstands the mathematical form.

Question 17

Two identical masses are attached to springs with different spring constants. Both systems are displaced by the same amount and released. Which statement about their motions is correct?

  1. Both exhibit simple harmonic motion with identical periods since the masses and displacements are the same
  2. Both exhibit simple harmonic motion but with different frequencies determined by their spring constants (correct answer)
  3. Only the system with larger spring constant exhibits true simple harmonic motion due to stronger restoring force
  4. Neither system exhibits simple harmonic motion because different spring constants create different force laws
Explanation: Both systems satisfy F=−kxF = -kx and exhibit SHM, but with different frequencies ω=k/m\omega = \sqrt{k/m}. The spring constant affects frequency but not the harmonic nature. Choice A ignores the effect of different spring constants. Choice C wrongly suggests only stronger springs produce SHM. Choice D incorrectly claims different spring constants prevent SHM.

Question 18

What is the significance of the negative sign in the simple harmonic motion force equation F=−kxF = -kx?

  1. The negative sign indicates that the force magnitude decreases as displacement increases, creating a stabilizing feedback mechanism
  2. The negative sign shows that the force direction is opposite to the displacement direction, always pointing toward equilibrium position (correct answer)
  3. The negative sign represents the mathematical convention for restoring forces and has no physical significance in the motion
  4. The negative sign indicates that the system loses energy over time due to the work done against internal friction
Explanation: The negative sign is crucial because it ensures the force is always directed opposite to displacement, creating a restoring force that pulls/pushes the system back toward equilibrium. This direction relationship is essential for oscillatory motion. Choice A incorrectly describes the magnitude relationship. Choice C dismisses the physical importance of the sign. Choice D confuses the restoring force sign with energy dissipation.

Question 19

For small oscillations, why can a simple pendulum be considered to undergo simple harmonic motion?

  1. The tension in the string provides a restoring force that is exactly proportional to the angular displacement for all angles
  2. The small angle approximation makes the restoring torque proportional to angular displacement, satisfying the SHM force condition (correct answer)
  3. The gravitational potential energy varies quadratically with angular displacement, creating the necessary restoring force relationship
  4. The centripetal acceleration required for circular motion automatically generates simple harmonic motion in the radial direction
Explanation: For small angles, sin⁡θ≈θ\sin\theta \approx \theta, making the restoring torque τ=−mglsin⁡θ≈−mglθ\tau = -mgl\sin\theta \approx -mgl\theta, which is proportional to angular displacement. This satisfies the SHM condition in rotational form. Choice A incorrectly attributes the restoring effect to tension alone. Choice C mentions energy but doesn't explain the force relationship correctly. Choice D confuses circular motion with oscillatory motion.

Question 20

In analyzing whether a system exhibits simple harmonic motion, which approach is most fundamental?

  1. Examine whether the motion can be described by sine or cosine functions with constant amplitude and frequency parameters
  2. Determine if the system's total mechanical energy remains constant throughout the oscillation cycle under ideal conditions
  3. Check whether the net restoring force acting on the system is directly proportional to its displacement from equilibrium (correct answer)
  4. Verify that the system exhibits periodic motion with a well-defined period that remains constant for different initial conditions
Explanation: The fundamental definition of SHM is based on the force law: F=−kxF = -kx. This is the starting point from which all other properties (sinusoidal motion, constant energy, etc.) can be derived. Choice A examines consequences rather than causes. Choice B describes energy conservation, which applies to many systems beyond SHM. Choice D identifies periodic motion but doesn't distinguish SHM from other periodic motions.