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

A block on a frictionless horizontal track is attached to a spring. When displaced a distance xx from equilibrium, the spring exerts a restoring force Fx=−kxF_x=-kx toward equilibrium. The block is released from rest. Which statement best determines whether the resulting motion is simple harmonic?

Yes; the restoring force is proportional to displacement and opposite in direction.
Yes; any motion that repeats is simple harmonic.
No; the restoring force must be proportional to velocity for SHM.
No; equilibrium occurs at maximum displacement, not at x=0x=0.
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AP Physics 1 Quiz

AP Physics 1 Quiz: Defining Simple Harmonic Motion Shm

Practice Defining Simple Harmonic Motion Shm in AP Physics 1 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 block on a frictionless horizontal track is attached to a spring. When displaced a distance xx from equilibrium, the spring exerts a restoring force Fx=−kxF_x=-kx toward equilibrium. The block is released from rest. Which statement best determines whether the resulting motion is simple harmonic?

  1. Yes; the restoring force is proportional to displacement and opposite in direction. (correct answer)
  2. Yes; any motion that repeats is simple harmonic.
  3. No; the restoring force must be proportional to velocity for SHM.
  4. No; equilibrium occurs at maximum displacement, not at x=0x=0.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires a restoring force that is directly proportional to the displacement from equilibrium and points in the opposite direction, mathematically expressed as F = -kx. In this scenario, the spring provides exactly this type of force: F_x = -kx, where the negative sign indicates the force opposes the displacement. The proportionality to x (not x² or x³) and the opposing direction are both essential criteria for SHM. Choice B incorrectly suggests any repeating motion is SHM, but periodicity alone is insufficient without the specific force relationship. To identify SHM, always check if the restoring force or acceleration follows the form proportional to -x.

Question 2

A small cart is attached to a device that provides a force Fx=−bvF_x=-bv opposite its velocity vv. When displaced and released, the cart returns toward equilibrium and overshoots repeatedly. Does this motion qualify as simple harmonic?

  1. Yes, because the force is always directed toward equilibrium.
  2. No, because SHM requires a restoring force proportional to displacement, not velocity. (correct answer)
  3. Yes, because equilibrium is located at the turning points.
  4. No, because SHM cannot be periodic.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires a restoring force proportional to displacement from equilibrium and opposite in direction: F = -kx. In this problem, the device provides a force Fx = -bv that is proportional to velocity, not displacement. While this force opposes motion and can cause oscillations (as the cart overshoots equilibrium repeatedly), it does not meet the fundamental requirement for SHM. Choice A incorrectly focuses only on the force direction, while choices C and D contain factual errors about SHM. To identify SHM, always verify that the restoring force depends on position (F ∝ -x), not on velocity, time, or other variables.

Question 3

A buoy oscillates vertically in water. Measurements show that for small vertical displacements yy from equilibrium, the net upward force is F=−kyF= -ky (downward when y>0y>0). Ignoring drag, is the buoy's motion simple harmonic?

  1. No; because SHM is only possible with springs, not buoyancy.
  2. Yes; because the restoring force is proportional to −y-y. (correct answer)
  3. No; because the restoring force must be proportional to velocity.
  4. No; because equilibrium occurs at maximum displacement where the buoy turns around.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) with buoyancy forces. Simple harmonic motion requires a restoring force proportional to the negative displacement from equilibrium, F = -ky. The measurements show exactly this relationship for the buoy's vertical motion, where the net upward force is F = -ky (negative when displaced upward, positive when displaced downward). This linear restoring force will produce SHM regardless of whether it originates from a spring, buoyancy, or any other physical mechanism. Choice A incorrectly limits SHM to spring systems, missing that any linear restoring force produces SHM. To identify SHM, focus on the mathematical form of the force law F = -ky, not the specific physical mechanism creating that force.

Question 4

A mass on a vertical spring oscillates about its equilibrium position. If yy is displacement from equilibrium, the net force is measured as Fy=−kyF_y=-ky (gravity already accounted for). Is the motion simple harmonic?

  1. Yes; because the net restoring force is proportional to −y-y. (correct answer)
  2. No; because gravity prevents SHM in vertical motion.
  3. No; because equilibrium must be where the mass momentarily stops at maximum yy.
  4. Yes; because the restoring force is proportional to velocity.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the net force be proportional to the negative of displacement from equilibrium. For a vertical spring system, when gravity is already accounted for in the equilibrium position, the net force F_y = -ky shows the required proportionality to displacement y. The negative sign ensures the force always points toward equilibrium, and the linear relationship with y satisfies the SHM criterion. Choice B incorrectly suggests gravity prevents vertical SHM, but gravity only shifts the equilibrium position without affecting the oscillatory behavior about that point. To verify SHM in vertical systems, measure forces relative to the equilibrium position (where spring force balances weight) and check for F = -ky.

Question 5

A glider on an air track is attached to a spring and experiences a damping force Fd=−bvF_d=-bv in addition to the spring force Fs=−kxF_s=-kx. The glider is released from rest at x=Ax=A. Which statement best describes whether the motion is simple harmonic?

  1. Yes; because the damping force is proportional to displacement.
  2. No; because the net force includes a term proportional to velocity, so the motion is not SHM. (correct answer)
  3. Yes; any oscillation with decreasing amplitude is SHM.
  4. No; because equilibrium occurs at x=Ax=A where the object starts.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) with damping. Simple harmonic motion requires that the net force be proportional only to the negative displacement from equilibrium, F = -kx. In this system, the net force is F = -kx - bv, which includes both a displacement term and a velocity term. The presence of the velocity-dependent damping force -bv means the net force is not purely proportional to displacement, violating the SHM requirement. Choice A incorrectly focuses on the damping force alone rather than considering the net force. When analyzing oscillatory motion, remember that true SHM requires the net force to depend only on position, not on velocity or other variables.

Question 6

A mass on a vertical spring is displaced downward by yy from equilibrium. Experiments show the net force relative to equilibrium is F=−kyF=-ky (gravity and spring stretch at equilibrium already balance). Neglecting air resistance, is the motion simple harmonic?

  1. No; because gravity prevents SHM in vertical motion.
  2. No; because the restoring force is proportional to y2y^2.
  3. Yes; because the net restoring force about equilibrium is proportional to −y-y. (correct answer)
  4. No; because equilibrium occurs where the speed is zero and displacement is maximum.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) for vertical spring systems. Simple harmonic motion requires that the net restoring force be proportional to the negative displacement from equilibrium. The key insight is that equilibrium for a vertical spring is where gravity and spring force balance, not at the natural spring length. When displaced by y from this equilibrium, the net force is F = -ky, exactly the form required for SHM. The fact that gravity is present doesn't prevent SHM; it merely shifts the equilibrium position downward. Choice A incorrectly claims gravity prevents vertical SHM, not recognizing that we measure displacement from the gravity-adjusted equilibrium. For vertical spring systems, always measure displacement from the equilibrium position where all constant forces balance.

Question 7

A cart of mass mm is attached to a horizontal spring on a frictionless track. The cart is displaced a distance xx from equilibrium and released. A force probe shows the spring force on the cart is always F=−kxF=-kx, opposite the displacement. Which statement best describes whether the cart's motion is simple harmonic?

  1. No; it is periodic but not SHM because the restoring force is constant.
  2. Yes; the restoring force is proportional to −x-x, so the motion is SHM. (correct answer)
  3. No; it is not SHM because the restoring force is proportional to velocity.
  4. No; it is not SHM because equilibrium occurs at maximum displacement.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires a restoring force that is directly proportional to the displacement from equilibrium and directed opposite to that displacement, mathematically expressed as F = -kx. In this problem, the force probe shows exactly this relationship: F = -kx, where the negative sign indicates the force opposes the displacement. The spring provides a restoring force that increases linearly with displacement, satisfying the fundamental requirement for SHM. Choice A incorrectly claims the restoring force is constant, which would produce uniform acceleration, not SHM. To identify SHM, always check if the restoring force or acceleration follows the form F = -kx or a = -ω²x.

Question 8

A cart moves back and forth between two bumpers. Between collisions, it travels at constant speed with zero net force; at each bumper it reverses direction quickly. The motion repeats with a fixed period. Is the cart's motion simple harmonic?

  1. Yes; because the motion is periodic, it must be SHM.
  2. No; because the acceleration is not proportional to −x-x and is mostly zero. (correct answer)
  3. Yes; because the restoring force is greatest at equilibrium.
  4. No; because equilibrium occurs at the bumpers where displacement is maximum.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) versus general periodic motion. Simple harmonic motion requires a restoring force proportional to the negative displacement from equilibrium throughout the motion. In this cart-bumper system, the cart experiences zero net force (and thus zero acceleration) between bumpers, traveling at constant velocity. The restoring force only acts during the brief collisions at the bumpers, not continuously throughout the motion. While the motion is periodic, it lacks the continuous position-dependent restoring force required for SHM. Choice A incorrectly equates all periodic motion with SHM, missing that SHM is a specific type of periodic motion. To identify SHM, ensure the restoring force acts continuously and is proportional to displacement, not just at isolated points.

Question 9

A mass is attached to a spring on a horizontal surface with kinetic friction. When displaced by xx from equilibrium, the spring exerts Fs=−kxF_s=-kx but friction adds a constant-magnitude force Ff=μkmgF_f=\mu_k mg opposite the velocity. Is the resulting motion simple harmonic?

  1. Yes; because the spring force is proportional to −x-x.
  2. No; because the net force is not strictly proportional to −x-x due to friction. (correct answer)
  3. Yes; because friction makes the period constant.
  4. No; because in SHM the object never passes through equilibrium.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) with friction. Simple harmonic motion requires that the net force be proportional only to the negative displacement from equilibrium, F = -kx. In this system, the net force is F = -kx ± μkmg, where the friction force has constant magnitude but changes direction with velocity. This constant friction term means the net force is not purely proportional to displacement—it has an additional constant term that shifts depending on motion direction. The resulting motion will be oscillatory but with decreasing amplitude, and the force-displacement relationship is not the linear F = -kx required for SHM. Choice A incorrectly considers only the spring force while ignoring how friction affects the net force. When analyzing oscillations, always consider the net force, not just individual force components.

Question 10

A small-angle pendulum bob is displaced so its arc-length displacement from equilibrium is ss. Measurements show the tangential acceleration is at=−gsin⁡θa_t=-g\sin\theta (with s=Lθs=L\theta). For small angles, which choice correctly evaluates whether the motion is simple harmonic?

  1. Yes; for small angles, sin⁡θ≈θ\sin\theta\approx\theta, so at≈−(g/L)sa_t\approx-(g/L)s and the motion is SHM. (correct answer)
  2. No; it cannot be SHM because acceleration must be constant in magnitude.
  3. No; it is periodic but not SHM because the restoring acceleration is proportional to sin⁡θ\sin\theta, not displacement.
  4. Yes; because equilibrium is at the maximum angle where speed is greatest.
Explanation: This question tests understanding of defining simple harmonic motion (SHM) for a pendulum. Simple harmonic motion requires that the restoring force or acceleration be proportional to the negative of the displacement from equilibrium. For a pendulum, the tangential acceleration is at = -g sin θ, but for small angles, sin θ ≈ θ (in radians). Since the arc length s = Lθ, we can write θ = s/L, giving at ≈ -g(s/L) = -(g/L)s. This shows the acceleration is proportional to -s, satisfying the SHM requirement. Choice C incorrectly states that the acceleration being proportional to sin θ prevents SHM, missing the small-angle approximation. When analyzing pendulum motion, remember that SHM occurs only for small angles where sin θ ≈ θ.

Question 11

A cart oscillates on a track. Its measured acceleration satisfies ax=−βx3a_x=-\beta x^3 where xx is displacement from equilibrium. Is the cart's motion simple harmonic?

  1. Yes, because any restoring acceleration toward equilibrium produces SHM
  2. No, because the acceleration is not proportional to −x-x (correct answer)
  3. Yes, because the motion will still be periodic for small enough amplitudes
  4. No, because equilibrium occurs where speed is maximum, not where x=0x=0
Explanation: This question asks whether motion with acceleration a_x = -βx³ qualifies as simple harmonic motion. SHM specifically requires that the restoring acceleration be directly proportional to the negative of the displacement, meaning a = -ωx where ω is a constant. Here, the acceleration depends on x³, not x, which means the restoring force is nonlinear. While this motion may still be periodic and have a restoring force toward equilibrium, it doesn't satisfy the strict proportionality requirement for SHM. Choice A incorrectly suggests any restoring acceleration produces SHM, but the linear relationship is essential. Remember that SHM requires a = -constant × x, not any other power of x.

Question 12

A ball rolls back and forth in a smooth bowl-shaped track. Near the bottom, the horizontal acceleration is measured as ax=−gRxa_x=-\frac{g}{R}x for small displacements xx from the lowest point. Is the motion simple harmonic for small oscillations?

  1. No; because rolling motion cannot be SHM.
  2. Yes; because the acceleration is proportional to −x-x near equilibrium. (correct answer)
  3. No; because axa_x must be proportional to velocity to be SHM.
  4. Yes; because equilibrium occurs at maximum displacement where v=0v=0.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the acceleration be directly proportional to the negative of displacement from equilibrium. For the ball in the bowl, the measured relationship a_x = -(g/R)x shows exactly this proportionality for small displacements, where x is the horizontal distance from the lowest point and the negative sign indicates acceleration toward equilibrium. The proportionality constant g/R depends on gravity and bowl curvature but remains constant for a given system. Choice A incorrectly assumes rolling motion prevents SHM, but the type of motion matters less than the force relationship. To identify SHM in curved surfaces, verify that acceleration near equilibrium follows a = -(constant)x for small displacements.

Question 13

A cart oscillates on a track. Measurements show its acceleration satisfies ax=−βx3a_x=-\beta x^3 where xx is displacement from equilibrium. The cart is released from rest at x=Ax=A. Is the motion simple harmonic?

  1. Yes; the acceleration always points toward equilibrium.
  2. No; simple harmonic motion requires axa_x proportional to −x-x, not −x3-x^3. (correct answer)
  3. Yes; any restoring acceleration guarantees SHM.
  4. No; SHM requires equilibrium at x=Ax=A where speed is zero.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the acceleration be directly proportional to the negative of the displacement from equilibrium, expressed as a = -ωx where ω is a constant. In this problem, the acceleration follows a_x = -βx³, which means acceleration is proportional to the cube of displacement, not displacement itself. This cubic relationship creates a different type of oscillation that is not simple harmonic, even though it still provides a restoring force toward equilibrium. Choice A incorrectly focuses only on direction without considering proportionality. To verify SHM, always check that force or acceleration is proportional to -x (first power only), not higher powers or other functions.

Question 14

A small-angle pendulum of length LL swings with angular displacement heta heta from vertical. For the angles used, the tangential acceleration is measured to satisfy at=−gLhetaa_t=-\frac{g}{L} heta. Does the motion qualify as SHM?

  1. No; because the bob's path is circular, it cannot be SHM.
  2. Yes; because ata_t is proportional to −heta- heta for small angles. (correct answer)
  3. No; because acceleration is greatest at equilibrium in SHM.
  4. Yes; because the acceleration is proportional to velocity.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the restoring force or acceleration be directly proportional to the negative of the displacement from equilibrium. For a pendulum with small angles, the tangential acceleration a_t = -(g/L)θ shows exactly this relationship, where θ represents the angular displacement and the negative sign indicates the acceleration opposes the displacement. The proportionality constant g/L ensures the acceleration increases linearly with angle for small oscillations. Choice C incorrectly states that acceleration is greatest at equilibrium in SHM, when actually acceleration is zero at equilibrium and maximum at the extremes. To identify SHM in rotational systems, verify that angular acceleration is proportional to -θ for small angles.

Question 15

A glider oscillates between two magnets. Measurements show ax=−kmxa_x=-\frac{k}{m}x for ∣x∣≤2 cm|x|\le 2\text{ cm}. Is the motion SHM in that interval?

  1. No, because magnetic forces cannot produce SHM
  2. Yes, because acceleration is proportional to −x-x over that range (correct answer)
  3. No, because equilibrium occurs where axa_x is maximum
  4. No, because the restoring force must be proportional to vv
Explanation: This question asks whether motion satisfying a_x = -k/m × x qualifies as simple harmonic motion. SHM is defined by having acceleration directly proportional to the negative of displacement from equilibrium, which is exactly what a_x = -k/m × x represents. The negative sign ensures the acceleration points toward equilibrium, and k/m provides the constant of proportionality (equal to ω² in standard SHM notation). The fact that this relationship only holds for |x| ≤ 2 cm doesn't disqualify it as SHM within that range. Choice A incorrectly claims magnetic forces cannot produce SHM, but any force following F = -kx can create SHM regardless of its origin. Always verify SHM by checking if a = -constant × x, regardless of the physical mechanism creating the force.

Question 16

A cart oscillates on a track. Its measured acceleration satisfies ax=−βx3a_x=-\beta x^3 for displacement xx from equilibrium. Is the motion simple harmonic?

  1. Yes; any acceleration directed toward equilibrium produces SHM.
  2. No; the acceleration is not proportional to xx. (correct answer)
  3. Yes; the acceleration is zero at maximum displacement.
  4. No; SHM requires acceleration proportional to velocity.
Explanation: This question assesses the understanding of defining simple harmonic motion (SHM) in AP Physics 1. Simple harmonic motion requires a restoring force that is linearly proportional to the displacement from equilibrium, meaning F = -kx for some constant k. This force must act opposite to the displacement, resulting in acceleration a = - (k/m) x. In this case, the acceleration a_x = -β x^3 is proportional to x^3, not x, so it does not meet the criteria for SHM. One distractor, choice A, wrongly claims that any acceleration toward equilibrium produces SHM, ignoring the need for linear proportionality. To identify SHM in various systems, always check if the net force or acceleration follows the form F = -kx or a = -(k/m)x relative to equilibrium.

Question 17

A 0.40 kg0.40\,\text{kg} block on a frictionless track is attached to a spring. When displaced a distance xx from equilibrium, the spring exerts Fx=−kxF_x=-kx toward equilibrium. The block is released from rest and oscillates back and forth. Which statement best determines whether the motion is simple harmonic?

  1. It is SHM because the restoring force is proportional to displacement and opposite in direction. (correct answer)
  2. It is SHM because the block's speed is maximum at the turning points.
  3. It is not SHM because the motion is periodic but the force is constant.
  4. It is SHM because the restoring force is proportional to velocity, F∝−vF\propto -v.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion occurs when the restoring force is proportional to the displacement from equilibrium and opposite in direction, mathematically expressed as F = -kx. In this problem, the spring exerts exactly this type of force: Fx = -kx, where the negative sign indicates the force points toward equilibrium. This proportional relationship ensures that the acceleration a = F/m = -(k/m)x is also proportional to displacement, which is the defining characteristic of SHM. Choice B incorrectly states that speed is maximum at turning points (it's actually zero there), while choices C and D misunderstand the force requirements for SHM. To identify SHM, always check if the restoring force or acceleration is proportional to negative displacement: F ∝ -x or a ∝ -x.

Question 18

A glider on an air track is pulled toward equilibrium by a force that always has magnitude 2.0 N2.0\,\text{N} and points toward x=0x=0, regardless of displacement. The glider oscillates between x=±0.50 mx=\pm 0.50\,\text{m}. Is the motion SHM?

  1. Yes, because the motion repeats with a constant period.
  2. No, because SHM requires F∝−xF\propto -x, not a constant-magnitude restoring force. (correct answer)
  3. Yes, because equilibrium occurs at the turning points.
  4. No, because SHM requires F∝−vF\propto -v.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires the restoring force to be proportional to displacement from equilibrium: F ∝ -x. In this problem, the force has constant magnitude (2.0 N) regardless of displacement, only changing direction to point toward equilibrium. This means F is not proportional to x - it doesn't increase with larger displacements or decrease near equilibrium. While this constant-magnitude force can produce periodic oscillation, it doesn't satisfy the proportionality requirement for SHM. Choice A incorrectly assumes periodic motion implies SHM, while choice D confuses the force requirement. To identify SHM, verify that force magnitude varies linearly with displacement: |F| = k|x|, not constant.

Question 19

A cart oscillates in one dimension with measured acceleration ax=−(9.0 s−2)xa_x=-(9.0\,\text{s}^{-2})x for all displacements tested. The cart repeatedly passes through x=0x=0 with maximum speed. Based on the acceleration relationship, is the motion SHM?

  1. No, because the cart's speed changes during the motion.
  2. Yes, because acceleration is proportional to −x-x, satisfying the SHM condition. (correct answer)
  3. No, because periodic motion cannot be simple harmonic.
  4. Yes, because the restoring force is proportional to velocity.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion is defined by acceleration being proportional to negative displacement from equilibrium: a = -ω²x. The given relationship ax = -(9.0 s⁻²)x exactly matches this form, with ω² = 9.0 s⁻². This proportionality ensures the restoring force F = ma is also proportional to -x, creating the conditions for SHM. The fact that the cart passes through x = 0 with maximum speed further confirms SHM behavior. Choice A incorrectly suggests changing speed disqualifies SHM, while choice C makes the false claim that periodic motion cannot be SHM. To verify SHM from experimental data, check if acceleration follows a = -(positive constant)×x throughout the motion.

Question 20

A mass on a horizontal surface is attached to a spring and moves back and forth. The displacement from equilibrium is xx. A motion sensor shows the acceleration is ax=−ω2x3a_x=-\omega^2 x^3 where ω\omega is a constant with appropriate units. The mass passes through equilibrium repeatedly with its maximum speed. Neglect friction.

Is the motion simple harmonic?

  1. Yes, because the acceleration always points toward equilibrium.
  2. No, because the acceleration is not proportional to displacement. (correct answer)
  3. Yes, because the object is periodic and returns to the same position.
  4. No, because the force must be proportional to velocity for SHM.
Explanation: This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the acceleration be proportional to the first power of displacement and directed toward equilibrium. The given relationship a_x = -ω²x³ shows acceleration proportional to the cube of displacement, not to x itself. While the negative sign correctly indicates acceleration opposes displacement (pointing toward equilibrium), the x³ dependence violates the strict proportionality requirement for SHM. Choice D incorrectly claims the force must be proportional to velocity for SHM, when in fact SHM requires force proportional to displacement. To identify SHM, check that the restoring force or acceleration follows F = -kx or a = -ω²x exactly, with no higher powers or additional terms.