AP Physics 1 Quiz: Defining Simple Harmonic Motion Shm

What this quiz covers

This quiz focuses on Defining Simple Harmonic Motion Shm, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Physics 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. No, because acceleration must be proportional to velocity for SHM.
2. Yes, because the acceleration is proportional to $-x$. (correct answer)
3. No, because equilibrium is at maximum displacement.
4. Yes, because the speed is constant throughout the motion.

Explanation: This question assesses the skill of defining simple harmonic motion (SHM) in AP Physics 1. SHM is characterized by a restoring force proportional to and opposite the displacement, resulting in acceleration a = -ω²x. The given acceleration a = -ω²x directly matches this form, confirming SHM for the block. This equation implies sinusoidal motion with constant frequency ω. Choice A is a distractor because SHM requires acceleration proportional to displacement, not velocity. A useful strategy is to recall that SHM solutions satisfy the differential equation d²x/dt² = -ω²x, and check if the system fits.

Question 2

1. Yes; because the force is proportional to displacement from equilibrium, $x-x_0$. (correct answer)
2. No; because the force is not proportional to the position $x$ measured from the origin.
3. No; because equilibrium must occur at $x=0$ for SHM.
4. Yes; because the force is proportional to velocity when passing through $x_0$.

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 proportional to the displacement from the equilibrium position, not necessarily from the origin. This force must be opposite in direction, so F = -k (x - x_eq) qualifies, where x_eq is the equilibrium point. In this case, F_x = -k (x - x_0) shows equilibrium at x = x_0, and the force is proportional to the displacement from there, making it SHM. One distractor, choice C, wrongly insists equilibrium must be at x=0, but SHM can occur around any equilibrium point. 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 3

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 $x=0$.

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 4

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 5

1. No; because SHM is only possible with springs, not buoyancy.
2. Yes; because the restoring force is proportional to $-y$. (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 6

1. Yes; because the net restoring force is proportional to $-y$. (correct answer)
2. No; because gravity prevents SHM in vertical motion.
3. No; because equilibrium must be where the mass momentarily stops at maximum $y$.
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 7

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 $x=A$ 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 8

1. No; because gravity prevents SHM in vertical motion.
2. No; because the restoring force is proportional to $y^2$.
3. Yes; because the net restoring force about equilibrium is proportional to $-y$. (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 9

1. No; it is periodic but not SHM because the restoring force is constant.
2. Yes; the restoring force is proportional to $-x$, 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 10

1. Yes; because the motion is periodic, it must be SHM.
2. No; because the acceleration is not proportional to $-x$ 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 11

1. Yes; because the spring force is proportional to $-x$.
2. No; because the net force is not strictly proportional to $-x$ 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 12

1. Yes; for small angles, $\sin\theta\approx\theta$, so $a_t\approx-(g/L)s$ 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 $\sin\theta$, 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 13

1. Yes, because the force is always opposite the displacement.
2. No, because the proportionality between force and displacement is not the same for all $x$. (correct answer)
3. Yes, because equilibrium occurs at maximum displacement.
4. No, because the restoring force is proportional to velocity.

Explanation: This question assesses the skill of defining simple harmonic motion (SHM) in AP Physics 1. SHM necessitates a restoring force proportional to and opposite the displacement, with the same constant of proportionality throughout. In this setup, the force differs for stretching (F = -2kx) and compression (F = -kx), lacking consistent proportionality. Thus, the motion is oscillatory but not simple harmonic due to the asymmetry. Choice A is a distractor as it overlooks the need for uniform proportionality across all displacements. To evaluate such systems, plot or analyze the force vs. displacement graph for linearity through the origin.

Question 14

1. Yes, because acceleration always points toward equilibrium.
2. No, because SHM requires $a=-\omega^2 x$ (signed proportionality), not $\omega^2|x|$. (correct answer)
3. Yes, because equilibrium occurs at maximum displacement.
4. No, because acceleration must be proportional to velocity.

Explanation: This question assesses the skill of defining simple harmonic motion (SHM) in AP Physics 1. SHM requires the restoring force to be proportional to -x, including the negative sign for direction opposite displacement. Here, a = ω²|x| directs toward equilibrium but lacks the signed proportionality, as it's always positive magnitude regardless of x's sign. This would not yield the sinusoidal solutions of SHM. Choice A distracts by emphasizing direction without the linear signed dependence. To verify SHM, ensure the acceleration equation includes the negative sign and linear term in x, not |x|.

Question 15

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, $F\propto -v$.

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 16

1. Yes, because the motion repeats with a constant period.
2. No, because SHM requires $F\propto -x$, not a constant-magnitude restoring force. (correct answer)
3. Yes, because equilibrium occurs at the turning points.
4. No, because SHM requires $F\propto -v$.

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 17

1. No, because the cart’s speed changes during the motion.
2. Yes, because acceleration is proportional to $-x$, 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 18

1. Yes, because any sinusoidal force guarantees SHM
2. No, because the net acceleration is not solely proportional to $-x$ (correct answer)
3. No, because equilibrium occurs at maximum displacement
4. Yes, because the motor force is proportional to velocity

Explanation: This question tests whether external driving forces are compatible with simple harmonic motion. SHM specifically requires that the net acceleration be proportional only to -x. With both spring and motor forces present, the total force is F_total = -kx + F₁cos(Ωt), giving acceleration a = -kx/m + F₁cos(Ωt)/m. This acceleration depends on both position x and time t explicitly, violating the SHM requirement that acceleration depend only on position. The time-dependent driving force creates forced oscillation, which generally produces steady-state motion at the driving frequency Ω rather than natural SHM at frequency ω = √(k/m). Choice A incorrectly assumes any sinusoidal force guarantees SHM, ignoring that SHM requires no explicit time dependence. Always check that acceleration depends only on position for true SHM.

Question 19

1. No, because acceleration must be constant for SHM
2. Yes, because the acceleration is proportional to $- heta$ (correct answer)
3. No, because the restoring force is proportional to velocity
4. No, because equilibrium is at the turning points where $heta$ is maximum

Explanation: This question tests recognition of simple harmonic motion in rotational systems. SHM requires that the restoring acceleration be proportional to the negative of the displacement from equilibrium. For a small-angle pendulum, the tangential acceleration a_t = -gθ shows exactly this relationship, with angular displacement θ playing the role of linear displacement. The negative sign ensures the acceleration always points back toward equilibrium (θ = 0), and g provides the constant of proportionality. Choice C incorrectly claims the restoring force depends on velocity, confusing force with acceleration. When analyzing rotational SHM, look for angular acceleration proportional to -θ, just as linear SHM has linear acceleration proportional to -x.

Question 20

1. No; because gravity prevents simple harmonic motion in vertical systems.
2. Yes; because the net restoring force is proportional to $y$ and directed toward equilibrium. (correct answer)
3. No; because the restoring force must be proportional to velocity.
4. Yes; because equilibrium occurs where the displacement is maximum.

Explanation: This question assesses the understanding of defining simple harmonic motion (SHM) in AP Physics 1. Simple harmonic motion requires a net restoring force proportional to displacement from equilibrium and opposite in direction. For vertical systems, gravity shifts the equilibrium, but the net force can still be F = -ky relative to that point. Here, the net force F_y = -ky satisfies the proportionality and opposition, qualifying as SHM. One distractor, choice A, incorrectly states that gravity prevents SHM in vertical systems, but the net effect is still harmonic. 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.