Defining Simple Harmonic Motion (SHM)
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AP Physics 1 › Defining Simple Harmonic Motion (SHM)
A block oscillates on a frictionless surface. For displacement $x$, its acceleration is $a=-\omega^2 x$. Is the motion SHM?
No, because acceleration must be proportional to velocity for SHM.
Yes, because the acceleration is proportional to $-x$.
No, because equilibrium is at maximum displacement.
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.
A mass on a spring experiences a restoring force $F_x=-k(x-x_0)$, where $x_0$ is a constant offset. Does the motion about equilibrium qualify as SHM?
No; because the force is not proportional to the position $x$ measured from the origin.
No; because equilibrium must occur at $x=0$ for SHM.
Yes; because the force is proportional to velocity when passing through $x_0$.
Yes; because the force is proportional to displacement from equilibrium, $x-x_0$.
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.
A block on a frictionless horizontal track is attached to a spring. When displaced a distance $x$ from equilibrium, the spring exerts a restoring force $F_x=-kx$ 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.
No; the restoring force must be proportional to velocity for SHM.
Yes; any motion that repeats is simple harmonic.
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.
A small cart is attached to a device that provides a force $F_x=-bv$ opposite its velocity $v$. When displaced and released, the cart returns toward equilibrium and overshoots repeatedly. Does this motion qualify as simple harmonic?
Yes, because the force is always directed toward equilibrium.
Yes, because equilibrium is located at the turning points.
No, because SHM requires a restoring force proportional to displacement, not velocity.
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.
A buoy oscillates vertically in water. Measurements show that for small vertical displacements $y$ from equilibrium, the net upward force is $F= -ky$ (downward when $y>0$). Ignoring drag, is the buoy’s motion simple harmonic?
No; because the restoring force must be proportional to velocity.
No; because SHM is only possible with springs, not buoyancy.
Yes; because the restoring force is proportional to $-y$.
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.
A mass on a vertical spring oscillates about its equilibrium position. If $y$ is displacement from equilibrium, the net force is measured as $F_y=-ky$ (gravity already accounted for). Is the motion simple harmonic?
No; because gravity prevents SHM in vertical motion.
Yes; because the restoring force is proportional to velocity.
Yes; because the net restoring force is proportional to $-y$.
No; because equilibrium must be where the mass momentarily stops at maximum $y$.
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.
A glider on an air track is attached to a spring and experiences a damping force $F_d=-bv$ in addition to the spring force $F_s=-kx$. The glider is released from rest at $x=A$. Which statement best describes whether the motion is simple harmonic?
No; because equilibrium occurs at $x=A$ where the object starts.
Yes; any oscillation with decreasing amplitude is SHM.
No; because the net force includes a term proportional to velocity, so the motion is not SHM.
Yes; because the damping force is proportional to displacement.
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.
A mass on a vertical spring is displaced downward by $y$ from equilibrium. Experiments show the net force relative to equilibrium is $F=-ky$ (gravity and spring stretch at equilibrium already balance). Neglecting air resistance, is the motion simple harmonic?
No; because gravity prevents SHM in vertical motion.
No; because equilibrium occurs where the speed is zero and displacement is maximum.
Yes; because the net restoring force about equilibrium is proportional to $-y$.
No; because the restoring force is proportional to $y^2$.
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.
A cart of mass $m$ is attached to a horizontal spring on a frictionless track. The cart is displaced a distance $x$ from equilibrium and released. A force probe shows the spring force on the cart is always $F=-kx$, opposite the displacement. Which statement best describes whether the cart’s motion is simple harmonic?
No; it is periodic but not SHM because the restoring force is constant.
No; it is not SHM because equilibrium occurs at maximum displacement.
No; it is not SHM because the restoring force is proportional to velocity.
Yes; the restoring force is proportional to $-x$, so the motion is SHM.
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.
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?
Yes; because the motion is periodic, it must be SHM.
Yes; because the restoring force is greatest at equilibrium.
No; because equilibrium occurs at the bumpers where displacement is maximum.
No; because the acceleration is not proportional to $-x$ and is mostly zero.
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.