Defining Simple Harmonic Motion (SHM) - AP Physics 1

A cart oscillates on a track. Measurements show its acceleration satisfies $a_x=-\beta x^3$ where $x$ is displacement from equilibrium. The cart is released from rest at $x=A$ . Is the motion simple harmonic?
Answer: No; simple harmonic motion requires $a_x$ proportional to $-x$ , not $-x^3$ . 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.
A mass on a spring moves while experiencing a resistive force $F_r=-bv$ in addition to a spring force. Is its motion simple harmonic?
Answer: No, because a force proportional to velocity prevents ideal SHM This problem examines whether damped oscillation qualifies as simple harmonic motion. True SHM requires that the only force affecting the motion be proportional to -x, giving acceleration a = -ωx. When a resistive force $F_r$ = -bv is present, the total force becomes F_total = -kx - bv, so the acceleration is a = -kx/m - bv/m. This acceleration now depends on both position and velocity, violating the SHM requirement that acceleration depend only on position. Choice A incorrectly focuses only on the direction of forces rather than their mathematical form. To identify SHM, verify that the net acceleration depends only on position, not on velocity or other variables.
A mass on a spring experiences a damping force $F_d=-bv$ in addition to the spring force $F_s=-kx$ . It is released from rest after being displaced. Is the motion simple harmonic?
Answer: No; because a velocity-proportional damping force makes acceleration not solely proportional to $-x$ . This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the net force be proportional only to the negative of displacement from equilibrium, F = -kx. In this system, the total force is F_total = -kx - bv, which includes both a displacement-dependent term and a velocity-dependent damping term. The presence of the -bv term means the net force is no longer purely proportional to displacement, violating the fundamental requirement for SHM. This creates damped oscillation, which is a different type of motion even though it may still be periodic. Choice C incorrectly focuses only on the spring force while ignoring the damping contribution. To verify SHM, always check that the net force depends only on position (proportional to -x), not on velocity or other variables.
A block on a horizontal spring oscillates with negligible friction. Sensors show the restoring force satisfies $F_x=-kx$ for all measured displacements $x$ from equilibrium. Does the motion qualify as simple harmonic motion?
Answer: Yes, because the restoring force is proportional to $-x$ This question tests whether you can identify simple harmonic motion (SHM) from the defining force law. SHM requires a restoring force that is directly proportional to the negative of the displacement from equilibrium, which means F = -kx where k is a positive constant. The given force $F_x$ = -kx exactly matches this requirement, with the negative sign ensuring the force always points back toward equilibrium. The spring constant k provides the proportionality between force and displacement. Choice A incorrectly claims the acceleration isn't related to displacement, but F = ma gives a = -kx/m, which is proportional to -x. To identify SHM, always check if the restoring force or acceleration follows the form proportional to -x.
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?
Answer: Yes; the restoring force is proportional to $-x$ , so the motion is SHM. 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 particle moves back and forth on a line. Data show its acceleration is always $a_x=-\omega^2 x$ relative to equilibrium. Is this motion simple harmonic?
Answer: Yes, because acceleration is proportional to $-x$ This problem directly presents the defining equation of simple harmonic motion. SHM requires that acceleration be proportional to the negative of displacement from equilibrium, which is exactly what $a_x$ = -ω²x represents. The constant ω² determines the angular frequency of oscillation, and the negative sign ensures the acceleration always points toward equilibrium, creating the restoring effect. This mathematical relationship alone is sufficient to guarantee SHM, producing sinusoidal position and velocity functions. Choice B incorrectly claims SHM requires constant speed, when actually speed varies sinusoidally, reaching maximum at equilibrium and zero at turning points. To confirm SHM, verify that acceleration follows a = -constant × x; this mathematical form is both necessary and sufficient.
A mass slides in a smooth, horizontal track shaped like a parabola. Let $x$ be the horizontal displacement from the lowest point (equilibrium). For small displacements, measurements show the horizontal component of the acceleration satisfies $a_x\approx-\beta x$ , where $\beta$ is a positive constant. The mass is released from rest at small $x$ . Is the motion simple harmonic?
Answer: Yes, because the acceleration is approximately proportional to displacement and opposite in direction. This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that acceleration be proportional to displacement and directed toward equilibrium. The given relationship $a_x$ ≈ -βx perfectly satisfies these requirements: acceleration is proportional to x with constant β, and the negative sign ensures acceleration opposes displacement (pointing toward equilibrium). The parabolic track shape naturally provides this linear restoring acceleration for small displacements. Choice A incorrectly claims SHM is only possible with springs, but any system providing a linear restoring force exhibits SHM, including pendulums, floating objects, and masses in shaped tracks. To identify SHM, look for the defining characteristic: restoring force or acceleration proportional to displacement, regardless of the physical mechanism.
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?
Answer: Yes; the restoring force is proportional to displacement and opposite in direction. 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 pendulum swings through large angles where the tangential acceleration is $a = -g \sin \theta$ . Is its motion simple harmonic?
Answer: No, because $a$ is not proportional to displacement for large angles. This question assesses the skill of defining simple harmonic motion (SHM) in AP Physics 1. SHM requires the restoring force or acceleration to be proportional to -x, the displacement from equilibrium. For large angles, $a = -g \sin \theta$ does not approximate $- (g/L) \theta$ , as $\sin \theta \neq \theta$ , so proportionality to displacement fails. The motion is periodic but anharmonic for large swings. Choice A distracts by claiming $\sin \theta \propto \theta$ for all angles, which is only true for small $\theta$ . When dealing with pendulums, remember to assess if the small-angle approximation applies to confirm SHM.
A mass oscillates on a spring, but the spring force is measured as $F_x=-k(x-x_0)$ where $x_0$ is a constant offset. Is the motion simple harmonic?
Answer: Yes, because the restoring force is proportional to displacement from equilibrium This problem tests understanding that simple harmonic motion depends on the force relative to equilibrium position. The force $F_x$ = -k(x - x₀) can be rewritten as $F_x$ = -kx + kx₀, which represents a spring force plus a constant force. This shifts the equilibrium position to x = x₀ (where $F_x$ = 0), and if we define a new coordinate x' = x - x₀, then $F_x$ = -kx'. The motion is still SHM because the force remains proportional to displacement from the (shifted) equilibrium. Choice B incorrectly suggests any offset prevents SHM, but SHM is defined relative to equilibrium, wherever that may be. To identify SHM, always measure displacement from the equilibrium position where the net force is zero.
A glider oscillates on an air track between two magnets. Let $x$ be the glider's displacement from equilibrium. A sensor indicates the net force is approximately constant in magnitude and always points toward equilibrium: $F_x=-F_0\,\text{sgn}(x)$ . The glider repeatedly passes through $x=0$ with its greatest speed. Ignore friction. Is this motion simple harmonic?
Answer: No, because the restoring force is not proportional to displacement. This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires a restoring force that is proportional to displacement from equilibrium and directed opposite to that displacement. The given force $F_x$ = -F₀ sgn(x) has constant magnitude F₀ and always points toward equilibrium (due to the negative sign and sign function), but crucially, the force magnitude does not vary with displacement. This constant-magnitude force violates the proportionality requirement for SHM. Choice A incorrectly focuses only on the direction of the force, ignoring the proportionality requirement that distinguishes SHM from other types of oscillatory motion. To identify SHM, ensure both conditions are met: the restoring force must be proportional to displacement AND directed toward equilibrium.
A small-angle pendulum of length $L$ swings with angular displacement $\theta$ from vertical. For the angles used, the tangential acceleration is measured to satisfy $a_t=-\frac{g}{L}\theta$ . Does the motion qualify as SHM?
Answer: Yes; because $a_t$ is proportional to $-\theta$ for small angles. 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.
A small block is attached to a spring and oscillates on a frictionless table. The displacement from equilibrium is $x$ . The net force on the block is measured to be $F_x=-kx-\alpha x^3$ , where $k$ and $\alpha$ are positive constants. The block is released from rest at $x=+0.08\ \text{m}$ . Is the block's motion simple harmonic?
Answer: No, because the restoring force is not strictly proportional to $x$ . This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires a restoring force that is strictly proportional to displacement: F = -kx. The given force $F_x$ = -kx - αx³ contains both a linear term (-kx) and a cubic term (-αx³). While the linear term satisfies the SHM requirement, the additional cubic term means the total force is not strictly proportional to x. This nonlinear force law produces oscillatory motion that is not simple harmonic. Choice C incorrectly suggests that having maximum speed at equilibrium is sufficient for SHM, but many non-SHM oscillators also have this property. To identify SHM, verify that the restoring force contains only terms proportional to the first power of displacement.
A pendulum bob is displaced a small angle so that $a\approx-(g/L)x$ along the arc from equilibrium. Does it undergo SHM?
Answer: Yes, because the acceleration is proportional to $-x$ for small displacements. This question assesses the skill of defining simple harmonic motion (SHM) in AP Physics 1. SHM requires a restoring force that is proportional to the displacement from equilibrium and directed opposite to it. For small angles, the pendulum's acceleration approximates a = -(g/L)x, where x is the arc displacement, satisfying the SHM condition of proportionality to -x. This approximation holds because sinθ ≈ θ for small θ, making the motion harmonic. Choice A is a distractor because constant magnitude acceleration, like in free fall, does not imply SHM without the specific -x proportionality. A transferable strategy is to check if the system's equation of motion reduces to the SHM form a = -ω²x for the given conditions.
A mass on a horizontal surface is attached to a spring and moves back and forth. The displacement from equilibrium is $x$ . A motion sensor shows the acceleration is $a_x=-\omega^2 x^3$ where $\omega$ is a constant with appropriate units. The mass passes through equilibrium repeatedly with its maximum speed. Neglect friction. Is the motion simple harmonic?
Answer: No, because the acceleration is not proportional to displacement. 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.
A pendulum swings with a maximum angular displacement of $40^\circ$ . The tangential acceleration is $a_t=-g\sin\theta$ , where $\theta$ is the angular displacement from equilibrium. Does this motion satisfy the condition for SHM?
Answer: No, because SHM requires acceleration proportional to displacement, and $\sin\theta$ is not proportional to $\theta$ at large angles. This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires acceleration to be proportional to negative displacement: a ∝ -x. For a pendulum, the tangential acceleration is at = -g sin θ, where θ is angular displacement. For SHM, we need at ∝ -θ, which requires sin θ ∝ θ. This proportionality only holds for small angles where sin θ ≈ θ (the small angle approximation). At 40°, sin(40°) ≈ 0.643 while 40° ≈ 0.698 radians, showing significant deviation from proportionality. Choice A incorrectly claims sin θ always equals θ, while choice C only considers force direction without checking proportionality. To determine if pendulum motion is SHM, verify that the amplitude is small enough (typically < 15°) for the small angle approximation to be valid.
A glider oscillates between two magnets. Measurements show $a_x=-\frac{k}{m}x$ for $|x|\le 2\text{ cm}$ . Is the motion SHM in that interval?
Answer: Yes, because acceleration is proportional to $-x$ over that range 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.
A cart is attached to a spring that is stretched beyond its linear range. For displacements $x$ from equilibrium, the measured restoring force is $F_x=-k x - c x^2$ (for $x>0$ ). The cart is released from rest. Is the motion SHM?
Answer: No; because the restoring force is not strictly proportional to $x$ . This question tests understanding of defining simple harmonic motion (SHM). Simple harmonic motion requires that the restoring force be strictly proportional to the first power of displacement from equilibrium, F = -kx. In this nonlinear spring system, the force $F_x$ = -kx - cx² contains both a linear term (-kx) and a quadratic term (-cx²). The presence of the x² term means the force is no longer purely proportional to x, violating the fundamental requirement for SHM. This creates anharmonic oscillation where the period depends on amplitude, unlike true SHM where period is independent of amplitude. Choice C incorrectly suggests the extra term preserves SHM while only changing frequency. To verify SHM, check that force or acceleration contains only terms linear in displacement, with no higher powers or other functional forms.
A cart oscillates on a track. Its measured acceleration satisfies $a_x=-\beta x^3$ for displacement $x$ from equilibrium. Is the motion simple harmonic?
Answer: No; the acceleration is not proportional to $x$ . 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.
A cart oscillates in one dimension with measured acceleration $a_x=-(9.0\,\text{s}^{-2})x$ for all displacements tested. The cart repeatedly passes through $x=0$ with maximum speed. Based on the acceleration relationship, is the motion SHM?
Answer: Yes, because acceleration is proportional to $-x$ , satisfying the SHM condition. 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.

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