Defining Simple Harmonic Motion (SHM) - AP Physics C Mechanics

A physical system undergoes oscillatory motion described by $x(t) = A e^{-\gamma t} \cos(\omega t)$ . Is this simple harmonic motion?
Answer: No, because the exponentially decreasing amplitude indicates the presence of damping forces that violate the SHM force condition The exponential decay indicates damping, meaning additional forces beyond the ideal restoring force act on the system. True SHM requires only the restoring force $F = -kx$ ; 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.
A particle moves along a straight line with position given by $x(t) = A\cos(\omega t + \phi)$ . The particle's acceleration as a function of position can be written as $a = -kx$ where $k$ is a positive constant. Which statement correctly identifies the relationship between $k$ and the other parameters, and explains why this motion constitutes simple harmonic motion?
Answer: $k = \omega^2$ ; this is SHM because the acceleration is proportional to displacement with a negative constant of proportionality Taking the second derivative of $x(t) = A\cos(\omega t + \phi)$ gives $a = -A\omega^2\cos(\omega t + \phi) = -\omega^2 x$ . Comparing with $a = -kx$ , we get $k = \omega^2$ . Simple harmonic motion is defined as motion where acceleration is proportional to displacement with a negative constant of proportionality. Choice B is wrong because $k \neq \omega$ . Choice C incorrectly includes amplitude in the constant. Choice D has incorrect units and reasoning.
For a mass-spring system undergoing simple harmonic motion, which statement correctly describes the equilibrium position?
Answer: The equilibrium position is where the spring force exactly balances all other forces, resulting in zero net force on the mass 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.
Three masses oscillate with position functions: Mass 1: $x_1 = 2\cos(5t)$ , Mass 2: $x_2 = 3\cos^2(5t)$ , and Mass 3: $x_3 = 4\cos(5t + \pi/6)$ . Which mass(es) exhibit simple harmonic motion?
Answer: Only masses 1 and 3 exhibit SHM because their accelerations are proportional to their respective displacements with negative constants SHM requires $a = -kx$ . For Mass 1: $a_1 = -50\cos(5t) = -25x_1$ , so it's SHM. For Mass 3: $a_3 = -100\cos(5t + \pi/6) = -25x_3$ , so it's SHM. For Mass 2: $x_2 = 3\cos^2(5t) = \frac{3}{2}(1 + \cos(10t))$ , so $a_2 = -150\cos(10t)$ . Since $\cos(10t) = 2\cos^2(5t) - 1 = \frac{2x_2}{3} - 1$ , we get $a_2 = -100x_2 + 150$ , which is not proportional to $x_2$ alone. Choice A ignores the mathematical requirement. Choices C and D focus on irrelevant aspects of phase constants.
A particle's motion is described by the differential equation $\frac{d^2x}{dt^2} + 9x = 12\cos(3t)$ . To determine if any component of the motion exhibits simple harmonic motion, which analysis is most appropriate?
Answer: The homogeneous equation $\frac{d^2x}{dt^2} + 9x = 0$ represents SHM, while the driving term prevents the total motion from being simple harmonic The homogeneous equation $\frac{d^2x}{dt^2} + 9x = 0$ has solutions of the form $x_{hom} = A\cos(3t + \phi)$ , which represents SHM with $\omega = 3$ 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.
Which mathematical condition must be satisfied for an oscillating system to be classified as undergoing simple harmonic motion?
Answer: The second derivative of position with respect to time must be proportional to the negative of the position coordinate The differential equation for SHM is $\frac{d^2x}{dt^2} = -\omega^2 x$ , 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.
What is the significance of the negative sign in the simple harmonic motion force equation $F = -kx$ ?
Answer: The negative sign shows that the force direction is opposite to the displacement direction, always pointing toward equilibrium position 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.
An object moves according to $x(t) = 4e^{-0.1t}\cos(3t)$ . A student claims this represents damped simple harmonic motion. Which statement best evaluates this claim?
Answer: Incorrect, because the acceleration includes terms proportional to both position and velocity, violating the SHM condition $a \propto -x$ For SHM, we need $a = -kx$ exactly. Taking the second derivative: $a = \frac{d^2x}{dt^2} = 4e^{-0.1t}[-9\cos(3t) + 0.6\sin(3t) - 0.01\cos(3t)] = 4e^{-0.1t}[-9.01\cos(3t) + 0.6\sin(3t)]$ . This cannot be written as $a = -kx$ because of the $\sin(3t)$ term from the product rule applied to the damping factor. True SHM requires $a \propto -x$ exactly. While this motion oscillates with decreasing amplitude, it's not mathematically simple harmonic motion. Choice A confuses physical behavior with mathematical definition. Choice B is correct about constant amplitude but wrong reasoning. Choice C ignores the mathematical requirement.
A particle oscillates such that its position varies as $x(t) = A\cos(\omega t + \phi)$ . What can be concluded about the motion?
Answer: The motion is simple harmonic because the acceleration is proportional to negative displacement when differentiated twice Taking the second derivative: $\frac{d^2x}{dt^2} = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x$ , which satisfies the SHM condition $a = -\omega^2 x$ . 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.
A pendulum swings back and forth with small angular displacements. What makes this motion an example of simple harmonic motion?
Answer: The gravitational force component tangent to the arc is approximately proportional to the angular displacement for small angles For small angles, $\sin\theta \approx \theta$ , so the restoring torque $\tau = -mgl\sin\theta \approx -mgl\theta$ 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.
Which characteristic distinguishes simple harmonic motion from other types of periodic motion?
Answer: Simple harmonic motion requires linear restoring force while other motions may have nonlinear restoring forces The defining characteristic of SHM is the linear restoring force ( $F = -kx$ ). 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.
Consider two oscillating systems: System X has restoring force $F_X = -kx$ and System Y has restoring force $F_Y = -kx^3$ where $k > 0$ 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?
Answer: The argument is incorrect because only System X satisfies $a \propto -x$ ; System Y has $a \propto -x^3$ , violating the SHM definition Simple harmonic motion specifically requires acceleration proportional to displacement: $a = -\omega^2 x$ . System X: $F = ma = -kx$ , so $a = -\frac{k}{m}x$ (SHM). System Y: $F = ma = -kx^3$ , so $a = -\frac{k}{m}x^3$ (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.
A particle undergoes periodic motion. Which condition is necessary and sufficient to classify this motion as simple harmonic motion?
Answer: The net force must be proportional to displacement from equilibrium and directed toward the equilibrium position at all times The defining characteristic of SHM is $F = -kx$ 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.
A mass on a spring oscillates vertically. When the mass is at position $y = +0.10$ m above equilibrium, its acceleration is $a = -4.0$ m/s² (downward). When the mass is at position $y = -0.05$ m below equilibrium, its acceleration is $a = +2.0$ m/s² (upward). What can be concluded about this motion?
Answer: This is simple harmonic motion with angular frequency $\omega = 2\sqrt{10}$ rad/s because acceleration is proportional to displacement from equilibrium When analyzing oscillatory motion, you need to determine whether the acceleration is proportional to displacement from equilibrium. For simple harmonic motion, the relationship is $a = -\omega^2 y$ , where the acceleration is always directed toward equilibrium. Let's check if this relationship holds by calculating the ratio $a/y$ at both positions. At $y = +0.10$ m, $a = -4.0$ m/s², so $a/y = -4.0/0.10 = -40$ s⁻². At $y = -0.05$ m, $a = +2.0$ m/s², so $a/y = +2.0/(-0.05) = -40$ s⁻². Since this ratio is constant, we have $-\omega^2 = -40$ , giving us $\omega = \sqrt{40} = 2\sqrt{10}$ 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 $y = 0$ . The fact that acceleration points toward $y = 0$ in both cases (downward when above, upward when below) confirms that $y = 0$ is indeed equilibrium. Study tip: Always check if $a/y$ is constant when determining whether motion is simple harmonic. If the ratio is constant and negative, you have SHM with $\omega = \sqrt{|a/y|}$ .
In simple harmonic motion, the phase relationship between displacement and acceleration is such that:
Answer: Displacement and acceleration are exactly $180°$ out of phase, with acceleration being maximum when displacement is maximum Since $a = -\omega^2 x$ , acceleration is always proportional to displacement but opposite in direction, creating a $180°$ phase difference. When displacement is maximum positive, acceleration is maximum negative, and vice versa. Choice A incorrectly suggests they're in phase. Choice B describes the relationship between displacement and velocity. Choice D is incorrect because the phase relationship is always $180°$ regardless of initial conditions.
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?
Answer: Only at the equilibrium position where the net force on the mass is zero and no restoring force acts In SHM, acceleration is proportional to displacement: $a = -\frac{k}{m}x$ . Acceleration equals zero only when displacement $x = 0$ , 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).
Two objects undergo motion described by the following equations: Object 1: $x_1(t) = 3\cos(4t)$ and Object 2: $x_2(t) = 2\sin(4t + \pi/3) + 5$ . Which statement correctly characterizes these motions?
Answer: Both objects exhibit simple harmonic motion with identical periods, but Object 2 oscillates about a displaced equilibrium position Simple harmonic motion requires $a \propto -x$ relative to equilibrium position. For Object 1: equilibrium at $x = 0$ , so $a_1 = -16 \cdot 3\cos(4t) = -16x_1$ . For Object 2: equilibrium at $x = 5$ , displacement from equilibrium is $(x_2 - 5) = 2\sin(4t + \pi/3)$ , so $a_2 = -16 \cdot 2\sin(4t + \pi/3) = -16(x_2 - 5)$ . Both exhibit SHM with period $T = \pi/2$ . 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.
A mass attached to a vertical spring oscillates up and down. How does the presence of gravity affect the simple harmonic motion?
Answer: Gravity shifts the equilibrium position downward but does not change the harmonic nature of small oscillations about this new equilibrium Gravity shifts the equilibrium to where $kx_0 = mg$ , but oscillations about this new equilibrium still follow $F = -k(x - x_0)$ , maintaining SHM. The motion remains harmonic with the same frequency as horizontal oscillation. Choice A incorrectly suggests gravity prevents SHM. Choice C is wrong because the motion remains symmetric about the new equilibrium. Choice D incorrectly claims gravity changes the spring constant or frequency.
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?
Answer: Both exhibit simple harmonic motion but with different frequencies determined by their spring constants Both systems satisfy $F = -kx$ and exhibit SHM, but with different frequencies $\omega = \sqrt{k/m}$ . 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.
A mass-spring system is displaced from equilibrium and released. Which statement about the forces during the subsequent motion is correct?
Answer: The net force on the mass is always directed toward equilibrium and varies linearly with the displacement from equilibrium In SHM, the net force (not necessarily a single force) must be proportional to displacement and directed toward equilibrium: $F_{net} = -kx$ . This net force varies linearly with displacement. Choice A oversimplifies by considering only the spring force. Choice C incorrectly assumes other forces always cancel. Choice D is partially correct but less precise than B about the linear relationship and direction.

Start this quiz