MCAT topics → Translational Motion, Forces, Work, and Energy → Simple Harmonic Motion (Springs, Pendulums)

Simple Harmonic Motion (Springs, Pendulums)

MCAT trap: Thinks larger amplitude means longer period for a spring oscillator. The period of a spring-mass system (T = 2π√(m/k)) is independent of amplitude; only mass and spring constant determine the period.

Simple harmonic motion is what happens when a restoring force pulls an object back toward equilibrium in proportion to displacement — mathematically, F = -kx. The MCAT exploits two intuitions that are both wrong: that a bigger amplitude takes longer (it does not — period is independent of amplitude for springs), and that a heavier pendulum bob takes longer (it does not — mass cancels from the pendulum equation entirely). Springs and pendulums are the canonical examples, and the exam tests both in ways that go beyond plugging into formulas.

The exam hits this topic from several angles. Pure recall questions ask you to identify SHM conditions or state what the period depends on. Application questions give you a modified system — double the mass, cut the string length in half — and ask how the period changes. Passage-based questions might show you a graph of position, velocity, or acceleration versus time and ask you to extract phase relationships or identify where energy is maximized. That last category is where most students lose points, because it requires you to hold the energy picture and the graph picture in your head simultaneously.

What makes SHM tricky is that intuition routinely misleads you. Students expect that a bigger swing or a heavier bob must take longer — it feels right. It isn't. The period formulas are counterintuitive until you internalize why mass and amplitude drop out of the pendulum equation, and why amplitude drops out of the spring equation entirely. The MCAT will absolutely exploit these intuitions, so you need to understand the mechanics behind each formula, not just memorize it.

Common misconceptions

Common mistake
Wrong: Increasing the amplitude of a spring-mass system increases its period.
Right: The period of a spring-mass system (T = 2π√(m/k)) is independent of amplitude; only mass and spring constant determine the period.
The period formula T = 2π√(m/k) contains no amplitude term — this is not an accident or an approximation. A larger amplitude means the object travels farther, but the restoring force also scales up proportionally (because F = -kx), so the object moves faster and the extra distance is covered in exactly the same time. These two effects cancel perfectly, leaving period independent of amplitude. If you find yourself thinking 'bigger swing, longer period,' remind yourself that F = -kx makes the force and displacement grow together.
Common mistake
Wrong: A heavier pendulum bob swings with a longer period.
Right: The period of a simple pendulum (T = 2π√(L/g)) depends only on length and gravitational acceleration, not on the mass of the bob.
The restoring force on a pendulum bob is a component of gravity, which scales with mass (F = mg sinθ). But the inertia resisting that force also scales with mass. These two mass dependencies cancel exactly in the equation of motion, which is why T = 2π√(L/g) has no m in it. A heavier bob is harder to accelerate, but it's also pulled back harder — it's the same physics that makes all objects fall at the same rate in a gravitational field regardless of mass.
Common mistake
Wrong: Kinetic energy is maximum at the extremes of oscillation where the restoring force is greatest.
Right: Kinetic energy is maximum at the equilibrium position (where PE = 0 and speed is greatest); potential energy is maximum at the extremes.
At the extremes of oscillation the object is momentarily stopped — velocity is zero, so KE is zero. The restoring force is indeed greatest there (maximum displacement means maximum |F| = k|x|), but a large force doesn't mean large kinetic energy; it means large acceleration. The object is about to speed up from rest. KE peaks at the equilibrium position, where all the stored potential energy has been converted and speed is at its maximum. Don't conflate 'strongest force' with 'most kinetic energy.'
Common mistake
Wrong: Velocity and acceleration are both zero at the equilibrium position during SHM.
Right: At equilibrium, velocity is at its maximum and acceleration is zero (no net restoring force); acceleration is maximum at the extremes.
Acceleration is determined by net force (a = F/m = -kx/m). At the equilibrium position x = 0, so the restoring force is zero and acceleration is zero — but velocity is at its peak, not zero. The object is moving fastest through equilibrium. Zero acceleration means no force is changing the velocity at that instant; it does not mean the object is stopped. Velocity and acceleration are 90° out of phase in SHM: when one is at its maximum magnitude, the other is zero.
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What the exam tests

  1. Recognize that SHM requires a restoring force that is directly proportional to displacement and always directed toward equilibrium (F = -kx), and identify whether a described system qualifies as SHM.
  2. Calculate or compare periods for spring-mass systems using T = 2π√(m/k) and for simple pendulums using T = 2π√(L/g), including predicting how period changes when mass, spring constant, or string length is altered.
  3. Explain the continuous exchange between kinetic and potential energy during oscillation — knowing that KE is maximum and PE is zero at equilibrium, and that PE is maximum and KE is zero at the extremes.
  4. Read and interpret sinusoidal x(t), v(t), and a(t) graphs, including identifying the phase offset between them (v leads x by 90°, a is 180° out of phase with x) and linking graph features to physical quantities like amplitude and period.

Can you avoid these mistakes?

A spring-mass system oscillates with period T. If you double the amplitude while keeping the same spring and mass, what is the new period? Explain why using the force relationship, not just the formula.
Pendulum A has a 1 kg bob on a 0.5 m string. Pendulum B has a 4 kg bob on a 0.5 m string. Compare their periods and explain which physical principle is responsible for your answer.
At what point in its oscillation does a spring-mass system have (a) maximum kinetic energy, (b) maximum potential energy, and (c) maximum acceleration? Are any of these points the same location, and why?
A graph shows x(t) for a mass on a spring as a cosine wave starting at maximum displacement. Sketch what v(t) and a(t) look like on the same time axis. At t = 0, is acceleration positive, negative, or zero? Justify your answer using F = -kx.

Related topics

Kinetic and Potential Energy; Conservation of Energy One-Dimensional Kinematics Two-Dimensional Kinematics and Projectile Motion Newton's Three Laws of Motion Friction (Static and Kinetic)

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