Simple Harmonic Motion Equations

Why This Matters

Simple harmonic motion (SHM) is the foundation for understanding nearly every oscillating system you'll encounter in physics, from masses on springs to pendulums, sound waves, and even the behavior of atoms in solids. Mastering these equations gives you the toolkit to analyze periodic motion, energy conservation, and wave behavior across multiple units of your course.

What you're really being tested on: the relationships between position, velocity, acceleration, and energy as a system oscillates, and how physical parameters like mass and spring constant determine the motion's characteristics. Don't just memorize each equation in isolation. Know what concept each equation captures and how they connect to each other. When you see a spring problem on an exam, you should immediately recognize which equation applies based on what quantity you're solving for.

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System Parameters: What Determines the Motion

Before you can describe how something oscillates, you need to understand what controls the oscillation. These equations define the fundamental characteristics of any SHM system based on its physical properties. The spring constant $k$ measures stiffness (how hard the spring pushes back), while mass $m$ provides inertia (how much the object resists being accelerated).

Angular Frequency

• $\omega = 2\pi f = \sqrt{k/m}$ gives the rate of oscillation in radians per second
• $k$ in the numerator means stiffer springs oscillate faster
• $m$ in the denominator means heavier objects oscillate slower. This inverse relationship is heavily tested.

Period

• $T = 2\pi\sqrt{m/k} = 1/f$ is the time for one complete oscillation cycle
• Heavier masses increase the period because more inertia means a slower response to the restoring force
• Stiffer springs decrease the period because the stronger restoring force accelerates the mass back toward equilibrium more quickly

Frequency

• $f = 1/T = \frac{1}{2\pi}\sqrt{k/m}$ gives cycles per second, measured in Hertz (Hz)
• Inversely related to period. If you know one, you immediately know the other.
• Independent of amplitude. This is a defining SHM property. A system oscillating with a large amplitude has the same frequency as one oscillating with a small amplitude, as long as the motion remains simple harmonic.

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Position, Velocity, and Acceleration: Describing the Motion

These three equations form a connected family. Each is the time derivative of the one before it (velocity is the derivative of position, acceleration is the derivative of velocity). Understanding their phase relationships, meaning when each reaches its maximum or zero, is essential for exam success.

Displacement

• $x(t) = A\cos(\omega t + \phi)$ gives position as a function of time
• Amplitude ($A$) is the maximum displacement from equilibrium, determined by initial conditions (how far you pull the spring, for example)
• Phase constant ($\phi$) sets the starting position. If $\phi = 0$, the object starts at maximum positive displacement ($x = A$ at $t = 0$). If the object starts at equilibrium and moves in the positive direction, you'd use $\phi = -\pi/2$ (which effectively turns the cosine into a sine).

Velocity

• $v(t) = -A\omega\sin(\omega t + \phi)$ is the time derivative of position
• Maximum speed occurs at equilibrium ($x = 0$), where all the system's energy is kinetic
• Velocity equals zero at the turning points ($x = \pm A$), where the object momentarily stops before reversing

Acceleration

• $a(t) = -A\omega^2\cos(\omega t + \phi)$ always points toward equilibrium. This is the defining feature of SHM.
• Proportional to displacement but opposite in sign. You can also write this as $a = -\omega^2 x$.
• Maximum acceleration occurs at maximum displacement, where the restoring force is strongest.

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Maximum Values: The Extremes of Motion

These simplified equations give you the magnitude of the largest velocity and acceleration without worrying about time dependence. They're useful for quick calculations when you only need the extreme values.

Maximum Velocity

• $v_{max} = A\omega$ occurs as the object passes through equilibrium
• Directly proportional to both amplitude and angular frequency. Double either one, and you double the max speed.
• Useful for energy calculations since all energy is kinetic at this instant: $KE_{max} = \frac{1}{2}mv_{max}^2 = E_{total}$

Maximum Acceleration

• $a_{max} = A\omega^2$ occurs at the turning points ($x = \pm A$)
• Proportional to $\omega^2$, so angular frequency has a stronger effect on acceleration than on velocity
• Connects directly to Newton's second law: $F_{max} = ma_{max} = m(A\omega^2) = kA$

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Energy in SHM: Conservation and Exchange

Energy analysis often provides the fastest path to solving SHM problems. Total mechanical energy stays constant throughout the motion, continuously converting between potential and kinetic forms. No energy is lost (assuming no friction or damping).

Total Mechanical Energy

• $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ is constant throughout the motion, set entirely by amplitude
• Depends on amplitude squared. Double the amplitude, and you quadruple the total energy.
• The two equivalent forms let you calculate energy using either the spring constant or the angular frequency, depending on what information you're given.

Potential Energy

• $PE = \frac{1}{2}kx^2$ is the elastic energy stored in the spring due to displacement from equilibrium
• Maximum at the turning points ($x = \pm A$), where $PE = \frac{1}{2}kA^2 = E_{total}$
• Zero at equilibrium ($x = 0$), where the spring is at its natural length

Kinetic Energy

• $KE = \frac{1}{2}mv^2$ is the energy of motion, maximum when the object moves fastest
• Maximum at equilibrium, where $KE = \frac{1}{2}mv_{max}^2 = E_{total}$
• Zero at the turning points, where the object momentarily stops

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The Restoring Force: What Makes It "Simple Harmonic"

This single equation is the defining condition for simple harmonic motion. Any system where the net restoring force is proportional to displacement (and opposite in direction) will exhibit SHM.

Spring Force (Hooke's Law)

• $F = -kx$ says force is proportional to displacement and opposite in direction
• The negative sign is critical. It tells you the force always pushes or pulls the object back toward equilibrium. Displace the object in the positive direction, and the force acts in the negative direction (and vice versa).
• The linear relationship between $F$ and $x$ is what makes the motion "simple." If the restoring force depended on $x^2$ or some other nonlinear function, the oscillation would be more complex and these neat equations wouldn't apply.

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Quick Reference Table

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Self-Check Questions

1. If you double the mass attached to a spring, what happens to the period? What happens to the maximum velocity if amplitude stays the same?

2. At what position in the oscillation cycle are velocity and acceleration both at their maximum magnitudes? (Trick question. Think carefully about the phase relationship.)

3. Compare the total energy, PE, and KE at $x = 0$ versus $x = A$. What's the same? What's different?

4. You're given mass, spring constant, and amplitude, then asked for the speed when $x = A/2$. Which approach is faster: using the velocity equation $v(t)$ or using energy conservation? Set up both and compare. (Hint: the energy method doesn't require you to find $t$.)

5. Why does a negative sign appear in $F = -kx$, $v(t) = -A\omega\sin(\omega t + \phi)$, and $a(t) = -A\omega^2\cos(\omega t + \phi)$? What physical meaning does each negative sign carry?