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. When you master these equations, you're not just learning formulas; you're building the toolkit to analyze periodic motion, energy conservation, and wave behavior across multiple units of your course.

Here's 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 illustrates 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, while mass m provides inertia.

Angular Frequency

• $\omega = 2\pi f = \sqrt{k/m}$—the rate of oscillation measured in radians per second
• Spring constant (k) in the numerator means stiffer springs oscillate faster
• Mass (m) in the denominator means heavier objects oscillate slower—this inverse relationship is heavily tested

Period

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

Frequency

• $f = 1/T = \frac{1}{2\pi}\sqrt{k/m}$—cycles per second, measured in Hertz (Hz)
• Inversely related to period—if you know one, you immediately know the other
• Independent of amplitude—a key SHM property that distinguishes it from other types of motion

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Position, Velocity, and Acceleration: Describing the Motion

These three equations form a connected family—each is the derivative of the one before it. Understanding their phase relationships (when each reaches maximum or zero) is essential for exam success.

Displacement

• $x(t) = A\cos(\omega t + \phi)$—position as a function of time, where A is amplitude
• Amplitude (A) represents maximum displacement from equilibrium, set by initial conditions
• Phase constant ($\phi$) determines starting position—if $\phi = 0$, the object starts at maximum displacement

Velocity

• $v(t) = -A\omega\sin(\omega t + \phi)$—the derivative of position with respect to time
• Maximum velocity occurs at equilibrium ($x = 0$) where all energy is kinetic
• Velocity equals zero at 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—can also be written as $a = -\omega^2 x$
• Maximum acceleration 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—perfect 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, double the max speed
• Useful for energy calculations since all energy is kinetic at this instant

Maximum Acceleration

• $a_{max} = A\omega^2$—occurs at the turning points ($x = \pm A$)
• Proportional to $\omega^2$—angular frequency has a stronger effect than on velocity
• Equals the maximum restoring force divided by mass—connects directly to $F = ma$

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Energy in SHM: Conservation and Exchange

Energy analysis often provides the fastest path to solving SHM problems. Total mechanical energy remains constant throughout the motion, continuously converting between potential and kinetic forms.

Total Mechanical Energy

• $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$—constant throughout the motion, set entirely by amplitude
• Depends on amplitude squared—double the amplitude, quadruple the energy
• Two equivalent forms let you calculate energy using either spring constant or angular frequency

Potential Energy

• $PE = \frac{1}{2}kx^2$—energy stored in the spring due to displacement from equilibrium
• Maximum at turning points ($x = \pm A$) where $PE = \frac{1}{2}kA^2 = E_{total}$
• Zero at equilibrium ($x = 0$) where the spring is neither stretched nor compressed

Kinetic Energy

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

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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 restoring force is proportional to displacement will exhibit SHM.

Spring Force (Hooke's Law)

• $F = -kx$—force is proportional to displacement and opposite in direction
• Negative sign is critical—it indicates the force always pushes/pulls toward equilibrium
• Linear relationship between F and x is what makes the motion "simple"—nonlinear restoring forces produce more complex oscillations

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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!)

3. Compare and contrast the energy at $x = 0$ versus $x = A$. What's the same? What's different?

4. An FRQ gives you mass, spring constant, and amplitude, then asks for the speed when $x = A/2$. Which approach is faster: using the velocity equation $v(t)$ or using energy conservation? Set up both methods.

5. Why does the 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 principle does each negative sign represent?