Simple Harmonic Motion Examples

Why This Matters

Simple harmonic motion (SHM) is one of the most elegant patterns in physics—and one of the most heavily tested on AP Physics 1. When you see oscillating systems on the exam, you're being tested on your understanding of restoring forces, energy transformations, and the mathematical relationships that govern periodic motion. The key insight is that SHM occurs whenever a restoring force is proportional to displacement, whether that force comes from a spring, gravity, or even fluid pressure.

Don't just memorize the period formulas for each system. Instead, focus on why each system oscillates and how energy moves between kinetic and potential forms. Every SHM problem connects back to the same core principle: a displaced object experiences a force that pulls it back toward equilibrium, overshoots, and repeats. Master this concept, and you'll recognize SHM whether it appears as a pendulum, a spring, or a completely novel scenario on the FRQ.

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Spring-Based Oscillators

These systems demonstrate SHM in its purest form. The restoring force follows Hooke's Law ($F = -kx$), making the motion perfectly sinusoidal. The negative sign indicates the force always opposes displacement—this is what creates oscillation rather than runaway motion.

Mass on a Horizontal Spring

• Period formula $T = 2\pi\sqrt{\frac{m}{k}}$—notice that period depends only on mass and spring constant, not amplitude
• Hooke's Law provides the restoring force with $F = -kx$, where the negative sign indicates force opposes displacement
• Energy oscillates between elastic potential energy ($U_s = \frac{1}{2}kx^2$) and kinetic energy ($K = \frac{1}{2}mv^2$) with total mechanical energy conserved

Mass on a Vertical Spring

• Same period formula $T = 2\pi\sqrt{\frac{m}{k}}$ as horizontal—gravity only shifts the equilibrium position, not the oscillation characteristics
• New equilibrium position occurs where spring force balances weight: $kx_{eq} = mg$
• Both gravitational and elastic potential energy contribute, but when measured from the new equilibrium, the math simplifies to match the horizontal case

Bungee Jumper

• Combines free fall and spring-like oscillation—the cord only provides restoring force when stretched beyond its natural length
• Non-ideal SHM because the restoring force is asymmetric: gravity acts throughout, but elastic force only acts when the cord is taut
• Real-world complexity includes damping and variable $k$ as the cord stretches, making this an excellent conceptual extension of ideal spring systems

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Gravity-Driven Oscillators

In these systems, gravity provides the restoring force rather than elasticity. The key requirement for SHM is that the restoring force must be proportional to displacement—this only holds for small oscillations in pendulum systems.

Simple Pendulum

• Period formula $T = 2\pi\sqrt{\frac{L}{g}}$—depends on length and gravitational acceleration, but not on mass or amplitude (for small angles)
• SHM approximation valid for small angles (less than ~15°) where $\sin\theta \approx \theta$ in radians
• Restoring force is the tangential component of gravity ($F = -mg\sin\theta$), which becomes proportional to displacement only when the angle is small

Oscillating Liquid in a U-Tube

• Period approximates $T = 2\pi\sqrt{\frac{L}{g}}$ where $L$ is the total length of the liquid column
• Gravity creates the restoring force—when liquid rises on one side, the pressure difference pushes it back toward equilibrium
• Demonstrates SHM in fluids by following the same mathematical pattern as a simple pendulum, connecting mechanics to fluid behavior

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Rotational and Torsional Oscillators

These systems extend SHM into rotational motion, where angular displacement replaces linear displacement and torque replaces force. The restoring torque must be proportional to angular displacement for the motion to be simple harmonic.

Torsional Pendulum

• Period formula $T = 2\pi\sqrt{\frac{I}{\kappa}}$ where $I$ is moment of inertia and $\kappa$ is the torsional constant of the wire
• Restoring torque $\tau = -\kappa\theta$ is the rotational analog of Hooke's Law, with torque proportional to angular displacement
• Used experimentally to measure moment of inertia—by measuring period and knowing $\kappa$, you can calculate $I$ for complex objects

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Wave-Producing Oscillators

These systems demonstrate how localized SHM creates traveling waves. Each point on the oscillator undergoes simple harmonic motion, but the collective behavior produces wave phenomena like standing waves and harmonics.

Vibrating String (Guitar String)

• Wave speed $v = \sqrt{\frac{T}{\mu}}$ where $T$ is tension and $\mu$ is mass per unit length—connects string properties to wave behavior
• Standing waves form with nodes (no motion) at fixed ends and antinodes (maximum motion) between them
• Fundamental frequency and harmonics depend on string length, tension, and linear mass density—changing any of these tunes the instrument

Vibrating Tuning Fork

• Produces a pure, consistent frequency determined by the fork's material properties, dimensions, and geometry
• Prongs undergo SHM as they flex back and forth, with the restoring force coming from the elasticity of the metal
• Standard frequency reference because the oscillation is highly stable and minimally affected by external conditions

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Electrical Oscillators

The LC circuit extends SHM concepts beyond mechanics into electromagnetism. Energy oscillates between electric field (capacitor) and magnetic field (inductor) just as mechanical energy oscillates between potential and kinetic forms.

LC Circuit

• Natural frequency $f = \frac{1}{2\pi\sqrt{LC}}$ where $L$ is inductance and $C$ is capacitance—the electrical analog of mechanical oscillation
• Energy oscillates between capacitor (storing energy in electric field, $U_E = \frac{1}{2}CV^2$) and inductor (storing energy in magnetic field, $U_B = \frac{1}{2}LI^2$)
• Damping occurs from resistance in real circuits, causing oscillations to decay—analogous to friction in mechanical systems

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Quick Reference Table

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Self-Check Questions

1. A simple pendulum and a mass-spring system have the same period on Earth. If both are taken to the Moon (where $g$ is smaller), which system's period changes, and why?

2. Compare the energy transformations in a mass-spring system and an LC circuit. What plays the role of kinetic energy in the electrical system? What plays the role of potential energy?

3. Why does the period of a vertical mass-spring system equal that of a horizontal mass-spring system, even though gravity acts on the vertical one?

4. A student claims that doubling the amplitude of a simple pendulum will double its period. Explain why this claim is incorrect for small amplitudes, and under what conditions it might become partially true.

5. Identify two systems from this guide where the period formula has the form $T = 2\pi\sqrt{\frac{\text{something}}{g}}$. What do these systems have in common that explains this similarity?