Simple Harmonic Motion Examples

Why This Matters

Simple harmonic motion (SHM) is one of the most common patterns in physics and one of the most heavily tested topics 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. 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. 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, which 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. A larger amplitude means the object moves faster through the middle to compensate for the longer path, keeping the period the same.
• Hooke's Law provides the restoring force with $F = -kx$. The force is zero at equilibrium and strongest at maximum displacement.
• Energy oscillates between elastic potential energy ($U_s = \frac{1}{2}kx^2$) and kinetic energy ($K = \frac{1}{2}mv^2$). At maximum displacement, all energy is potential. At equilibrium, all energy is kinetic. Total mechanical energy stays constant throughout.

Mass on a Vertical Spring

• Same period formula: $T = 2\pi\sqrt{\frac{m}{k}}$. Gravity only shifts the equilibrium position downward; it doesn't change the oscillation characteristics.
• New equilibrium position occurs where the spring force balances the object's weight: $kx_{eq} = mg$. The object oscillates symmetrically around this lower point.
• Both gravitational and elastic potential energy contribute to the total energy. But when you measure displacement from the new equilibrium, the math simplifies to match the horizontal case exactly.

Bungee Jumper

• Combines free fall and spring-like oscillation. The cord only provides a restoring force when stretched beyond its natural length.
• Not true SHM because the restoring force is asymmetric: gravity acts throughout the motion, but the elastic force only acts when the cord is taut. During the portion of the bounce where the cord is slack, the jumper is simply in free fall.
• Real-world complexity includes damping and a potentially variable $k$ as the cord stretches, making this a useful conceptual extension of ideal spring systems but not a clean SHM example.

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Gravity-Driven Oscillators

In these systems, gravity provides the restoring force rather than elasticity. For the motion to qualify as SHM, the restoring force must be proportional to displacement. This only holds for small oscillations in pendulum systems.

Simple Pendulum

• Period formula: $T = 2\pi\sqrt{\frac{L}{g}}$. This depends on string length and gravitational acceleration, but not on mass or amplitude (for small angles). That mass-independence surprises many students, but it follows directly from the fact that both the inertia and the restoring force scale with $m$.
• SHM approximation is valid for small angles (roughly less than 15°) where $\sin\theta \approx \theta$ in radians. At larger angles, the restoring force grows more slowly than displacement, so the period gets longer.
• The restoring force is the tangential component of gravity: $F = -mg\sin\theta$. This only becomes proportional to angular displacement $\theta$ when the small-angle approximation applies.

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 in the tube.
• Gravity creates the restoring force. When liquid is pushed higher on one side, the weight difference between the two columns creates a pressure imbalance that pushes the liquid back toward equilibrium.
• This demonstrates SHM in fluids and follows 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 the moment of inertia and $\kappa$ (kappa) is the torsional constant of the wire. A stiffer wire (larger $\kappa$) gives a shorter period, just as a stiffer spring gives a shorter period.
• Restoring torque: $\tau = -\kappa\theta$. This is the rotational analog of Hooke's Law, with torque proportional to angular displacement.
• Used experimentally to measure moment of inertia. By measuring the period and knowing $\kappa$, you can calculate $I$ for complex objects that would be difficult to analyze geometrically.

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Wave-Producing Oscillators

These systems demonstrate how localized SHM creates traveling waves. Each small segment of 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 linear mass density (mass per unit length). Higher tension or lower mass density means faster waves.
• Standing waves form with nodes (points of zero motion) at the fixed ends and antinodes (points of maximum motion) between them. The simplest pattern has just one antinode in the middle.
• Fundamental frequency and harmonics depend on string length, tension, and linear mass density. Tightening a guitar string raises the tension, increases wave speed, and raises the pitch.

Vibrating Tuning Fork

• Produces a pure, consistent frequency determined by the fork's material properties, dimensions, and geometry.
• The prongs undergo SHM as they flex back and forth, with the restoring force coming from the elasticity of the metal itself.
• Serves as a frequency reference because the oscillation is highly stable and minimally affected by external conditions like temperature or humidity.

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Electrical Oscillators

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

Note: LC circuits are not part of the AP Physics 1 curriculum, but they appear frequently in Physics C and serve as a powerful conceptual bridge for understanding SHM universality.

LC Circuit

• Natural frequency: $f = \frac{1}{2\pi\sqrt{LC}}$ where $L$ is inductance and $C$ is capacitance.
• Energy oscillates between the capacitor (storing energy in an electric field, $U_E = \frac{1}{2}CV^2$) and the inductor (storing energy in a magnetic field, $U_B = \frac{1}{2}LI^2$). This is directly analogous to energy swapping between potential and kinetic forms in a mass-spring system.
• Damping occurs from resistance in real circuits, causing oscillations to decay over time, 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?