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Physics C: Mechanics Unit 8: Oscillations and Simple Harmonic Motion

Whenever a restoring force is proportional to displacement, the motion is simple harmonic. This unit solves the SHM equation, relates period and energy to physical properties, and applies the method to pendulums and small oscillations.

Covers: Conditions for simple harmonic motion, Mass–spring systems, Solving x(t) and phase, Energy in SHM, Simple, physical and torsion pendulums, Small oscillations about a potential minimum, Damping

Study guide

Simple harmonic motion happens when the net force (or torque) is a linear restoring force: F = −kx. Newton's second law gives d²x/dt² = −(k/m)x, whose general solution is x(t) = A cos(ωt + φ) with ω = √(k/m). The amplitude A and phase constant φ come from the initial position and velocity. The period T = 2π/ω and frequency f = 1/T do not depend on amplitude. The key test for SHM is to write the equation of motion in the form d²x/dt² = −ω²x; whatever multiplies x is ω².

Differentiating x(t) gives v(t) = −Aω sin(ωt + φ) and a(t) = −Aω² cos(ωt + φ) = −ω²x. So the maximum speed is Aω, reached at equilibrium, and the maximum acceleration is Aω², reached at the turning points. Velocity leads position by a quarter cycle. To find φ, use x(0) = A cos φ and the sign of v(0) = −Aω sin φ; two phases fit any starting position, and the velocity tells them apart.

Energy is a powerful shortcut. The total energy is E = ½kA², constant in an ideal oscillator, and it trades between kinetic and potential energy: ½mv² + ½kx² = ½kA². That gives v = ω√(A² − x²) at any position. Potential energy is proportional to x², so at x = A/2 only a quarter of the energy is potential. For a vertical spring, gravity just shifts the equilibrium by mg/k and leaves the period unchanged. Springs in parallel add constants; springs in series add reciprocals.

Pendulums oscillate under a restoring torque. A simple pendulum has T = 2π√(L/g) for small angles, independent of mass. A physical pendulum, an extended object swinging about a pivot a distance d from its center of mass, has T = 2π√(I/(Mgd)), where I is about the pivot. A torsion pendulum, twisted by a wire with τ = −κθ, has T = 2π√(I/κ). More generally, any particle near a stable minimum of a potential U(x) undergoes SHM with effective spring constant k = d²U/dx² at the minimum.

Real oscillators lose energy. With a light damping force −bv, the motion is still nearly sinusoidal but the amplitude decays as e^(−bt/2m) and the energy as e^(−bt/m). Heavy damping prevents oscillation entirely; critical damping returns to equilibrium fastest without overshooting. A driven oscillator responds most strongly at resonance, when the driving frequency is close to the natural frequency, and lighter damping gives a taller, sharper resonance peak. Collisions with oscillators combine ideas: use momentum for the collision, then find the new amplitude from the new speed and new ω.

Key ideas

• SHM results from a linear restoring force or torque.
• Write the equation of motion as d²x/dt² = −ω²x to read off ω.
• Period does not depend on amplitude in SHM.
• Maximum speed is Aω at equilibrium; maximum acceleration is Aω² at the turning points.
• Total energy ½kA² is shared between kinetic and potential energy.
• Physical and torsion pendulums use I in place of mass.
• Small oscillations about any potential minimum are SHM with k = U″.

Formulas and rules

Common mistakes

• Confusing ω (rad/s) with f (Hz) and forgetting the factor 2π.
• Thinking a larger amplitude changes the period.
• Using the simple pendulum formula for an extended object.
• Thinking a vertical spring's period depends on g.
• Choosing the wrong phase by ignoring the sign of the initial velocity.

Before the test, make sure you can

• Show that a system undergoes SHM and identify ω from its equation of motion.
• Write x(t), v(t) and a(t), and find A and φ from initial conditions.
• Calculate period, frequency, maximum speed and maximum acceleration.
• Use energy to find speed at any position.
• Find periods of simple, physical and torsion pendulums.
• Find the small-oscillation period about a minimum of U(x).
• Combine collisions with SHM to find a new amplitude.
• Describe the effects of damping on amplitude and energy.

Words to know

Simple harmonic motion

Oscillation produced by a restoring force proportional to displacement.

Amplitude

The maximum displacement from equilibrium.

Period

The time for one complete cycle, T = 2π/ω.

Frequency

Cycles per second, f = 1/T, measured in hertz.

Angular frequency

ω = 2πf, in rad/s; for a spring, ω = √(k/m).

Phase constant

The angle φ in x = A cos(ωt + φ), set by initial conditions.

Restoring force

A force that points back toward equilibrium.

Simple pendulum

A point mass on a light string; T = 2π√(L/g) for small angles.

Physical pendulum

An extended rigid body swinging about a pivot; T = 2π√(I/(Mgd)).

Torsion pendulum

A body twisting on a wire with restoring torque −κθ; T = 2π√(I/κ).

Effective spring constant

The curvature U″ at a potential minimum, which sets small-oscillation frequency.

Damping

Energy loss from a resistive force, which makes the amplitude decay.

Critical damping

The damping level that returns a system to equilibrium fastest without oscillating.

Resonance

Large-amplitude response when the driving frequency matches the natural frequency.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.