Difficulty: Intermediate | Prerequisites: basic mechanics (Newton's laws, energy conservation), trigonometry, unit conversions
This material forms the backbone of PHYS 116's first midterm. Oscillatory motion and simple harmonic motion (SHM) appear everywhere in physics, from springs and pendulums to electrical circuits and quantum mechanics. You need to be comfortable converting between period, frequency and angular frequency, and you should know how position, velocity and acceleration relate to one another in SHM. If you are behind on energy conservation or basic trigonometric identities, review those before diving in here.
SHM describes any system where the restoring force is proportional to displacement. The motion is sinusoidal, fully described by amplitude, angular frequency and phase. Energy swaps back and forth between kinetic and potential, and the total mechanical energy stays constant in the absence of damping.
Period (T)
The time for one complete cycle of oscillation, measured in seconds. Think of it as: how long until the motion repeats itself exactly.
Frequency (f)
The number of complete cycles per second, measured in hertz (Hz). In simple terms, this means how many times the object goes back and forth each second. Related to period by f = 1/T.
Angular frequency (ω)
The rate of change of phase angle, measured in radians per second. ω = 2πf = 2π/T. Think of it as: frequency expressed in radians rather than full cycles.
Amplitude (A)
The maximum displacement from the equilibrium position. In simple terms, this means how far the object moves from centre at its greatest extent.
Simple harmonic motion (SHM)
Oscillatory motion in which the restoring force is directly proportional to the displacement from equilibrium and directed towards it: F = −kx. Think of it as: the idealised version of any back-and-forth motion near a stable equilibrium.
Spring constant (k)
A measure of a spring's stiffness, in N/m. The larger k is, the stiffer the spring and the greater the restoring force for a given displacement.
Physical pendulum
Any rigid body that oscillates about a pivot point that is not at its centre of mass. Its period depends on the moment of inertia about the pivot and the distance from the pivot to the centre of mass.
Damping constant (τ)
The time constant that describes how quickly the amplitude of a damped oscillator decays. The amplitude envelope follows A(t) = A₀ e^(−t/2τ). A larger τ means slower energy loss.
Beats per minute (bpm)
A unit of frequency commonly used in music. 90 bpm = 90 beats / 60 s = 1.5 Hz.
Period T = 1/f. Frequency f = 1/T. Angular frequency ω = 2πf = 2π/T.
When given bpm, convert: f = bpm / 60. Then T = 60 / bpm.
When given Hz, T = 1/f directly.
When given laps/minute, that is already a frequency in units of min⁻¹. Convert to seconds: T = 60 / (laps per minute).
When given steps/second, that is frequency in Hz directly.
To rank periods from largest to smallest, convert everything to T in seconds first, then order.
Standard form: x(t) = A cos(ωt + φ)
Velocity: v(t) = −Aω sin(ωt + φ)
Acceleration: a(t) = −Aω² cos(ωt + φ) = −ω² x(t)
If you are given a(t) = −(7.40 m/s²) cos(1.57 rad/s · t), you can read off ω = 1.57 rad/s and Aω² = 7.40 m/s².
The amplitude is A = 7.40 / ω² = 7.40 / (1.57)² ≈ 3.0 m.
The period is T = 2π/ω = 2π/1.57 ≈ 4.0 s.
The position graph must be a cosine with amplitude ≈ 3.0 m and period ≈ 4.0 s. Because a(t) has a negative cosine, x(t) is a positive cosine (they share the same phase but a(t) = −ω²x).
If x(0) = +A, the position function is x(t) = A cos(ωt) with no phase shift.
Speed is |v(t)| = Aω |sin(ωt)|. The velocity is v(t) = −Aω sin(ωt).
To find when the speed first equals Aω/4: solve |sin(ωt)| = 1/4.
Pay attention to whether the problem asks for speed (always positive) or velocity (signed). A "speed of −Aω/4" likely means a velocity of −Aω/4, so sin(ωt) = +1/4 (since v = −Aω sin).
ωt = arcsin(1/4) ≈ 0.2527 rad. Then t = 0.2527/ω = 0.2527 T/(2π) ≈ 0.04 T.
Total mechanical energy: E = ½kA² = ½mv²_max
At equilibrium (x = 0), all energy is kinetic: ½mv² = ½kA²
Solving for amplitude: A = v_max √(m/k)
Example: m = 0.250 kg, k = 15.0 N/m, v = 3.50 m/s at equilibrium. A = 3.50 × √(0.250/15.0) = 3.50 × 0.1291 ≈ 0.45 m.
Period: T = 2π √(I / (mgd)), where I is the moment of inertia about the pivot and d is the distance from the pivot to the centre of mass.
For a solid disc pivoted at R/2 from centre: I = ¾mR² (given), d = R/2.
T = 2π √( (¾mR²) / (mg · R/2) ) = 2π √( (¾R) / (g/2) ) = 2π √(3R / (2g)).
The amplitude of a damped oscillator decays as A(t) = A₀ e^(−t/(2τ)).
To find τ from a graph, read off two amplitude values at known times and use: A₂/A₁ = e^(−(t₂ − t₁)/(2τ)).
Taking the natural log: ln(A₂/A₁) = −(t₂ − t₁)/(2τ), so τ = −(t₂ − t₁) / (2 ln(A₂/A₁)).
From the practice exam graph, the amplitude drops from about 0.30 rad to roughly 0.10 rad over about 2.5 minutes (150 s). ln(0.10/0.30) = −1.099. τ = −150 / (2 × (−1.099)) ≈ 68 s. (The exact answer depends on precise readings; the exam choices suggest reading more carefully from the graph.)
Potential energy: U = ½kx². This is a parabola centred at x = 0.
Kinetic energy: K = E_total − U = ½kA² − ½kx². This is an inverted parabola, zero at x = ±A and maximum at x = 0.
From the energy graph, you can read the spring constant: k = 2U/x² at any point on the curve.
The oscillator with a steeper (narrower) parabola has a larger spring constant.
Maximum speed: v_max = A√(k/m). If two oscillators have the same total energy but different masses and spring constants, you need to know both k and m to compare their maximum speeds.
T = \frac{1}{f} = \frac{2\pi}{\omega}x(t) = A \cos(\omega t + \phi)v(t) = -A\omega \sin(\omega t + \phi)a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)T_{\text{spring}} = 2\pi \sqrt{\frac{m}{k}}T_{\text{phys. pend.}} = 2\pi \sqrt{\frac{I}{mgd}}E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}mv_{\max}^2A(t) = A_0\, e^{-t/(2\tau)}Spring-mass oscillators model everything from car suspensions to the vibration of atoms in a crystal lattice. Understanding the physical pendulum is how engineers design clock mechanisms and seismometers. Damping is central to any shock absorber: the goal is to dissipate energy at just the right rate so the system returns to rest without bouncing too long (underdamped) or too sluggishly (overdamped).
Students often confuse frequency and period. If something has a high frequency, it has a short period, not a long one.
Students sometimes forget that angular frequency ω is not the same as frequency f. ω = 2πf, so ω is always larger than f by a factor of 2π.
A common error is thinking that at maximum displacement the velocity is at its maximum. The opposite is true: velocity is zero at the turning points (x = ±A) and maximum at the equilibrium position.
When using the physical pendulum formula, students frequently plug in the moment of inertia about the centre of mass rather than about the pivot. The formula requires I about the pivot point.
⚠️ Unit conversion questions (bpm to Hz, laps/minute to period) are a near-certainty. Have the conversions drilled.
⚠️ Expect a question where you are given a(t) or v(t) and must identify the correct x(t) graph. Know that a = −ω²x, so position and acceleration are always opposite in sign.
⚠️ The physical pendulum formula is commonly tested. Memorise T = 2π√(I/(mgd)) and be ready to substitute given values for I and d.
⚠️ Energy problems where you solve for amplitude from v_max at equilibrium are a staple. The key equation is ½mv² = ½kA².
⚠️ Damped oscillation problems may give you a graph and ask for the damping time constant. Know how to read two amplitude values and use the exponential decay formula.
True or false: A pendulum with a longer period swings more slowly. (True)
Fill in the blank: The maximum speed of a mass on a spring occurs at the ______ position. (equilibrium)
True or false: Doubling the mass on a spring doubles the period. (False, it increases the period by a factor of √2)
Fill in the blank: For SHM, acceleration is always directed towards ______. (the equilibrium position)
True or false: In a damped system, the frequency of oscillation changes significantly with light damping. (False, lightly damped systems oscillate at nearly the natural frequency)
Q: Rank these by period from largest to smallest: (1) 90 bpm drum beat, (2) 80 Hz hummingbird wings, (3) 0.4 laps/min F1 car, (4) 3 steps/s running cadence.
A: Convert each to period. (1) T = 60/90 = 0.667 s. (2) T = 1/80 = 0.0125 s. (3) T = 60/0.4 = 150 s. (4) T = 1/3 = 0.333 s. Ranking: 3 > 1 > 4 > 2. Answer E.
Q: Given a(t) = −(7.40 m/s²) cos(1.57 rad/s · t), what does the position-vs-time graph look like?
A: Since a = −ω²x, we have x(t) = (7.40/1.57²) cos(1.57t) ≈ 3.0 cos(1.57t). The graph is a cosine wave with amplitude ≈ 3.0 m and period ≈ 4.0 s. Answer C (the graph showing amplitude ~3 m, period ~4 s, starting at a positive maximum).
Q: A block starts at x = +A at t = 0 with period T. When does the speed first reach Aω/4?
A: v(t) = −Aω sin(ωt). Speed = Aω|sin(ωt)| = Aω/4 when |sin(ωt)| = 0.25. ωt = arcsin(0.25) ≈ 0.2527 rad. t = 0.2527/(2π/T) = 0.2527T/(2π) ≈ 0.04T. Answer A.
Q: A 0.250 kg block oscillates on a spring (k = 15.0 N/m). Speed at equilibrium is 3.50 m/s. What is the amplitude?
A: At equilibrium, ½mv² = ½kA². A = v√(m/k) = 3.50 × √(0.250/15.0) = 3.50 × 0.1291 ≈ 0.45 m. Answer B.
Q: A solid disc (mass m, radius R) is pivoted at R/2 from centre. I about pivot = ¾mR². What is the period?
A: T = 2π√(I/(mgd)) = 2π√((¾mR²)/(mg(R/2))) = 2π√(3R/(2g)). Answer A.
This material connects directly to waves (the next major topic in PHYS 116) because a wave is essentially many coupled oscillators. The mathematics of SHM, particularly sinusoidal functions and angular frequency, carries over unchanged to describe travelling waves and standing waves.
Energy conservation in SHM is a specific case of the broader energy methods from mechanics. If you are comfortable with kinetic and potential energy swaps here, you will find wave energy and sound intensity more intuitive.
Damped oscillations lead naturally into driven oscillations and resonance, which explain phenomena from bridge collapses to how radios tune to a specific station.
simple harmonic motion, SHM, oscillation, spring-mass system, Hooke's law, restoring force, period, frequency, angular frequency, amplitude, phase, physical pendulum, moment of inertia, damped oscillation, damping constant, time constant, exponential decay, energy in SHM, potential energy parabola, kinetic energy, equilibrium position, turning points, bpm to Hz conversion, PHYS 116 midterm 1, University of Washington general physics