Difficulty: Intermediate | Prerequisites: Newton's laws, free-body diagrams, basic trigonometry, energy conservation (Chapters 5, 10, 11)
Chapter 14 introduces oscillatory motion, the back-and-forth movement you see in pendulums, vibrating strings and masses bouncing on springs. It sits at the intersection of Newton's second law and energy conservation: the restoring force that pulls an object back toward equilibrium, combined with the object's inertia, produces repetitive motion described by sine and cosine functions. You will need a solid grasp of free-body diagrams and the work-energy theorem from earlier chapters. Everything here feeds directly into wave motion (Chapter 15) and, further on, into AC circuits and quantum mechanics.
An oscillation happens whenever a restoring force pulls an object back toward equilibrium and inertia carries it past. If that restoring force is proportional to displacement (a linear restoring force), the result is simple harmonic motion (SHM), whose position follows a cosine curve in time. Period and frequency depend on the spring constant and mass (for springs) or length and gravity (for pendulums), but never on amplitude.
Oscillation
A repetitive combination of a restoring force and inertia that produces back-and-forth motion about an equilibrium position. In simple terms, any motion that keeps repeating the same path, like a swinging pendulum or a bobbing spring.
Equilibrium position
The location where the net force on the object is zero. In a mass-spring system, it is the point where the spring is at its natural (unstretched) length or, for a vertical spring, where the spring force balances gravity. Think of it as the "home base" the object keeps passing through.
Restoring force
The net force that always points back toward the equilibrium position. The farther the object is displaced, the stronger this force becomes. In simple terms, the force that pulls you back whenever you move away from centre.
Period (T)
The time, in seconds, to complete one full cycle of oscillation. Think of it as the round-trip time: out, back, and out the other way, then back again.
Frequency (f)
The number of complete cycles per second, measured in hertz (Hz). Related to period by f = 1/T. In simple terms, how many times per second the motion repeats.
Amplitude (A)
The maximum displacement from the equilibrium position. It is always positive and represents the farthest the object travels on either side. Think of it as how far the swing goes before turning around.
Simple harmonic motion (SHM)
Oscillation about an equilibrium position under a linear restoring force, meaning the force is directly proportional to displacement. The resulting motion traces out a sinusoidal (sine or cosine) curve over time. In simple terms, the "purest" kind of oscillation, where the position graph is a smooth wave.
Spring constant (k)
A measure of a spring's stiffness, in units of N/m. A larger k means a stiffer spring that exerts more force for a given stretch or compression. Think of it as how hard the spring pushes back per metre you pull it.
Angular frequency (omega, ω)
The rate of oscillation expressed in radians per second, defined as ω = 2πf = 2π/T. For a mass on a spring, ω = √(k/m). In simple terms, frequency scaled up by 2π so it fits neatly into trig functions.
Small-angle approximation
For a pendulum, the assumption that sin θ ≈ θ (in radians) when θ is roughly 10° or less. This turns the pendulum's restoring force into a linear restoring force, making it SHM. Think of it as the trick that lets you treat a pendulum like a spring, but only for small swings.
An object at its equilibrium position experiences zero net force. Displace it, and a restoring force appears, directed back toward equilibrium.
The restoring force grows with displacement: the farther the object is pulled away, the stronger the pull back.
Inertia carries the object through the equilibrium point and out the other side, where the restoring force reverses direction and the cycle repeats.
This combination of restoring force and inertia is what produces oscillation.
Period and frequency
Period T is measured in seconds per cycle.
Frequency f is measured in cycles per second (hertz, Hz).
The two are reciprocals: f = 1/T and T = 1/f.
To convert frequency in Hz to beats per minute, multiply by 60. For example, f = 1.45 Hz gives 1.45 × 60 = 87 beats/min.
Any oscillation about an equilibrium position with a linear restoring force (force proportional to displacement) is simple harmonic motion.
Examples of SHM: a mass on a spring, the bob of a pendulum (small angles), vibrations of eardrums.
The motion is sinusoidal: position, velocity and acceleration all follow sine or cosine functions of time.
SHM is the foundation for understanding more complex oscillations and waves.
Sinusoidal oscillation, step by step
The object starts at maximum positive displacement (position = +A). This is a turning point: velocity is zero, acceleration is at its most negative (pointing back toward equilibrium).
It accelerates toward equilibrium, gaining speed.
At the equilibrium position (x = 0), velocity reaches its maximum. Acceleration is zero here because the net force is zero.
Inertia carries it past equilibrium to maximum negative displacement (position = -A). Velocity is again zero; acceleration is at its most positive.
The cycle then reverses and repeats.
A block on a frictionless air track is attached to a spring. The equilibrium position (x = 0) is where the spring is at its natural length.
Displacing the block by Δx causes the spring to exert a restoring force: F_sp = -kΔx. The negative sign means the force opposes the displacement.
Because force is proportional to displacement, the motion is always SHM.
The amplitude A is the maximum distance the block reaches from equilibrium on either side.
The graph of position vs time is a smooth sinusoidal curve with peaks at +A and troughs at -A.
Position as a function of time
x(t) = A cos(2πt / T), equivalently x(t) = A cos(ωt), where ω = 2π/T = √(k/m).
The cosine form assumes the object starts at x = +A at t = 0. If it starts elsewhere, a phase constant is added, but most exam problems use the cosine form.
Velocity and acceleration
Velocity is the time derivative of position. It is zero at the turning points (x = ±A) and at its maximum magnitude when the object passes through equilibrium.
Acceleration is the time derivative of velocity. It is always opposite in direction to displacement: when x is positive, a is negative, and vice versa.
Maximum speed: v_max = ωA = (2π/T)A.
Maximum acceleration magnitude: a_max = ω²A.
A simple pendulum is a point mass m at the end of a light string of length L.
The arc length s is measured from the lowest point: s is positive to the right, negative to the left. The angle θ = s/L.
The tangential component of gravity provides the restoring force: F_net(tangential) = -mg sin θ.
Why the small-angle approximation matters
For angles of about 10° (roughly 0.17 rad) or less, sin θ ≈ θ (in radians).
The restoring force then becomes F_net ≈ -mgθ = -mg(s/L), which is proportional to displacement s. That makes it SHM.
Without this approximation, the pendulum equation is nonlinear and does not produce true SHM.
Always set your calculator to radians when working pendulum problems.
Pendulum frequency and period
f = (1/2π)√(g/L)
T = 2π√(L/g)
A longer pendulum swings more slowly (longer period). A shorter pendulum swings more quickly.
Pendulum frequency and period depend on length and gravitational acceleration, not on mass.
Physical (extended) pendulum
A physical pendulum is an extended object (not a point mass) swinging about a pivot that is not at its centre of mass.
Its period depends on the moment of inertia I about the pivot, the mass m, the distance d from the pivot to the centre of gravity, and g: T = 2π√(I / (mgd)).
For a simple pendulum (all mass at the end), I = mL² and d = L, recovering T = 2π√(L/g).
Vertical mass-spring system
A block of mass m hangs from a spring. Gravity stretches the spring by ΔL from its natural length until kΔL = mg. This new position is the equilibrium.
If you displace the block by y from that equilibrium, the net restoring force is F_net = -ky. Gravity has already been accounted for in setting the equilibrium point.
The motion is SHM with the same formulas as the horizontal case, just measured about the new (lower) equilibrium.
Position, velocity and acceleration graphs
Position: a cosine curve oscillating between +A and -A with period T.
Velocity: a negative sine curve (shifted 90° from position). It is zero at the turning points and at its peak magnitude at equilibrium.
Acceleration: a negative cosine curve (shifted 180° from position). It is always opposite to displacement.
These three graphs share the same period T, but they are out of phase with one another.
Key relationships to remember
When displacement is at a maximum (turning point), velocity = 0 and acceleration is at its maximum magnitude.
When displacement is zero (equilibrium), velocity is at its maximum and acceleration = 0.
Acceleration is always opposite the displacement: a = -ω²x.
The period does not depend on amplitude. A larger amplitude means the object travels farther but also moves faster, keeping T constant.
f = \frac{1}{T}, \qquad T = \frac{1}{f}x(t) = A \cos\!\left(\frac{2\pi t}{T}\right) = A \cos(\omega t)\omega = 2\pi f = \frac{2\pi}{T}Mass-spring system
F_{\text{sp}} = -k\,\Delta x\omega = \sqrt{\frac{k}{m}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}, \qquad T = 2\pi\sqrt{\frac{m}{k}}v_{\max} = \omega A, \qquad a_{\max} = \omega^2 ASimple pendulum (small angle)
f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}, \qquad T = 2\pi\sqrt{\frac{L}{g}}Physical pendulum
T = 2\pi\sqrt{\frac{I}{mgd}}Vertical spring equilibrium stretch
\Delta L = \frac{mg}{k}Car suspension springs are mass-spring oscillators. Engineers choose k and damping so the ride period feels comfortable and the car does not keep bouncing after a bump.
The quartz crystal in a wristwatch vibrates at a precise frequency (32 768 Hz), and the watch counts those oscillations to keep time. The frequency depends on the crystal's stiffness and mass, exactly as k/m governs a spring.
Pendulum clocks exploit the fact that the period of a pendulum depends only on length and gravity: adjust the length, and you set the tick rate.
Students often think a larger amplitude means a longer period. It does not. Period is independent of amplitude in SHM.
Students often confuse frequency and period, especially under time pressure. Remember: frequency is cycles per second, period is seconds per cycle. They are reciprocals.
Students often think the restoring force is constant. It is not: it changes with displacement. At equilibrium the restoring force is zero; at maximum displacement it is strongest.
Students often forget to set their calculator to radians for pendulum problems. The small-angle approximation sin θ ≈ θ only works in radians, not degrees.
⚠️ Know the relationship between position, velocity and acceleration at every key point in the cycle (turning points and equilibrium). This comes up repeatedly.
⚠️ Be able to read T, f, and A directly from a position-vs-time graph.
⚠️ For a vertical spring, expect a question that asks you to show the equilibrium position accounts for gravity, so the net restoring force is just -ky.
⚠️ Pendulum problems: know that period depends on L and g, not on mass. A common exam question changes the mass and asks what happens to the period (answer: nothing).
True or false: Doubling the amplitude of a mass-spring system doubles the period. (False)
Fill in the blank: At the equilibrium position, velocity is at its ______ and acceleration is ______. (maximum, zero)
True or false: A pendulum's period depends on the mass of the bob. (False)
Fill in the blank: The SI unit of frequency is the ______. (hertz)
True or false: In SHM, acceleration is always directed opposite to displacement. (True)
Q: A mass on a spring oscillates with a period of 2 s. At t = 0 the mass is at x = +A. What is the position at t = 0.5 s?
A: Half a period is 1 s, so at t = 0.5 s the mass is at T/4 of its cycle. x(T/4) = A cos(2π · 0.25) = A cos(π/2) = 0. The mass is at the equilibrium position.
Q: A pendulum has a frequency of 1.45 Hz. What is its period?
A: T = 1/f = 1/1.45 ≈ 0.69 s.
Q: A block on a spring has k = 200 N/m and m = 0.5 kg. What is the angular frequency ω?
A: ω = √(k/m) = √(200/0.5) = √400 = 20 rad/s.
Q: A simple pendulum has a period of 7.9 s. What is its length?
A: T = 2π√(L/g), so L = gT²/(4π²) = 9.8 × 7.9² / (4π²) ≈ 15.5 m.
Q: A mass-spring system has T = 2 s and at t = 5 s the position is x = 3 cm. What is the amplitude?
A: At t = 5 s, the number of complete periods elapsed is 5/2 = 2.5, so the phase is 2π × 2.5 = 5π. cos(5π) = cos(π) = -1, giving x = -A. Since x = 3 cm at that point, A = 3 cm (the sign indicates direction; the problem states the magnitude).
This material connects directly to Chapter 15 (wave motion): a wave is essentially many coupled oscillators, and the frequency, period and amplitude vocabulary carries over unchanged. Energy conservation in SHM (covered in 14.4) builds on the work-energy theorem from Chapter 10. Pendulum dynamics reappear in rotational mechanics whenever a rigid body swings about a pivot.
Simple harmonic motion, SHM, oscillation, oscillatory motion, restoring force, equilibrium position, amplitude, period, frequency, hertz, angular frequency, omega, spring constant, Hooke's law, mass-spring system, horizontal spring, vertical spring, pendulum, simple pendulum, physical pendulum, small-angle approximation, sinusoidal motion, cosine curve, linear restoring force, turning point, phase, displacement from equilibrium, PHYS 101, Chapter 14, University of Washington General Physics