Difficulty: Intermediate | Prerequisites: basic mechanics (Newton's laws, energy conservation, trigonometry)
Simple harmonic motion is the foundation for nearly everything else in PHYS 123, from mechanical waves to sound to optics. If you can write down the equation of motion for a mass on a spring and understand what each term means, you already have the toolkit for analysing pendulums, standing waves, and resonance. The maths here (sinusoidal functions, angular frequency, phase) carries straight through to the wave unit. You should already be comfortable with Newton's second law, Hooke's law, and conservation of energy before diving in.
Simple harmonic motion occurs whenever a restoring force is proportional to displacement from equilibrium. The motion is sinusoidal, fully described by amplitude, angular frequency, and phase constant. Energy swaps back and forth between kinetic and potential forms, and the total mechanical energy depends on amplitude and spring constant, not on mass.
Simple harmonic motion (SHM)
Oscillatory motion that results from a restoring force directly proportional to the displacement from equilibrium. Think of it as: any time the "pull back" force doubles when you double the stretch, you get SHM.
Restoring force
A force that always points back towards equilibrium, with magnitude proportional to displacement: F = -kx. In simple terms, this is the force that keeps yanking the object back to centre.
Spring constant (k)
The stiffness of a spring, measured in N/m. A larger k means a stiffer spring that pushes back harder for the same displacement.
Angular frequency (omega, ω)
The rate of oscillation in radians per second: ω = √(k/m) for a mass-spring system, or ω = √(g/L) for a simple pendulum. Think of it as: how quickly the oscillation cycles through its full motion.
Period (T)
The time for one complete oscillation cycle: T = 2π/ω. In simple terms, this is the stopwatch time from one peak back to the next peak.
Frequency (f)
The number of complete oscillations per second, measured in Hz: f = 1/T = ω/(2π).
Amplitude (A)
The maximum displacement from equilibrium. This sets the scale of the motion but does not affect the period or frequency.
Phase constant (φ)
The angle (in radians) that specifies where in the cycle the motion starts at t = 0. In simple terms, this tells you the "starting position" of the oscillation on the sine curve.
Equilibrium position
The position where the net force on the object is zero. For a vertical spring, this is the point where the spring force balances gravity, not the natural length of the spring.
A mass m on a spring with spring constant k oscillates with angular frequency ω = √(k/m).
The maximum velocity occurs as the mass passes through equilibrium: v_max = Aω.
Example: a 1.0 kg cart on a spring with k = 9.0 N/m, displaced 1.0 m, has ω = √(9/1) = 3 rad/s and v_max = (1.0)(3) = 3.0 m/s.
The period T = 2π√(m/k) depends on mass and spring constant only, not on amplitude.
For small angles, a pendulum of length L oscillates with ω = √(g/L) and period T = 2π√(L/g).
Frequency is proportional to 1/√L. Doubling the string length reduces the frequency by a factor of √2.
Example: if f = 0.5 Hz and you double L, the new frequency is 0.5/√2 ≈ 0.354 Hz.
The period does not depend on mass or amplitude (for small angles).
SHM requires a restoring force proportional to displacement: F(x) = -kx, where k is a positive constant.
A force like F(x) = -5x + 8 still produces SHM. Rewriting: F(x) = -5(x - 8/5). The equilibrium is shifted to x₀ = 8/5, but the motion about that point is simple harmonic.
A force like F(x) = +5x does not produce SHM because it pushes the object further from equilibrium rather than back towards it.
A torque τ(x) = -5x can also produce SHM in a rotational system, since it is a restoring torque proportional to angular displacement.
Total mechanical energy: E = ½kA². This depends on the spring constant and amplitude, not on mass directly.
At any point: E = ½kx² + ½mv² (potential + kinetic).
When comparing two systems with the same spring constant and the same total energy, the amplitudes must be equal, regardless of the masses attached.
Example: springs with identical k, masses m and 2m, same energy. Since E = ½kA² and k is the same, A₁ = A₂. The ratio A₁/A₂ = 1.
The general SHM equation: x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ).
The phase constant φ is determined by initial conditions (position and velocity at t = 0).
If using x(t) = d sin(ωt + φ) and the mass starts at maximum positive displacement (x = d) with zero velocity, then sin(φ) = 1, so φ = π/2.
This means the sine function starts at its peak, which is equivalent to a cosine function with φ = 0.
\omega = \sqrt{\frac{k}{m}} \quad \text{(mass-spring)}\omega = \sqrt{\frac{g}{L}} \quad \text{(simple pendulum)}v_{\max} = A\omegaE = \tfrac{1}{2}kA^2 = \tfrac{1}{2}m\omega^2 A^2x(t) = A\sin(\omega t + \phi)T = \frac{2\pi}{\omega}, \quad f = \frac{1}{T}SHM describes the motion of any system near a stable equilibrium, from the suspension in your car to the vibrations of atoms in a crystal lattice. The mass-spring model is how engineers design shock absorbers, and the pendulum model underpins the mechanics of grandfather clocks and seismometers.
Students often think that doubling the mass doubles the period. It does not. Period scales with the square root of mass: T ∝ √m.
Students often think amplitude affects frequency. It does not. For SHM, frequency and period are independent of amplitude.
Students often assume F(x) = -5x + 8 cannot produce SHM because of the +8 constant. It can. The constant merely shifts the equilibrium position; the motion about that new equilibrium is still simple harmonic.
Students often confuse the phase constant with the phase. The phase is (ωt + φ) and changes with time. The phase constant φ is fixed by initial conditions.
⚠️ Maximum velocity problems are a favourite. Remember v_max = Aω. You need to compute ω = √(k/m) first.
⚠️ Pendulum frequency scaling: know that ω ∝ 1/√L. If they change L by some factor, the frequency changes by the inverse square root of that factor.
⚠️ Identifying whether a force produces SHM: the test is whether F is proportional to -(displacement from some equilibrium). Constant offsets are fine; positive proportionality is not.
⚠️ Energy comparison problems: E = ½kA² is the key. When k is the same and energies are equal, amplitudes are equal regardless of mass.
⚠️ Phase constant from initial conditions: set t = 0 in x(t) = A sin(ωt + φ) and solve for φ using the given initial displacement and velocity.
True or false: the period of a mass-spring system depends on amplitude. (False)
Fill in the blank: the maximum speed of a mass on a spring is v_max = ______. (Aω)
True or false: F(x) = -3x + 7 can produce simple harmonic motion. (True, equilibrium shifted to x = 7/3)
Fill in the blank: if you quadruple the length of a pendulum, the period increases by a factor of ______. (2)
True or false: two springs with identical k but different masses, oscillating with the same total energy, must have the same amplitude. (True)
Q: A 2.0 kg block on a frictionless surface is attached to a spring with k = 8.0 N/m. It is displaced 0.5 m from equilibrium and released. What is the maximum velocity?
A: ω = √(8/2) = 2 rad/s. v_max = Aω = (0.5)(2) = 1.0 m/s.
Q: A simple pendulum has a period of 2.0 s. If the length is tripled, what is the new period?
A: T ∝ √L, so the new period is 2.0 × √3 ≈ 3.46 s.
Q: Does the force F(x) = -10x + 20 produce SHM? If so, where is the equilibrium position?
A: Yes. Rewrite as F(x) = -10(x - 2). The equilibrium is at x₀ = 2, and the effective spring constant is 10 N/m.
Q: Two identical springs (same k) support masses m and 3m. Both systems oscillate with the same total energy. What is the ratio of their amplitudes, A₁/A₂?
A: E = ½kA². Since k and E are the same for both, A₁ = A₂. The ratio is 1.
Q: A mass on a spring is at its maximum negative displacement at t = 0. If the motion is written as x(t) = A sin(ωt + φ), what is φ?
A: At t = 0, x = -A, so sin(φ) = -1, giving φ = -π/2.
SHM is the building block for mechanical waves. A travelling wave on a string is essentially SHM at every point along the string, linked together with a time delay. The energy expressions here (E = ½kA²) carry directly into wave intensity and power calculations. The phase constant concept reappears when you study standing waves and interference, where relative phase between two waves determines constructive or destructive superposition.
Simple harmonic motion, SHM, oscillation, mass-spring system, Hooke's law, restoring force, angular frequency, omega, period, frequency, amplitude, phase constant, phase angle, pendulum, small-angle approximation, spring constant, stiffness, energy in SHM, kinetic energy, potential energy, equilibrium position, sinusoidal motion, harmonic oscillator, PHYS 123, UW physics