Simple Harmonic Motion (SHM)
A specific kind of repeated motion where a restoring force pulls the object back toward equilibrium, with a force proportional to displacement (F = −kx). The position-time graph is sinusoidal. The central concept of Unit 7.
FoundationRestoring Force
A force that always points back toward the equilibrium position. For SHM specifically, the restoring force is LINEAR in displacement: F = −kx. The negative sign means it points opposite to the displacement.
FoundationEquilibrium Position
The point where the net force on the oscillator is zero. For a horizontal spring: the spring's natural length. For a vertical spring with a hanging mass: the spring stretched by mg/k. For a pendulum: hanging straight down.
7.1 Defining SHMDisplacement (in SHM)
The position of the oscillator relative to equilibrium. Usually called x. Positive on one side, negative on the other. Maximum |x| = amplitude A.
7.1 Defining SHMAmplitude (A)
The maximum displacement from equilibrium. The "size" of the oscillation. Importantly, in SHM, amplitude does NOT affect the period — only the energy.
7.1 Defining SHMLinear Restoring Force
F = −kx. The defining feature of SHM. The force is proportional to displacement AND in the opposite direction. Pull farther, get pushed back harder.
7.1 Defining SHMHooke's Law
The force exerted by an ideal spring stretched by an amount x: F = −kx, where k is the spring constant (N/m). Stiffer spring = larger k = stronger restoring force per unit stretch.
7.1 Defining SHMPeriod (T)
The time for one complete oscillation. Units: seconds. The time for the oscillator to return to its starting position AND moving in the same direction.
7.2 Period & FrequencyFrequency (f)
The number of oscillations per second. f = 1/T. Units: hertz (Hz) = 1/s. Higher frequency = faster wiggling.
7.2 Period & FrequencyAngular Frequency (ω)
ω = 2π/T = 2πf. The "rotational" frequency used in sin/cos formulas: x(t) = A·cos(ωt). Units: rad/s. NOT the same as the rotational ω from Units 5 and 6 — but the formulas look identical.
7.2 Period & FrequencyPeriod of a Mass-Spring System
T = 2π√(m/k). Depends only on mass and spring constant. NOT on amplitude. NOT on gravity (horizontal spring). Stiffer spring → smaller T → faster oscillation. Heavier mass → larger T → slower.
7.2 Period & FrequencyPeriod of a Pendulum
T = 2π√(L/g) (valid for small angles, less than ~15°). Depends only on length and gravity. NOT on mass of the bob! NOT on amplitude. Longer string → larger T.
7.2 Period & FrequencyPosition vs Time Graph (SHM)
A pure sine or cosine curve. x(t) = A·cos(ωt + φ). The amplitude is the height; the period is the horizontal distance between repeating points; phase shifts move the curve left or right.
7.3 Representing SHMPhase
The argument of the sine/cosine function: (ωt + φ). Tells you where in the cycle you are at time t. The phase constant φ is set by initial conditions (where you started).
7.3 Representing SHMVelocity in SHM
v(t) = −Aω·sin(ωt + φ). Maximum at equilibrium (x = 0), zero at the turning points (x = ±A). The velocity curve is a sine if position is a cosine (phase shifted by 90°).
7.3 Representing SHMAcceleration in SHM
a(t) = −Aω²·cos(ωt + φ) = −ω²·x. Always proportional and opposite to displacement. Maximum at turning points, zero at equilibrium. This is the OPPOSITE of velocity.
7.3 Representing SHMMaximum Speed
v_max = Aω = A·(2π/T). Achieved at equilibrium position. From energy conservation: ½mv_max² = ½kA², which gives v_max = A√(k/m).
7.3 Representing SHMMaximum Acceleration
a_max = Aω². Achieved at the turning points (where |x| = A). From F = ma and F_max = kA: a_max = kA/m = ω²A.
7.3 Representing SHMPhase Relationships (x, v, a)
x peaks when v = 0 and a is maximum (negative). v peaks when x = 0 and a = 0. The three quantities are 90° out of phase: x leads v by 90°; v leads a by 90°; a is 180° out of phase with x (opposite direction).
7.3 Representing SHMSinusoidal Motion
Motion that traces out a sine (or cosine) curve in time. All SHM is sinusoidal. The mathematical signature: position oscillates as a sinusoidal function of time.
7.3 Representing SHMPotential Energy in SHM
U(x) = ½kx² (for a spring). Maximum at the turning points (|x| = A); zero at equilibrium. Stored in the spring's compression or stretch.
7.4 Energy of SHOKinetic Energy in SHM
K = ½mv². Maximum at equilibrium (where v is largest); zero at the turning points (where v = 0). The "kinetic" part of the energy.
7.4 Energy of SHOTotal Mechanical Energy
E = K + U = ½mv² + ½kx² = ½kA². Conserved in ideal SHM. The constant ½kA² is the total energy of the oscillator — it's what you get when ALL the energy is potential (at the turning point).
7.4 Energy of SHOEnergy Conservation in SHM
As the oscillator moves, energy continuously swaps between PE and KE — but the sum stays constant: ½mv² + ½kx² = ½kA². Use this to find v at any position x.
7.4 Energy of SHOEnergy Exchange (PE ↔ KE)
During one period, energy converts: PE → KE → PE → KE → PE. At the extremes, all PE. At equilibrium, all KE. Halfway through (x = A/√2), the energy is split 50/50.
7.4 Energy of SHOMaximum KE at Equilibrium
K_max = ½kA² = ½mv_max². The oscillator is fastest at x = 0, where all the energy is kinetic. This is where you'd "tag" a moving pendulum's bob if you wanted maximum collision speed.
7.4 Energy of SHOMaximum PE at Extremes
U_max = ½kA². Occurs at the turning points, where the oscillator momentarily stops before reversing direction. All the energy is stored as potential.
7.4 Energy of SHODoubling the Amplitude Quadruples the Energy
E = ½kA². Since A is SQUARED, doubling A multiplies E by 4. The maximum speed also doubles (v_max = A√(k/m)). But the period stays the same — a key feature of SHM.
7.4 Energy of SHO