Simple Harmonic Motion
Defining SHM
Simple harmonic motion (SHM) is oscillatory motion where the restoring force is directly proportional to displacement from equilibrium and always directed towards it.
$$a = -\omega^2 x$$
Acceleration is proportional to displacement and opposite in direction — the defining signature of SHM.
$a$=Instantaneous acceleration(m/s²)
$\omega$=[Angular Frequency](rad/s)
$x$=Displacement from equilibrium(m)
$x = 0$ (equilibrium)
→Acceleration is zero, velocity is maximum.
$x = \pm A$ (extremes)
→Acceleration is maximum ($|a_{max}| = \omega^2 A$), velocity is zero.
Key Test: If $a \propto -x$, the motion is SHM. The negative sign means acceleration always points back to the mean position.
Amplitude ($A$): Maximum displacement from equilibrium — does NOT affect period or frequency in ideal SHM.
Dimensional Check: $[\omega^2 x] = (\text{rad/s})^2 \times \text{m} = \text{m/s}^2$ ✓ confirms acceleration units.
SHM Quantities
SHM is characterized by angular frequency $\omega$, time period $T$, and frequency $f$, all interrelated through simple formulas.
$$\omega = \frac{2\pi}{T} = 2\pi f$$
Angular frequency converts between the time domain (period) and the circular motion analogy (radians per second).
$\omega$=Angular frequency(rad/s)
$T$=Time for one complete oscillation(s)
$f$=Number of oscillations per second(Hz)
Vibration: One complete round trip — mean → extreme → opposite extreme → back to mean.
Unit Check: $[2\pi / T] = 1/\text{s} = \text{rad/s}$ since radians are dimensionless.
SHM and Circular Motion
SHM is the projection of uniform circular motion onto a diameter — a point moving in a circle at constant $\omega$ produces SHM when viewed from the side.
Displacement: $x = A \sin(\omega t)$ — the projection of the rotating point onto the diameter.
Velocity: $v = A\omega \cos(\omega t) = \omega\sqrt{A^2 - x^2}$ — maximum at center ($x = 0$), zero at extremes ($x = \pm A$).
Acceleration: $a = -A\omega^2 \sin(\omega t) = -\omega^2 x$ — always opposite to displacement.
Proportionality Shortcut: $v_{max} = A\omega$ and $a_{max} = A\omega^2$. If $\omega$ doubles, $v_{max}$ doubles but $a_{max}$ quadruples.
Phase of Oscillation
The phase angle $\theta = \omega t + \phi$ specifies both the displacement and direction of motion at any instant.
$$x = A \sin(\omega t + \phi)$$
The general SHM equation with initial phase $\phi$ — determines where in the cycle the motion begins.
$A$=Amplitude(m)
$\omega t$=Phase gained over time(rad)
$\phi$=Initial phase at $t = 0$(rad)
$\phi = 0$
→Starts from mean position: $x = A \sin(\omega t)$.
$\phi = \pi/2$
→Starts from positive extreme: $x = A \cos(\omega t)$.
Phase Determines State: At phase $0$ the body is at mean position moving forward; at $\pi/2$ it is at the positive extreme; at $\pi$ back at mean moving backward; at $3\pi/2$ at the negative extreme.
Phase Difference: Two oscillators with a phase difference of $\pi$ are always on opposite sides of equilibrium (anti-phase).
The Mass-Spring System
The mass-spring system is a fundamental oscillator where a restoring force proportional to displacement (Hooke's Law) drives motion.
$$T = 2\pi \sqrt{\frac{m}{k}}$$
The period depends on the ratio of inertia to stiffness.
$T$=Time period(s)
$m$=Oscillating mass(kg)
$k$=[Spring Constant](N/m)
Increasing $m$ by factor $n$
→$T$ increases by $\sqrt{n}$ — e.g. quadrupling mass doubles $T$.
Increasing $k$ by factor $n$
→$T$ decreases by $\sqrt{n}$ — stiffer spring means faster oscillation.
Vertical spring
→Same formula applies — gravity only shifts the equilibrium point downward, not the period.
Proportionality: $T \propto \sqrt{m}$ and $T \propto 1/\sqrt{k}$. These square-root relationships are a common source of errors.
Equilibrium Position: The point where net force is zero ($x = 0$); velocity is maximum here.
Restoring Force: Always directed towards equilibrium ($F = -kx$), opposing the displacement.
Finding $k$ from Static Extension: Hang a known mass and measure extension: $k = mg/x$. Use this $k$ with a different mass to find $T$.
Dimensional Check: $[2\pi\sqrt{m/k}] = \sqrt{\text{kg}/(\text{N/m})} = \sqrt{\text{s}^2} = \text{s}$ ✓
The instantaneous velocity of a mass on a spring depends on both position and the system's angular frequency.
$$v = \sqrt{\frac{k}{m}(A^2 - x^2)}$$
Velocity is maximum at center and zero at the extremes — it depends on how far the mass is from its turning point.
$v$=Instantaneous speed(m/s)
$A$=Amplitude(m)
$x$=Current displacement(m)
$x = 0$
→$v_{max} = A\sqrt{k/m} = A\omega$.
$x = \pm A$
→$v = 0$ — the mass momentarily stops at the extremes.
Maximum Speed: $v_{max} = A\omega = A\sqrt{k/m}$. Doubling amplitude doubles $v_{max}$.
Normalized Form: $v = v_{max}\sqrt{1 - x^2/A^2}$ — useful for finding speed at any fraction of amplitude.
The Simple Pendulum
A simple pendulum approximates SHM for small angles, where gravity acts as the restoring agent.
$$T = 2\pi \sqrt{\frac{L}{g}}$$
The period depends solely on geometry and local gravity, independent of mass.
$L$=Effective length of string(m)
$g$=[Gravitational Field Strength](m/s²)
$\theta > 15°$
→The Small Angle Approximation fails; motion is no longer strictly SHM.
$L \to \infty$
→$T \to \infty$ — an infinitely long pendulum would never complete a swing.
$g \to 0$ (free fall / space)
→$T \to \infty$ — no gravity means no restoring force, so no oscillation.
Small Angle Approximation: Valid when $\sin \theta \approx \theta$ in radians, ensuring force is linear with displacement.
Mass Independence: Increasing the bob's mass increases both inertia and weight, which cancel out in the period calculation.
Restoring Force Derivation: The tangential component $F = -mg\sin\theta \approx -mg\theta = -mg(x/L)$, giving $a = -(g/L)x$, confirming SHM with $\omega = \sqrt{g/L}$.
Proportionality: $T \propto \sqrt{L}$ and $T \propto 1/\sqrt{g}$. Quadrupling $L$ doubles $T$; taking the pendulum to a planet with $g/4$ also doubles $T$.
Energy in SHM
SHM systems continuously swap energy between kinetic and potential forms while maintaining a constant Mechanical Energy.
$$E_{total} = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$$
Total energy is proportional to the square of the amplitude.
$E_{total}$=Total mechanical energy(J)
$A$=Amplitude (maximum displacement)(m)
$v_{max}$=Maximum velocity at equilibrium(m/s)
$x = A/2$
→$PE = E/4$ and $KE = 3E/4$ — at half-amplitude, 75% of energy is kinetic.
$KE = PE$
→This occurs at $x = A/\sqrt{2} \approx 0.707A$.
Turning Points: At maximum displacement, velocity is zero and all energy is potential (elastic or gravitational).
Maximum Velocity: Occurs at the equilibrium point ($x=0$) where all potential energy has been converted to kinetic.
Energy at Any Point: $KE = \frac{1}{2}k(A^2 - x^2)$ and $PE = \frac{1}{2}kx^2$. Their sum is always $\frac{1}{2}kA^2$.
Proportionality Shortcut: $E \propto A^2$. Doubling amplitude quadruples total energy.