MechanicsHigh Yield (2 Qs in JEE)

Simple Harmonic Motion (SHM) Formulas

Kinematics and dynamics of SHM, spring-mass systems, simple pendulum, energy conservation, and superpositions.

8 Formulas•511+ Questions in Bank•

Concept Diagram

Simple Harmonic Motion (SHM) Phase & Energy

Restoring force F = -kx on mass m, equilibrium x = 0, extremes x = ±A, and phase relationship between displacement, velocity, and acceleration.

x = A\sin(\omega t + \phi), \quad v = \omega\sqrt{A^2 - x^2}, \quad a = -\omega^2 x

Displacement, velocity, and restoring acceleration in simple harmonic motion.

Examiner Tip: Maximum velocity v_max = ωA; maximum acceleration a_max = ω²A.

K = \frac{1}{2}m\omega^2(A^2 - x^2), \quad U = \frac{1}{2}m\omega^2 x^2, \quad E_{total} = \frac{1}{2}m\omega^2 A^2 = \text{constant}

Continuous interchange between kinetic and potential energy.

Examiner Tip: At x = A/√2, kinetic energy equals potential energy (K = U = E/2).

T = 2\pi\sqrt{\frac{m}{k}}, \quad k_{\text{series}} = \frac{k_1 k_2}{k_1 + k_2}, \quad k_{\text{parallel}} = k_1 + k_2

Oscillation period of spring with mass m.

Examiner Tip: Cutting a spring into n equal parts increases stiffness of each part to n*k.

T = 2\pi\sqrt{\frac{L}{g}}

Small-angle oscillation period of ideal pendulum.

Examiner Tip: Seconds pendulum has T = 2 s and length L ≈ 1 meter.

A_{\text{res}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos\delta}, \quad \tan\theta = \frac{A_2\sin\delta}{A_1 + A_2\cos\delta}

Vector phasor addition of two SHMs of identical frequency with phase difference δ.

Examiner Tip: Constructive interference at δ = 0 gives A = A₁ + A₂.

T = 2\pi\sqrt{\frac{I}{mgd}} = 2\pi\sqrt{\frac{k^2 + d^2}{gd}} \implies T_{\text{min}} = 2\pi\sqrt{\frac{2k}{g}} \quad (\text{when } d = k)

Rigid body swinging about horizontal suspension axis at distance d from CM (k is radius of gyration about CM).

Examiner Tip: Points of suspension and oscillation are interchangeable: interchanging d and k²/d yields the exact same time period.

A(t) = A_0 e^{-\frac{b t}{2m}}, \quad \omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}, \quad E(t) = E_0 e^{-\frac{b t}{m}}, \quad Q = \frac{\omega_0 m}{b}

Oscillations with viscous damping force F_d = -b v. Amplitude decays exponentially with damping constant b/(2m).

Examiner Tip: Critical damping occurs when b² = 4mk (no oscillations occur; body returns to equilibrium most rapidly).

T_{\text{U-tube}} = 2\pi\sqrt{\frac{h}{g}} = 2\pi\sqrt{\frac{L}{2g}}, \quad T_{\text{cylinder}} = 2\pi\sqrt{\frac{m}{A\rho g}} = 2\pi\sqrt{\frac{h_{\text{submerged}}}{g}}

Restoring buoyant and gravitational head pressures creating natural simple harmonic oscillations.

Examiner Tip: Both systems oscillate with effective length equal to the equilibrium submerged/arm depth h.

Frequently tested misconceptions in Simple Harmonic Motion (SHM)

•Velocity is MAXIMUM at mean position (x = 0), and acceleration is MAXIMUM at extreme positions (x = ±A).
•Frequency of kinetic energy is TWICE (2f) the frequency of displacement oscillation.