Oscillations

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Periodic motion is motion that repeats itself at regular intervals of time. The smallest interval of time after which the motion repeats is called its period (T). Examples: (1) Motion of a pendulum in…

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Frequency (ν) is the number of oscillations completed per unit time. It is the reciprocal of the period: ν = 1/T. The SI unit of frequency is hertz (Hz), where 1 Hz = 1 oscillation per second = 1 s⁻¹.

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Simple harmonic motion is oscillatory motion where the displacement varies sinusoidally with time. Mathematical expression: x(t) = A cos(ωt + φ), where A = amplitude, ω = angular frequency, φ = phase …

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Amplitude (A): Maximum displacement from equilibrium position. Angular frequency (ω): Related to period by ω = 2π/T, measured in rad/s. Phase constant (φ): Value of phase at t = 0, determines initial …

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The projection of uniform circular motion on any diameter of the circle follows simple harmonic motion. If a particle moves on a circle of radius A with angular speed ω, its projection on x-axis is x(…

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Starting with x(t) = A cos(ωt + φ), velocity v(t) = dx/dt = -ωA sin(ωt + φ). The maximum velocity is v_max = ωA, occurring at equilibrium position (x = 0). Velocity is zero at extreme positions (x = ±…

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Starting with v(t) = -ωA sin(ωt + φ), acceleration a(t) = dv/dt = -ω²A cos(ωt + φ) = -ω²x(t). This shows acceleration is directly proportional to displacement and always directed towards equilibrium p…

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Force law: F(t) = -kx(t), where k = mω². Using Newton's second law, F = ma = m(-ω²x) = -mω²x = -kx. The force is always directed towards equilibrium position, hence called restoring force.