Title Simple Harmonic Motion 4.0.pdf ✅

Oscillations (Simple Harmonic Motion) 1 4 Oscillations (Simple Harmonic Motion) 1. Periodic Motion • A motion which repeats itself after regular interval of time is called periodic motion and the time interval after which the motion is repeated is called time period. • A periodic motion can be either rectilinear or closed or open curvilinear (Fig.) • In case of periodic motion, force is always directed towards a fixed point. 2. Oscillatory Motion • The motion of body is said to be oscillatory or vibratory motion if it moves back and forth (to and fro) about a fixed point after regular interval of time. • The fixed point about which the body oscillates is called mean position or equilibrium position. Examples (i) Vibration of the wire of 'Sitar'. (ii) Oscillation of the mass suspended from spring. Note :- • Every free oscillatory motion is periodic but every periodic motion is not oscillatory. • Oscillatory (or vibratory) motion is a constrained periodic motion between certain precisely fixed limits. • For any body to undergo oscillation, the body must experience a force which is always directed towards mean position of the body. (Restoring force) It is given by F = –kx.


4 Oscillations (Simple Harmonic Motion) Solution: (a) Functions (vi) log(1 + ωt) and (vii) exp (–ωt) increase (or decrease) continuously with time and can never repeat themselves so these are aperiodic. (b) Functions (iv) sinωt + sin2ωt + cos2ωt and (v) sin3 ωt are periodic [i.e., f (t + T) = f(t)] with periodicity (π/ω), (2π/ω) & (2π/ω) respectively but not simple harmonic as for these functions (d2 y/dt2 ) is not ∝ – y. (c) Functions (i) sin2ωt, (ii) (1 + cos 2ωt) and (iii) asinωt + bcosωt, i.e., (a2 + b2 ) 1/2 sin[ωt + tan–1 (b/a)] are simple harmonic [with time period (π/ω), (π/ω) and (2π/ω) respectively] as for these (d2 y/dt2 ) ∝ – y. 3.4 Comparison Between Linear & Angular SHM Linear S.H.M Angular S.H.M • F ∝ – x • F = – kx Where k is the restoring force constant • k a–x m = • 2 2 dx k x 0 dt m + = It is known as differential equation of linear S.H.M. • x = A sinωt • a = –ω2 X where ω is the angular frequency • 2 k m ω = • k 2 2 n m T π ω= = = π where T is time period and n is frequency • m T 2 k = π • 1 k n 2 m = π This concept is valid for all types of linear S.H.M. • τ ∝ – θ • τ = – Cθ Where C is the restoring torque constant • C– I α= θ • 2 2 d C 0 dt I θ + θ= It is known as differential equation of angular S.H.M. • θ = θ0sinωt • α = –ω2 θ • 2 C I ω = • C 2 2 n I T π ω= = = π • I T 2 C = π • 1 C n 2 I = π This concept is valid for all types of angular S.H.M.