Question

A simple harmonic oscillator has amplitude 10 cm and it completes 120 oscillations in 60 s. i) Calculate its time period and angular frequency. ii) If the initial phase is π/2, write expressions for its displacement and velocity. iii) Calculate the values of maximum velocity and acceleration.

Answer 1

Given that

Amplitude = A = 10 cm

i)

The body completed 120 oscillations in 60 seconds

Hence Time period T = Time/number of oscillations

T = 60/120 = 0.5 s.

The angular frequency = ω = 2π/Time Period

ω = 2π / T = 2π / 0.5 = 4π rad/s.

ii)

Given that initial phase of the SHM = ϕ = π/2

Displace equation with respect to time = x(t) = A cos(ωt+ϕ)

= 10 cos(4πt+π/2) = (-10 sin4πt ) cm.

Displacement = -10sin4πt cm

Velocity equation with respect to time = v(t) = 1/4π(-10 * cos4πt )=(-5cos4πt/2π ) cm/s.

iii)

max velocity = V= Amplitude * Angular frequency

V = Aω = (10)(4π) = 40π cm/s

Max acceleration = a = Aω² = (10) (4π)²=160π² cm/s² ≈15.8 m/s²