Question

A point mass is executing simple harmonic motion with an amplitude of 10 mm and frequency of 4 Hz. The maximum acceleration (m/s²) of the mass is _______

Answer 1

Concept:

Equation of displacement of the simple harmonic motion is given by:

x = Asin ωt or x = Acos ωt 

where A is amplitude, ω is the angular velocity

Calculation:

Given:

A = 10 mm = 0.01 m, frequency f = 4 Hz

Angular velocity (ω)

ω = 2 × π × f = 2 × π × 4 = 8π rad/s

Let x = Acos ωt

Velocity (v)

\({\rm{v}} = \frac{{{\rm{dx}}}}{{{\rm{dt}}}} = - {\rm{Aω }}\sin {\rm{ω t}}\)

Acceleration (a)

\(a = \frac{{dv}}{{dt}} = - Aω \frac{{d(\sin ω t)}}{{dt}} = - A{ω ^2}\cos ω t\)

At t = 0, value of cos ωt is maximum i.e. 1

a_max = -Aω²

∴ |a_max| = Aω² = 0.01 × (8π)² = 6.31 m/s²

Note:

You can also solve by assuming x = A sin ωt, answer would be same.