A mass of 2 kg oscillating in simple harmonic motion has a maximum displacement of 20 mm and a time period of 1.57 s. At time t = 0, the displacement is 0. Then the acceleration is

Concept:

The equation of displacement of the simple harmonic motion is given by:

\(x(t) = A \cos(\omega t + ϕ)\) or \(x(t) = A \sin(\omega t + ϕ)\)

where A is amplitude, ω is the angular velocity and ϕ is the phase angle.

Calculation:

Given:

M = 2 kg, A = 20 mm = 0.02 m, time period t = 1.57 s

Angular velocity (ω)

\(\omega=\frac{2\pi}{T}=\frac{2\pi}{1.57}=4\)

Since at t = 0, the displacement x(0) = 0, this means the motion starts from the mean position and moves in the positive direction. This suggests a sine function:

\(x(t) = A \sin(\omega t)\)

\(x(t) = 0.02 \sin(4t)\)

The velocity v(t) is the derivative of the displacement with respect to time:

\(v(t) = \frac{dx(t)}{dt} = 0.02 \times 4 \cos(4t) = 0.08 \cos(4t)\)

The acceleration a(t) is the derivative of the velocity with respect to time or the second derivative of the displacement:

\(a = \frac{{dv}}{{dt}} = \frac{{d( 0.08~ \rm cos (4t)}}{{dt}} = −0.32 ~sin (4t)\)