Example

Question:

A body oscillates with SHM according to the equation \( x = 5 \cos\left[ 2\pi t + \pi/4 \right] \) (in SI units). At \( t = 1.5\,\text{s} \), calculate the (a) displacement, (b) speed and (c) acceleration of the body.

Solution:

The angular frequency \( \omega = 2\pi~\text{s}^{-1} \) and period \( T = 1~\text{s} \).
At \( t = 1.5~\text{s} \):

(a) Displacement: \[ x = 5 \cos[2\pi \times 1.5 + \pi/4] = 5 \cos(3\pi + \pi/4) = 5 \times (-0.707) = -3.535~\text{m} \] (b) Speed: \[ v = -A\omega \sin(\omega t + \phi) = -5 \times 2\pi \sin(3\pi + \pi/4) = -10\pi \times 0.707 \approx 22~\text{m/s} \] (c) Acceleration: \[ a = -\omega^2 x = -(2\pi)^2 \times (-3.535) = 4\pi^2 \times 3.535 \approx 140~\text{m/s}^2 \]