Question

The equation of motion of a particle started at t = 0 is given by x = 5 sin (20t + π/3), where x is in centimetre and t in second. When does the particle
(a) first come to rest
(b) first have zero acceleration
(c) first have maximum speed?

Answer 1

Explanation:

(a) The particle will first come to rest when velocity ‘v’ becomes zero and at extreme position, $\sin \left(20 t+\frac{\pi}{3}\right)=\sin \frac{\pi}{2}$. Thus, on solving we get $t=\frac{\pi}{120}$+`+ s at which the particle first comes to rest.

(b) The particle to have zero acceleration, a = 0.  We know that $a=\omega^{2} x$  and on substituting the given x value, $x=5 \sin \left(20 t+\frac{\pi}{3}\right)$ we get $t=\frac{\pi}{30}$s at which the particle will first have zero acceleration.

(c)  The particle to have maximum speed,$v=A \omega \cos \left(\omega t+\frac{\pi}{3}\right)$. Therefore, on substituting the known values, we get $t=\frac{\pi}{30}$s at which the particle will get maximum velocity.