Question

A DISPLACEMENT OF AN OBJECT ATTACHED TO A SPRING AND EXECUTING SIMPLE HARMONIC MOTION IS A GIVEN BY X=2*10^-2cos(pie)t METERS. THE TIME AT WHICH THE MAXIMUM SPEED OCCURS

Answer 1

The position of the particle is given to you as a function of time:
\[x = 2*10^{-2}*\cos(\pi*t)\]
To find the velocity, differentiate x with respect to time
\[v = dx/dt = 2*10^{-2}*\pi*[-\sin(\pi*t)]\]
The maximum and minimum of the velocity are decided by the sin function in square brackets. The speed is maximum whenever the velocity is either maximum or minimum since the speed is just the magnitude of the velocity. This happens when the argument of the sin function is pi/2, 3*pi/2, 5*pi/2,...... So put the srgument of the sin function equal to these values:
\[\pi*t = \pi/2, 3*\pi/2, 5*\pi/2, .....\]
Solve for t:
\[t = 1/2, 3/2, 5/2,......\]
That is, the maximum speed occurs at t = odd multiples of 1/2 second