Question

A particle is oscillating according to the equation $x=7\cos 0.5\pi t$ where t is in seconds. The point moves from the position of equilibrium to maximum displacement in time:
A. 4.0 s
B. 2.0 s
C. 1.0 s
D. 0.5 s

Answer 1

Hint: Given a particle is oscillating. We have to find the time from which the point moves from the position of equilibrium to maximum displacement. As the particle is oscillating, so the motion is simple harmonic motion. We know maximum displacement can be found by using $x=A\cos wt$. By putting the values in the equation and solving it, we get the maximum displacement.

Formula Used:
Formula to find out the maximum displacement is:
$x=A\cos wt$
Where A is the amplitude, T is the time taken and w is the angular velocity.

Complete step by step solution:
A particle is oscillating according to the equation $x=7\cos 0.5\pi t$………………… (1)
As the particle is oscillating, so the motion is SHM.
We have to find the maximum displacement.
So we use the formula $x=A\cos wt$…………………….. (2)
Comparing equation (1) with equation (2), we get
$w=0.5\pi $
Where $w=\dfrac{2\pi }{T}$
So $T=\dfrac{2\pi }{w}$
Where T is the total time taken
$T=\dfrac{2\pi }{0.5\pi }$
$\Rightarrow T=4$ sec
To move from the position of equilibrium to maximum displacement time needed is,
$\dfrac{T}{4}=\dfrac{4}{4}=1 \\ $
$\therefore T=1$ sec

Thus, option C is the correct answer.

Note: In these types of questions, students get confused in using the formula. As in these questions, sin and cos both function but they must know the difference. If at t = 0, at the starting of motion, displacement is zero. That means it is at its normal position then we use a sin graph because $x=A\sin t$ at $t=0,x=0$. If at t = 0 if the displacement is at maximum position then we used cos graph because $x=A\cos t$ at $t=0,x=1$