Question

The maximum displacement of a particle executing SHM is 1 m and maximum acceleration is 1.57 $m/{s^2}$. Its time period is:
(A) 4 sec
(B) 2 sec
(C) 1.57 sec
(D) (1/1.57) sec

Answer 1

To answer this question we have to first know the formula for finding the maximum acceleration. Once we know that we can compare the values provided in the question with the expressions that are derived. Once the comparison is over and the equation is solved, we will be able to answer this question.

Complete step by step answer:
We know that maximum acceleration is given by: $A{\omega ^2}$
Therefore, we can put the values in the equation and obtain that:
$\Rightarrow {\left( {\dfrac{\pi }{2}} \right)^2} = 1{\left( {\dfrac{{2\pi }}{T}} \right)^2}$
The above expression is written as such because we know that 1.57 = $\dfrac{\pi }{2}$.
Hence we can write that:
$\dfrac{{{\pi ^2}}}{4} = \dfrac{{4{\pi ^2}}}{{{T^2}}} \\\ \Rightarrow {T^2} = \dfrac{{4 \times 4{\pi ^2}}}{{{\pi ^2}}} \\\ \Rightarrow {T^2} = 16 \\\ \Rightarrow T = 4s \\\$
So we know that the time period will be 4s.

Hence option A is correct.

Note: In the answer we have come across the term SHM or simple harmonic motion. For better understanding we need to define it. By simple harmonic motion we mean a repetitive movement back and forth which occurs through an equilibrium, or the central position. The reason behind this is that the maximum displacement on one side of its position becomes equal to the maximum displacement on the other side.
When a particle is in motion which is along a straight line and is in an acceleration, the direction of which is always towards a fixed point on a specific line and the magnitude is proportional to the distance from a fixed point, is also known as SHM or simple harmonic motion.