Question

What is constant in S.H.M?
A. Restoring force
B. Kinetic energy
C. Potential energy
D. Periodic time

Answer 1

Hint:In this question first we will discuss the simple harmonic motion and its condition then derive the equation of SHM mathematically so that we can justify our result. Every SHM motion is periodic motion but every periodic motion is not SHM. So we will justify the motion then identify the constant term in SHM.

Complete step by step answer:
The acceleration of a particle executing the simple harmonic motion is:
\[ \Rightarrow \;a\left( t \right) = - {\omega ^{2\;}}x\left( t \right)\]
Where \[\omega \] is angular velocity of a particle.
To find the answer of the above question if simple harmonic motion is there in the spring block system.If the SHM start with its mean position, so we can write the displacement equation as follows:
\[ \Rightarrow x = A\sin \omega t\]

From above equation we find velocity equation
\[ \Rightarrow v = \dfrac{{dx}}{{dt}} = A\omega \cos \omega t\]
Now we get the acceleration by using velocity equation
\[ \Rightarrow a = \dfrac{{dv}}{{dt}} = - A{\omega ^2}\cos \omega t\]
\[ \Rightarrow a = - {\omega ^2}x\]
Now we multiply both by \[m\]as shown
\[ \Rightarrow am = - m{\omega ^2}x\]
\[\therefore F = - m{\omega ^2}x\]
From the above equation, it’s clearly given that force is directly proportional to negative of the displacement.So the Simple harmonic motion is constant in periodic time .

When both the mass and spring constant is constant there is no change in time period.The minimum time after which the particle keeps on repeating its motion is known as the time period.The formula of time period is:
\[T = 2\Pi \sqrt {\dfrac{m}{k}} \]
 Where \[k\] is spring constant.

Therefore, the correct option is D.

Note:To and fro motion of a particle about a mean position is called an oscillatory motion in which a particle moves on either side of equilibrium (or) mean position is an oscillatory motion. The object will keep on moving between two extreme points about a fixed point called mean position. In an oscillatory motion, the net force on the particle is zero at the mean position. The mean position is a stable equilibrium position.