Question: The speed of a particle at \[x = 20cm\]and \[t = 1\sec \]when it is at mean position. (A) \[0.2\p...

Question

The speed of a particle at $x = 20cm$ and $t = 1\sec$ when it is at mean position.
(A) $0.2\pi m/s$
(B) $0.6\pi m/s$
(C) $0.4\pi m/s$
(D) $0$

Answer 1

Solution

Since it is given as a particle, let us assume that the particle is undergoing a simple harmonic motion. Now, we know that in mean position. Use the equations of SHM to find the angular velocity of the object. And using this angular velocity, you can identify the time period. Applying the mean position condition, conclude your answer in terms of velocity of SHM.

Complete Step by step Solution:
In a simple harmonic motion, we consider the particle whose acceleration is pointed towards a fixed point and is proportional to the distance of the particle from the fixed point. When the particle moves away from the fixed point, the particle will slow down and stop at a point since the direction of acceleration is towards the fixed point, and will return back to the fixed point.
So, the general SHM equation for the acceleration of the particle is given as
$a(t) = {\omega ^2}x(t)$
Now, it is given that the object is at a mean position, which means that there is no external force acting on the particle to move. It is said that at time period of $t = 1\sec$ , the angular velocity is given as ,
$\omega = \dfrac{{2\pi }}{T}$
When T is given as 1 sec , the value of angular velocity is $\omega = 2\pi$ .
Now, at mean position,
The particle will be in rest and won’t have any designated motion or displacement. So at mean position, the angle will be zero and hence the acceleration of SHM will also be equal to zero. Since, the acceleration is zero, the Amplitude will also become zero, as mathematically acceleration is dependent on displacement.
$x = A\sin \omega t = 0$
So, now the velocity formula is given as ,
$\Rightarrow v = \omega \sqrt {{A^2} - {x^2}}$
Which gives us on substituting the values,
$\Rightarrow v = \omega (0)$
Hence, velocity will also be equal to zero when the particle is at mean position.

Hence, option (d) is the right answer for the given question.

Note: Simple Harmonic motion is the repetitive movement of a body of mass back and forth through an specified equilibrium so that the maximum displacement at one side of its position is equal to the maximum displacement on the other side. Oscillation of pendulum is one of the examples of SHM.