Question

Assertion: In $x = A\cos \omega t$, the dot product of acceleration and velocity is positive for time interval $0 < t < d\dfrac{\pi }{{2\omega }}$.
Reason: Angle between them is ${0^ \circ }$.
A. Both assertion and reason are correct and reason is the correct explanation for assertion.
B. Both assertion and reason are correct but reason is not the correct explanation for assertion.
C. Assertion is correct but reason is incorrect.
D. Both assertion and reason are incorrect.

Answer 1

Hint:Here we have to see whether the statement is correct or not. Then we have to see whether the reason provided for the statement is correct or not. If both are correct then, we have to see whether the first statement is explained by the second statement or not.

Complete answer:
First let us see what a simple harmonic motion is:
Simple harmonic motion, in mechanics, repeated movement back and forth by alignment, or centre point, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side.

Conditions for generating simple harmonic motion require that the net force must be defined as $F = - kx$, where $F$ is the restoring force, $x$ is the displacement, and $k$ is the constant force. In order to achieve basic harmonic motion, the velocity shift rate must be equal to the displacement. We know that angular frequency is equal to:
$
\omega = \dfrac{{2\pi }}{T} \\
\Rightarrow T = \dfrac{\pi }{{2\omega }} \\
\therefore T = \dfrac{T}{4} \\
$
Given,
$x = A\cos \omega t$, $0 < t < \dfrac{\pi }{{2\omega }}$
When the particle is at the extreme position:
When $t = 0$,$x = A$

When the particle is at the mean position:
When $t = \dfrac{T}{4}$, $x = 0$
For the interval $0 < t < \dfrac{T}{4}$, both the velocity and the acceleration of the particle are in the same direction, which means that the angle between them is 0. Therefore, their dot product is positive.

Hence, option A is correct. Both assertion and reason are correct and reason is the correct explanation for assertion.

Note:We have to see what angle is given in the reason. If another angle had been given or the time limit would have been different, then the reason would have not explained the assertion.Also remember that all the Simple Harmonic Motions are oscillatory and also periodic but not all oscillatory motions are SHM. Oscillatory motion is also called the harmonic motion of all the oscillatory motions wherein the most important one is simple harmonic motion (SHM).