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A body rotates about a fixed axis with angular acceleration of $1rad/{{\sec }^{2}}$. Through what does it rotate during this time in which its angular velocity increases from 5rad/s to 15rad/s.

Last updated date: 13th Jun 2024
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Hint: In the question it is clearly given to us that the body rotates about an axis. Hence we can determine the parameters of the rotational motion of the body using rotational dynamic equations. It is given to us that the body moves with an angular acceleration $1rad/{{\sec }^{2}}$. We are asked to determine the angle covered by the body when its velocity increases from 5rad/s to 15rad/s. Therefore using the rotational dynamic equation of displacement we will determine the angle traced by the body.
Formula used:
${{\omega }^{2}}-{{\omega }_{o}}^{2}=2\alpha \theta $

Complete answer:
Let us say a body moves along a circular path with constant angular acceleration $\alpha $. Let us say we want to determine the angle traced by the particle i.e. $\theta $ during the time interval when its velocity changes from ${{\omega }_{o}}$ to $\omega $. Hence the relation between all the above parameters of motion is given by,
${{\omega }^{2}}-{{\omega }_{o}}^{2}=2\alpha \theta $
It is given in the question that the velocity of the body increases from 5rad/s to 15rad/s when it undergoes a constant acceleration of $1rad/{{\sec }^{2}}$. Hence using that above equation we get the angle traced by the body as,
  & {{\omega }^{2}}-{{\omega }_{o}}^{2}=2\alpha \theta \\
 & \Rightarrow {{(15rad/s)}^{2}}-{{(5rad/s)}^{2}}=2\times 1rad/{{s}^{2}}\theta \\
 & \Rightarrow \theta =\dfrac{225-25}{2}rad \\
 & \Rightarrow 100rad \\

So, the correct answer is “Option A”.

It is always to be verified whether the body is undergoing a uniform acceleration. If this condition is not satisfied, then the rotational dynamic equations are no longer valid. If the body is undergoing a non uniform acceleration, first we have to figure out how the acceleration varies with time and accordingly integrate it in the equation of rotational dynamics to determine the other parameters.