Question:

The moment of inertia of a body about a given axis is $1.2 \,kg - m ^{2}$. To produce a rotational kinetic energy of $1500\, J$ an angular acceleration of $25\, rad / s ^{2}$ must be applied for :

Updated On: Jun 20, 2022
  • 8.5 s
  • 5 s
  • 2 s
  • 1 s
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The Correct Option is C

Solution and Explanation

Key Idea : Kinetic energy of rotation is half the product of the moment of inertia (I) of the body and the square of the angular velocity $(\omega)$ of the body.
Kinetic energy of rotation $=\frac{1}{2} \times$ moment of inertia $\times$ angular velocity.
i.e. $ K =\frac{1}{2} I \omega^{2} $
$ \Rightarrow \omega^{2} =\frac{2 K}{I} $
Given, $I= 1.2\, kgm ^{2} , $
$K=1500\, J $
$\therefore \omega^{2} =\frac{2 \times 1500}{1.2}$
$\omega^{2} =2500$
$\Rightarrow \omega =50 \,rad / s$
From the equation of angular motion, we have
$\omega=\omega_{0}+\alpha t$
where $\omega_{0}$ is initial angular velocity,
$\alpha$ is angular acceleration and $t$ is time.
Given, $\omega_{0}=0, $
$\omega=50 \,rad / s , $
$\alpha=25 \,rad / s ^{2}$
$\therefore t=\frac{\omega}{\alpha}=\frac{50}{25}=2 s$
Note : The equation of kinetic energy of rotation is similar to kinetic energy expression in linear form $\left(\frac{1}{2} m v^{2}\right)$ and equation of angular motion is similar to Newton's equation of motion $(v=u+a t)$
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Concepts Used:

System of Particles and Rotational Motion

  1. The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
  2. The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
  3. The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.