Question

The rotational energy of a body is ${{K}_{rot}}$ and its moment of inertia is I. The angular momentum of body is:
$\begin{align}
  & (A)I{{K}_{rot}} \\
 & (B)2\sqrt{I{{K}_{rot}}} \\
 & (C)\sqrt{2I{{K}_{rot}}} \\
 & (D)2I{{K}_{rot}} \\
\end{align}$

Answer 1

Hint: The rotational kinetic energy and angular momentum are analogical terms to translational kinetic energy and linear momentum, with moment of inertia replacing mass and angular velocity replacing linear velocity. Using the formulas for rotational kinetic energy and angular momentum, we can deduce the required relation between, angular momentum, moment of inertia and rotational kinetic energy.

Complete step-by-step answer:
Let the angular velocity of the object be denoted by $\omega $. And it has been given that the moment of inertia of the object is given by ‘I’ and its rotational kinetic energy is given by ${{K}_{rot}}$ .
Also, let the angular momentum of the body be denoted by ‘L’.
Now, the rotational kinetic energy of a body is written as:
$\Rightarrow {{K}_{rot}}=\dfrac{1}{2}I{{\omega }^{2}}$
From the above equation, we can calculate the angular velocity as:
$\Rightarrow \omega =\sqrt{\dfrac{2{{K}_{rot}}}{I}}$ [Let this expression be equation number (1)]
Now, the expression for angular momentum is given by:
$\Rightarrow L=I\omega $
Now, using the value of angular velocity from equation number (1), we have:
$\begin{align}
  & \Rightarrow L=I\sqrt{\dfrac{2{{K}_{rot}}}{I}} \\
 & \therefore L=\sqrt{2I{{K}_{rot}}} \\
\end{align}$
Hence, the angular momentum of the body in terms of its rotational kinetic energy and moment of inertia comes out to be $\sqrt{2I{{K}_{rot}}}$ .

So, the correct answer is “Option C”.

Note: These are some basic problems in which one parameter is to be calculated in terms of some other given parameters. Also, all the formulas in rotation can be derived and explained with the help of terms used in linear motion. For example, moment of inertia to mass, angular velocity to linear velocity, angular momentum to linear momentum, etc. We should always remember these as they can be helpful in solving a complex problem.