Question

Angular momentum $\left(L\right)=mvr$
Moment of inertia $\left(I\right)=m{r}^2$
($m$ is the mass; $v$ is the velocity; $r$ is the radius)
Then the energy is equal to:
(A) ${\displaystyle \frac{L^2}{2I^2}}$
(B) $2L^{2}I$
(C) ${\displaystyle \frac{L^2}{2I}}$
(D) ${\displaystyle \frac{2I}{L^2}}$

Answer 1

Hint The energy can be determined by using the two relations, one is the angular momentum equation and the other is angular velocity equation by using these two equations, the energy can be determined easily in the terms of $L$ and $I$.
Useful formula
The angular momentum equation is given as,
$L=I\omega$
Where, $L$ is the angular momentum of the object, $I$ is the moment of the inertia of the object and $\omega$ is the angular velocity of the object.
The angular velocity equation is given as,
$\omega =\sqrt{{\displaystyle \frac{2E}{I}}}$
Where, $\omega$ is the angular velocity of the object, $E$ is the energy of the object and $I$ is the moment of the inertia of the object.

Complete step by step solution
Given that,
Angular momentum $\left(L\right)=mvr$
Moment of inertia $\left(I\right)=m{r}^2$
($m$ is the mass; $v$ is the velocity; $r$ is the radius)
Now,
The angular momentum equation is given as,
$L=I\omega .......................\left(1\right)$
Where, $L$ is the angular momentum of the object, $I$ is the moment of the inertia of the object and $\omega$ is the angular velocity of the object.
Now,
The angular velocity equation is given as,
$\omega =\sqrt{{\displaystyle \frac{2E}{I}}}......................\left(2\right)$
Where, $\omega$ is the angular velocity of the object, $E$ is the energy of the object and $I$ is the moment of the inertia of the object.
By substituting the equation (2) in the equation (1), then the equation (1) is written as,
$L=I\sqrt{{\displaystyle \frac{2E}{I}}}$
By cancelling the terms in the above equation, then the above equation is written as,
$L=\sqrt{2EI}$
By squaring on both sides, then the above equation is written as,
$L^2=2EI$
By rearranging the terms in the above equation, then the above equation is written as,
$E={\displaystyle \frac{L^2}{2I}}$

Hence, the option (C) is the correct answer.

Note The energy of the object is directly proportional to the square of the angular momentum and the energy of the object is inversely proportional to the momentum of the inertia. As the angular momentum increases, then the energy of the object is also increasing.