Question

Two bodies have their moments of inertia I and 2I respectively about their axis of rotation. If their kinetic energies of rotation are equal, their angular momentum will be in the ratio:
A. \[1:2\]
B. \[\sqrt 2 :1\]
C. \[1:\sqrt 2 \]
D. \[2:1\]

Answer 1

Hint:The total kinetic energy of a body in motion is the sum of its rotational and translational kinetic energies. The translational kinetic energy depends upon the mass and linear velocity of the object. The rotational kinetic energy can be calculated for a solid if the moment of inertia about the axis of rotation and the angular velocity of that solid is known.

Formula Used:
Total kinetic energy,
\[K.E. = K.E{._{rot}} + K.E{._{tran{s_{}}}}\]
Here, for translational motion part,
\[K.E{._{tran{s_{}}}} = \dfrac{1}{2}m{v^2}\]
Here, for rotational motion part,
\[K.E{._{rot}} = \dfrac{1}{2}I{\omega ^2}\]
Thus, \[K.E. = \dfrac{1}{2}m{v^2} + \dfrac{1}{2}I{\omega ^2}\]
Where m = mass of the object, v = linear velocity, \[\omega \]= angular velocity, I = moment of inertia and R is the radius of the object.

Complete step by step solution:
Given: Two bodies (let's assume A and B) with their moments of inertia I and 2I respectively about their axis of rotation. Their rotational kinetic energies are equal.
\[{I_A} = \dfrac{1}{2}{I_B}\]---- (1)
Let their angular velocities be \[{\omega _A}\] and \[{\omega _B}\] respectively. According to the question,
\[{(K.E{._{rot}})_A} = {(K.E{._{rot}})_B}\]----(2)
And,
\[K.E{._{rot}} = \dfrac{1}{2}I{\omega ^2}\]-----(3)

From equations (2) and (3),
\[\dfrac{1}{2}{I_A}{\omega _A}^2 = \dfrac{1}{2}{I_B}{\omega _B}^2\]----- (4)
Substituting from equation (1) in (4),
\[\dfrac{1}{2}{I_A}{\omega _A}^2 = \dfrac{1}{4}{I_A}{\omega _B}^2\]
\[\Rightarrow {\omega _A}^2 = \dfrac{2}{4}{\omega _B}^2\]
\[\Rightarrow \dfrac{{{\omega _A}}}{{{\omega _B}}} = \dfrac{1}{{\sqrt 2 }}\]
\[\therefore {\omega _A}:{\omega _B} = 1:\sqrt 2 \]

Hence option C is the correct answer.

Note: For a given system with rotational motion, the potential energy of the system is the same as in case with linear motion. The kinetic energy of such a system is different from the kinetic energy of a linear system. There are two components of kinetic energy. One component is the kinetic energy corresponding to translational motion with linear velocity v. The rotational motion component with angular velocity \[\omega \]. The moment of inertia for rotational kinetic energy depends upon the physical properties of the object.