Question

) Two bodies have their moments of inertia I and 21
respectively, about their axis of rotation. If their
kinetic energies of rotation are equal, then the ratio
of their angular momenta will be (AIPMT 2005)
(a) 2:1 (b) 1:2
(c) 2:1 (d) 1 : 2​

Answer 1

Given :

▪ Moment of inertia of body A = I

▪ Moment of inertia of body B = 2I

▪ Their kinetic energy of rotation are equal.

To Find :

▪ The ratio of their angular momentum.

Concept :

↗ As for linear motion,

$\bigstar\:\underline{\boxed{\bf{\red{KE=\dfrac{P^2}{2m}}}}}$

↗ Similarly, for rotational motion,

$\bigstar\:\underline{\boxed{\bf{\blue{(KE)_{rot}=\dfrac{L^2}{2I}}}}}$

Terms indication :

• P denotes linear momentum
• L denotes angular momentum
• m denotes mass
• I denotes moment of inertia

Calculation :

→ As said, (KE)rot remains same.

$\dashrightarrow\sf\:\dfrac{1}{2}I_A{\omega_A}^2=\dfrac{1}{2}I_B{\omega_B}^2\\ \\ \dashrightarrow\sf\:\dfrac{1}{2I_A}(I_A\omega_A)^2=\dfrac{1}{2I_B}(I_B\omega_B)^2\\ \\ \dashrightarrow\sf\:\dfrac{{L_A}^2}{I_A}=\dfrac{{L_B}^2}{I_B}\\ \\ \dashrightarrow\sf\:\dfrac{L_A}{L_B}=\sqrt{\dfrac{I_A}{I_B}}\\ \\ \dashrightarrow\sf\:\dfrac{L_A}{L_B}=\sqrt{\dfrac{I}{2I}}\\ \\ \dashrightarrow\sf\:\dfrac{L_A}{L_B}=\dfrac{1}{\sqrt{2}}\\ \\ \dashrightarrow\underline{\boxed{\bf{\purple{L_A:L_B=1:\sqrt{2}}}}}\:\gray{\bigstar}$