The ratio of the moment of inertia of the two objects A and B is 1 : 2. If the ratio of the angular velocity of the object A to object B is 2 : 1, then find the ratio of the angular momentum of the object A to object B.

CONCEPT:

Angular momentum:

• Angular momentum is the property of any rotating object given by moment of inertia times angular velocity.
  • It is a vector quantity.
  • Its SI unit is kg-m2/sec.
• If I and ω are the moment of inertia and the angular velocity respectively, then the angular momentum is given as,

⇒ L = Iω

⇒ L = rP

Where r = radius of rotation and P = linear momentum

CALCULATION:

Given \(\frac{I_A}{I_B}=\frac{1}{2}\), and \(\frac{\omega_A}{\omega_B}=\frac{2}{1}\)

• Angular momentum is the property of any rotating object given by moment of inertia times angular velocity.
• If I and ω are the moment of inertia and the angular velocity respectively, then the angular momentum is given as,

⇒ L = Iω     -----(1)

• By equation 1 the angular momentum of the object A is given as,

⇒ L_A = I_Aω_A     -----(2)

• By equation 1 the angular momentum of the object B is given as,

⇒ LB = IBωB     -----(3)

By equation 2 and equation 3,

\(\Rightarrow \frac{L_A}{L_B}=\frac{I_A\omega_A}{I_B\omega_B}\)

\(\Rightarrow \frac{L_A}{L_B}=\frac{1}{2}\times\frac{2}{1}\)

\(\Rightarrow \frac{L_A}{L_B}=\frac{1}{1}\)

• Hence, option 1 is correct.