JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Rotational Mechanics

A flat surface of a thin uniform disk

A

of radius

R

is glued to a horizontal table. Another thin uniform disk

B

of mass

M

and with the same radius

R

rolls without slipping on the circumference of

A

, as shown in the figure. A flat surface of

B

also lies on the plane of the table. The center of mass of

B

has fixed angular speed

ω

about the vertical axis passing through the center of

A

. The angular momentum of

B

n M ω R^{2}

with respect to the center of

A

. Which of the following is the value of

n

Answer: (B)

Solution

The center of mass of disk B moves in a circle of radius

2 R

with angular speed

ω

, so

v = 2 R ω

.

For rolling without slipping:

v = ω_{0} R

, giving

ω_{0} = 2 ω

(angular speed of B about its own center).

Angular momentum about center of A:

L = m r \times v_{c} + I_{c} ω_{0}

where

I_{c} = \frac{1}{2} M R^{2}

for a disk.

L = M (2 R) (2 R ω) + \frac{1}{2} M R^{2} (2 ω) = 4 M R^{2} ω + M R^{2} ω = 5 M R^{2} ω

Therefore,

n = 5

JEE Advanced · 2022 · 28 Aug 2022 · Paper 2 · Single correct option