An annular disk of mass $M$, inner radius $a$ and outer radius $b$ is placed on a horizontal surface with coefficient of friction $\mu$, as shown in the figure. At some time, an impulse $J_{0} \hat{x}$ is applied at a height h above the center of the disk. If $h=h_{m}$ then the disk rolls without slipping along the $x$-axis. Which of the following statement(s) is(are) correct ?

(A) For $\mu \neq 0$ and $a \rightarrow 0, h_{m}=b / 2$; (B) For $\mu \neq 0$ and $a \rightarrow b, h_{m}=b$; (C) For $h=h_{m}$, the initial angular velocity does not depend on the inner radius $a$.; (D) For $\mu=0$ and $h=0$, the wheel always slides without rolling.

JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Rotational Mechanics

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An annular disk of mass

M

, inner radius

a

and outer radius

b

is placed on a horizontal surface with coefficient of friction

μ

, as shown in the figure. At some time, an impulse

J_{0} \overset{x}{^}

is applied at a height h above the center of the disk. If

h = h_{m}

then the disk rolls without slipping along the

x

-axis. Which of the following statement(s) is(are) correct ?

(A)For $μ \neq = 0$ and $a \to 0, h_{m} = b /2$ ✓ Correct
(B)For $μ \neq = 0$ and $a \to b, h_{m} = b$ ✓ Correct
(C)For $h = h_{m}$ , the initial angular velocity does not depend on the inner radius $a$ .✓ Correct
(D)For $μ = 0$ and $h = 0$ , the wheel always slides without rolling.✓ Correct

Answer: (A), (B), (C), (D)

Solution

For rolling without slipping:

v = ω R

.

From impulse:

J_{0} = m v

and

J_{0} h_{m} = I_{c} ω

.

Combining:

h_{m} = \frac{I _{c}}{m R}

.

(A) For

a \to 0

I_{c} = \frac{1}{2} m b^{2}

and

R = b

, so

h_{m} = \frac{b}{2}

. ✓

(B) For

a \to b

I_{c} = m b^{2}

and

R = b

, so

h_{m} = b

. ✓

(C) Angular velocity

ω = \frac{J _{0}}{m R}

, which depends only on

R

(outer radius), not on

a

. ✓

(D) For

μ = 0

and

h = 0

: impulse acts at center of mass, producing only translation with no torque, so no rotation occurs — pure sliding. ✓

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JEE Advanced · 2023 · 04 Jun 2023 · Paper 2 · Rotational Mechanics

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