An annular disk of mass , inner radius and outer radius is placed on a horizontal surface with coefficient of friction , as shown in the figure. At some time, an impulse is applied at a height above the center of the disk. If , then the disk rolls wit

Question

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 $J_{0} \hat{x}$ 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 [2023]

Smart Achievers · Accepted Answer

(1, 2, 3, 4)

Impulse $a$ so

$J_{0} = m v ...(i)$

$J_{0} h_{m} = I_{c} ω \Rightarrow h_{m} = \frac{I_{c} ω}{J_{0}} = \frac{I_{c} ω}{m v} \Rightarrow h_{m} = \frac{I_{c}}{m R} [∵ V = R ω]$

$(1) If a = 0, I_{c} = \frac{1}{2} m b^{2} and R = b$ $∴$ $h_{m} = \frac{b}{2}$

$(2) If a = b, I_{c} = m b^{2} and R = b$ $∴$ $h_{m} = b$

$(3) v = \frac{J_{0}}{m} \Rightarrow 100 = \frac{V}{R} = \frac{J_{0}}{m R}$

$(4)$ As force is acting on centre of mass so no rotation for $μ = 0$ and $h = 0$ , the wheel always slides without rolling.

Solution

1 For $μ \neq 0$ and $a \to 0$ , $h_{m} = \frac{b}{2}$ .

2 For $μ \neq 0$ and $a \to 0$ , $h_{m} = b$ .

3 For $h = h_{m}$ the initial angular velocity does not depend on the inner radius $a$ .

4 For $μ = 0$ and $h = 0$ , the wheel always slides without rolling.

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Solution

1 For μ≠0 and a→0, hm=b2.

2 For μ≠0 and a→0, hm=b.

3 For h=hm the initial angular velocity does not depend on the inner radius a.

4 For μ=0 and h=0, the wheel always slides without rolling.

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1 For $μ \neq 0$ and $a \to 0$ , $h_{m} = \frac{b}{2}$ .

2 For $μ \neq 0$ and $a \to 0$ , $h_{m} = b$ .

3 For $h = h_{m}$ the initial angular velocity does not depend on the inner radius $a$ .

4 For $μ = 0$ and $h = 0$ , the wheel always slides without rolling.