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 J0x^ is applied at a height h above the center of the disk. If h = hm then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?

For h = h m , the initial angular velocity does not depend on the inner radius a , for μ ≠ 0 and a → 0 , h m = b / 2 , For μ ≠ 0 and a → b , h m = b , for μ = 0 and h = 0 the wheel always slides without rolling

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 J0x^ is applied at a height h above the center of the disk. If h = hm then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?

J 0 h = I cm ω ω = j 0 h I cm Linear impulse= p j 0 = P V = j 0 M Hence ω = j 0 M b When a→b i = Mb 2 ω = j 0 h Mb 2 h = b

for μ ≠ 0 and a → 0 , h m = b / 2

For μ ≠ 0 and a → b , h m = b

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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 $J_{0} \hat{x}$ 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?

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

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

For $μ \neq 0 and a \to b, h_{m} = b$

for $μ = 0 and h = 0$ the wheel always slides without rolling

answer is A, B, C, D.

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Detailed Solution

$J_{0} h = I_{cm} ω ω = \frac{j_{0} h}{I_{cm}}$
Linear impulse= p
$j_{0} = P V = \frac{j_{0}}{M}$
Hence $ω = \frac{j_{0}}{M_{b}}$
When a→b
$i = {Mb}^{2} ω = \frac{j_{0} h}{{Mb}^{2}} h = b$

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