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

For $h = h_{m}$ the initial angular velocity depends on the inner radius a.

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

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

answer is A, B, D.

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