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 ℐ0𝑥̂ 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?

• A. For 𝜇 ≠ 0 and 𝑎 → 0, ℎ𝑚 = 𝑏/2
• B. For 𝜇 ≠ 0 and 𝑎 → 𝑏, ℎ𝑚 = 𝑏
• C. For ℎ = ℎ𝑚, the initial angular velocity does not depend on the inner radius a
• D. For 𝜇 = 0 and ℎ = 0, the wheel always slides without rolling.

Answer 1

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

For 𝜇 ≠ 0 and 𝑎 → 0, ℎ𝑚 = 𝑏/2

Answer 2

(1) For 𝜇 ≠ 0 and 𝑎 → 0, $ℎ_𝑚 = \frac{𝑏}{2}$. This statement suggests that when the coefficient of friction is nonzero and the inner radius (a) tends to 0, the height (ℎ𝑚) at which the impulse is applied is equal to half the outer radius($\frac{b}{2}$). This statement is correct. When the inner radius approaches 0, the entire disk participates in the rolling motion, and the height at which the impulse is applied becomes equal to half the outer radius. Therefore, this option is correct.
(2) For 𝜇 ≠ 0 and 𝑎 → 𝑏, ℎ𝑚 = 𝑏. This statement suggests that when the coefficient of friction is nonzero and the inner radius (a) tends to the outer radius (b), the height (ℎ𝑚) at which the impulse is applied is equal to the outer radius (b). This statement is correct. When the inner radius approaches the outer radius, the entire disk participates in the rolling motion, and the height at which the impulse is applied becomes equal to the outer radius. Therefore, this option is correct.
(3) For ℎ = ℎ𝑚, the initial angular velocity does not depend on the inner radius a. This statement is correct. The initial angular velocity of the disk does not depend on the inner radius a. The initial angular velocity is determined by the impulse applied and the moment of inertia of the disk. The inner radius does not affect these factors. Therefore, this option is correct.
(4) For 𝜇 = 0 and ℎ = 0, the wheel always slides without rolling. This statement is correct. When the coefficient of friction is zero and the impulse is applied at the same height as the center of mass (ℎ = 0), the wheel will slide without rolling. In this case, there is no friction force to initiate rolling motion, so the wheel will simply slide. Therefore, this option is correct.
Therefore correct statements are (A), (B), (C) and (D)