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

The Correct Option is A, B, C, D

(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 (ℎ_𝑚) 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, ℎ_𝑚 = 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 = ℎ_𝑚, 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)

Answer 2

The moment of inertia 𝐼_cm​ of a ring-shaped disc, with an inner radius 𝑎a, an outer radius b, and a mass m, about an axis perpendicular to the disc and passing through its center C, is given by:

\(𝐼_{cm}=\frac{1}{𝑚}(𝑎^2+𝑏^2)\)
where 𝑎² and 𝑏² are the squares of the inner and outer radii, respectively.

The linear impulse 𝐽_𝑜​ changes the linear momentum of the disc. Right after the impulse is applied, the linear momentum can be represented as:

𝑚𝑣_cm=𝐽_𝑜​
where m is the mass of the disc and v_cm​ is the velocity of the center of mass resulting from the impulse.

The angular impulse about the center of mass C of the disc is applied in a clockwise direction, with its magnitude being 𝐽_𝑜,ang=ℎ𝐽_𝑜​. These impulse changes the angular momentum of the disc about point C, expressed as:

𝐼_cm𝜔=𝐽_𝑜,_ang=ℎ𝐽_𝑜​

where Icm​ is the moment of inertia of the disc about its center of mass, ω is the angular velocity resulting from the angular impulse, Jo​ is the linear impulse, and h is a factor relating the angular impulse to the linear impulse.

For the disc to roll without slipping, given that the friction coefficient μ is non-zero, the condition is 𝑣_cm=𝜔_𝑏. By substituting the expressions for 𝑣cm​ and ω and solving, we obtain the following relationship for the minimum value of h (denoted as hm​):

\(h_m \frac{I_{\text{cm}}}{m_b} = \frac{a^2 + b^2}{26}\)

where 𝐼_cm​ is the moment of inertia of the disc about its center of mass, m is the mass of the disc, a is the inner radius, and b is the outer radius of the disc.

By substituting 𝑎=0 into the formula, we find that \(h_m = \frac{1}{2}\)This implies that a solid disc will roll without slipping if the impulse is applied midway along the upper half of the disc.

Similarly, substituting 𝑎=𝑏 yields ℎ_𝑚=𝑏. This indicates that a ring (an annular disc with equal inner and outer radii) will roll without slipping when the horizontal impulse is applied at the very top of the ring.

For the condition where ℎ=ℎ_𝑚​, the initial angular velocity ω is given by:

\(\omega = \frac{h_mJ_o}{I _ {cm} \, m_b}\)

This expression for the angular velocity is independent of the inner radius a of the disc.

When a wheel is placed on a smooth surface with no friction (μ=0) and an impulse is applied directly at its center (h=0), there is no torque generated to initiate rotation. Consequently, under these conditions, if an impulse is delivered to the center of a wheel on a frictionless surface, the wheel will invariably slide and not roll.