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 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?
D. For μ = 0 and h = 0 , the wheel always slides without rolling
As we know, J → = ∆ P → ⇒   J 0 = M V                             . . .   i About center of the disk, we can also write J 0 × h = I C M ω ⇒   J 0 × h = M 2 a 2 + b 2 ω                             . . .   i i If h = h m ⇒ ω = V b pure rolling From equation(ii), we can write ⇒   J 0 × h m = M 2 a 2 + b 2 V b                     . . .   iii By dividing equation ( i ) with the equation ( iii ) , we get ⇒   h m = a 2 + b 2 2 b Now, considering all cases: If a → 0 ⇒ h m = b 2 If a → b ⇒ h m = b If h = h m ⇒ ω is independent of a (As we get ω = V b ) If μ = 0   and   h = 0 ⇒ Wheel will slide without rolling.
Related: Physics — Rotational Motion · All PYQ Banks