A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio R / h will be .

Question

A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio R / h will be . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Rolling is rotation about a point of contact.

Applying angular momentum equation about

P

,

M v (R + h) = I_{P} ω

As sphere rolls,

v = ω R

and

I_{P} = \frac{2}{5} m R^{2} + m R^{2} = \frac{7}{5} m R^{2}

∴ M ω R (R + h) = \frac{7}{5} M R^{2} ω

R + h = \frac{7}{5} R

h = \frac{2}{5} R

\frac{R}{h} = \frac{5}{2} = 2.5

Q. A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio R/h will be ________.

Rolling is rotation about a point of contact. Applying angular momentum equation about P, Mv(R+h)=IP​ω As sphere rolls, v=ωR and IP​=52​mR2+mR2=57​mR2 ∴MωR(R+h)=57​MR2ω R+h=57​R h=52​R hR​=25​=2.5

Q. A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height $h$ from centre. The sphere starts rolling just after the application of impulse. The ratio $R / h$ will be ________.