First slide

Rolling motion

Question

A spherical ball of radius r initially at rest on a rough horizontal surface is hit horizontally at a point at a distance x above the central line. Due to this sharp impulse the centre of the ball acquires a velocity v_c. After some time, the ball will start pure rolling with a velocity equal to

Difficult

A

$\frac{5}{7} v_{c} [\frac{x + r}{r}]$
B

$\frac{2}{7} v_{c} [\frac{x + r}{r}]$
C

$\frac{5}{7} v_{c} [\frac{x + r}{x}]$
D

$\frac{2}{7} v_{c} [\frac{x + r}{x}]$

Solution

Initially ball will gain both linear and angular velocity.
$\begin{array}{l} Linear impulse: I_{0} = m v_{C} \\ Angular impulse: I_{0} x = I ω_{0} \Rightarrow m v_{C} x = I ω_{0} \end{array}$
I is moment of Inertia about C.
Apply conservation of angular momentum about lowest point.
$I ω_{0} + m v_{C} r = I ω + m v r$
$where ω = \frac{v}{r} at the time of pure rolling$
$\begin{array}{l} \Rightarrow m v_{C} x + m v_{C} r = \frac{2}{5} m r^{2} \frac{v}{r} + m v r \\ \Rightarrow v_{C} (x + r) = \frac{7}{5} r v \\ \Rightarrow v = \frac{5}{7} v_{C} [\frac{x + r}{r}] \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App