First slide

Rolling motion

Question

A disc of radius r is rotating about its center with an angular speed $ω_{0}$ . It is gently placed on a rough horizontal surface. After what time it will start pure rolling?

Moderate

A

$\frac{ω_{0} r}{2 μ g}$
B

$\frac{ω_{0} r}{3 μ g}$
C

$\frac{ω_{0} r}{μ g}$
D

$\frac{3 ω_{0} r}{2 μ g}$

Solution

Given radius of disc= r
Angular speed= $ω_{0}$
Applying law of conservation of angular momentum about a point on the horizontal surface,
$\begin{array}{l} I ω_{0} = m v r + I ω \\ \Rightarrow \frac{m r^{2}}{2} ω_{0} = m v r + \frac{m r^{2}}{2} (\frac{v}{r}) \\ \Rightarrow v = \frac{ω_{0} r}{3} \end{array}$
Velocity of center of disc when it starts pure rolling is $(\frac{ω_{0} r}{3})$
From $v = u + a t$
$\begin{array}{l} \Rightarrow v = a t \\ \Rightarrow t = \frac{v}{a} (∵ a = μ g) \\ \Rightarrow t = \frac{ω_{0} r}{3 μ g} \end{array}$

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