First slide

Rolling motion

Question

A spool is rolling without slipping on a strip with velocity v as shown in figure. Then acceleration of point A relative to ground is

Moderate

A

$\frac{5 v^{2}}{R}$
B

$\frac{v^{2}}{2 R}$
C

$\frac{3 v^{2}}{R}$
D

$\frac{5 v^{2}}{2 R}$

Solution

Length $AO = \sqrt{R^{2} {(2 R)}^{2}} = \sqrt{5} R$
$∴$ Velocity of A relative to ground
= Velocity of A relative to O $= ω (OA) = (\frac{v}{R}) \sqrt{5 R}$
$= \sqrt{5} V$
Now acceleration of centre C, a_C = 0
Also, ${\vec{a}}_{A} = {\vec{a}}_{C} + {\vec{a}}_{A / C} = {\vec{a}}_{A / C} \Rightarrow a_{A} = a_{A / C} = \frac{v_{A}^{2}}{(CA)} = \frac{{(\sqrt{5} v)}^{2}}{2 R} = \frac{5 V^{2}}{2 R}$

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