First slide

Rolling motion

Question

A sphere is performing pure rolling with constant velocity V as shown in figure. The radius of curvature of trajectory of point A as shown in the figure is (OA = R/2)

Difficult

A

$\large \frac 94R$
B

$\large \frac R2$
C

$\large \frac 92R$
D

$\large \frac R4$

Solution

$\large \vec V_A=\vec V_O+\vec V_{A-O}$
$\large \therefore V_A= V+\omega (OA)$
$\large =V+\left ( \frac VR \right )\frac R2=\frac {3V}{2}$
Acceleration of A relative to ground $\large =\frac {(3V/2)^2}{r}=\frac {9V^2}{4r}$
Also, $\large \vec a_A=\vec a_O+\vec a_{A-O}\Rightarrow a_A=0+\omega^2(R/2)=\frac {V^2}{2R}$
$\large \therefore \frac {9V^2}{4r}=\frac {V^2}{2R}\Rightarrow r =\frac {9R}{2}$

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