Question

A ring rolls without slipping on the ground. Its centre C moves with a constant speed $u$. P is any point on the ring. The speed of P with respect to the ground is $v$.

• A. $0\le v\le 2u$
• B. $v=u$, if CP is horizontal
• C. $v=u$, if CP makes an angle of ${30}^0$ with the horizontal and P is below the horizontal level of C
• D. $v=\sqrt{2u}$, if CP is horizontal

Answer 1

Answer: (A), (C), (D)

The correct options are
A $0\le v\le 2u$
C $v=u$, if CP makes an angle of ${30}^0$ with the horizontal and P is below the horizontal level of C
D $v=\sqrt{2u}$, if CP is horizontal

The bottom point remains stationary at all times thereby having zero velocity.
Velocity at centre is $v$.
If the angular velocity is $\omega$, and radius as $R$ we have the velocity of the wheel at the top and bottom as-
$v_{top}=u+R\omega$ and $v_{bottom}=u-R\omega$
We know $v_{bottom}$ as zero giving $u=R\omega$

Thus we have $v_{top}=2u$
Thus we conclude that the velocity varies between $0$ and $2u$.
If CP is horizontal then we would have two components of the velocity, one is u in horizontal direction and the other is $v=R\omega$ in the vertical direction which is again equal to u. Thus the resultant is $\sqrt{2}u$.