Question

A sphere of radius $R$ is in contact with a wedge. The point of contact is from the ground as shown in the figure. Wedge is moving with velocity $20m{s}^{-1}$ toward left then the velocity of the sphere at this instant will be:

• A. $20m{s}^{-1}$
• B. $15m{s}^{-1}$
• C. $16m{s}^{-1}$
• D. $12m{s}^{-1}$

Answer 1

The correct option is B $15m{s}^{-1}$

Let velocity of center of mass be $v_{cm}$ as shown in the diagram.

The contact point A will have velocity as the sum of velocity of center of mass and due to the rotation.

Now, net velocity of point A is 20 m/s leftwards as it is constrained to have velocity in the left direction only. This means that the vertical component is zero.

From trigonometry, it can be proved that $\cos \theta =\frac{4R/5}{R}=\frac{4}{5}$

Making vertical component of point A as zero,

$R\omega \sin \theta =v_{cm}$

Using constraint of horizontal component of velocity of A,

$R\omega \cos \theta =20$

Eliminating $\omega ,v_{cm}=20\tan \theta$

$v_{cm}=15m/s$