Question

$A$ small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac{3v^2}{4g}$ with respect to the initial position. The object is :

• A. ring
• B. solid sphere
• C. hollow sphere
• D. disc

Answer 1

Answer: (D)

disc

Initially velocity of COM of the object is $v$ and its angular velocity be $w$. When the object reaches the maximum height, then its linear and angular velocity becomes zero.

Also $v=rw$ (pure rolling) where $r$ is the radius of the object

Now work-energy theorem: $W=\Delta K.E$

$-mgh=0-\frac{1}{2}m{v}^2-\frac{1}{2}I{w}^2$

$mgh=+\frac{1}{2}m{v}^2+\frac{1}{2}\times Km{r}^2{w}^2$

$mg\times \frac{3v^2}{4g}=+\frac{1}{2}m{v}^2+\frac{1}{2}\times Km{v}^2$

$⟹\frac{3}{2}=1+K$

Thus $K=\frac{1}{2}⟹$ the object is a disc