Question

A small object of uniform density rolls (without slipping) up a frictionless curved surface with an initial velocity $v$. It reaches upto 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

The correct option is D disc

As there is no non-conservative force acting on the body, therefore total mechanical energy will remain constant.


At the maximum height, object will stop momentarily i.e $(v=0\&\omega =0)$.
Applying mechanical energy conservation,
$\text{Loss in kinetic energy}=\text{Gain in potential energy}$
$\Rightarrow K.E_i-K.E_f=P.E_f-P.E_i$
Here, $K.E_i=K.E_{\text{Rot}}+K.E_{\text{Trans}}$
$\Rightarrow (\frac{1}{2}I{\omega }^2+\frac{1}{2}m{v}^2)-0=mgh-0...\left(i\right)$
$\because v=\omega r,$ for pure rolling and $h=\frac{3v^2}{4g}$
Considering reference of $PE=0$ at the initial position.
$\Rightarrow \frac{1}{2}I\times \frac{v^2}{r^2}+\frac{1}{2}m{v}^2=mg\times \frac{3v^2}{4g}$
$\therefore I=\frac{1}{2}m{r}^2$
This represents moment of inertia of a disc.
$\Rightarrow$The object is a disc.