Question

A solid sphere as shown is rolling without slipping. What is the maximum distance travelled by sphere on the inclined plane ?
Given $g=10{\text{m/s}}^2$

• A. $12\text{m}$
• B. $14\text{m}$
• C. $16\text{m}$
• D. $18\text{m}$

Answer 1

The correct option is B $14\text{m}$

Given that,
Mass of the sphere, $m=1\text{kg}$
Linear velocity of the sphere, $v=10\text{m/s}$
Let the sphere travel a height $h$ on the inclined plane.
On applying law of conservation of mechanical energy between bottom and top most point,
$M.E_i=M.E_f$
$\Rightarrow \frac{1}{2}m{v}^2+\frac{1}{2}I{w}^2=mgh$
$\Rightarrow \frac{1}{2}m{v}^2+\frac{1}{2}\times \frac{2}{5}m{r}^2\times \frac{v^2}{r^2}=mgh[I=\frac{2}{5}m{r}^2,v=wr]$
$\Rightarrow \frac{7}{10}{v}^2=gh$
$\Rightarrow h=\frac{7v^2}{10g}$
$\Rightarrow h=\frac{7\times {10}^2}{10\times 10}=7\text{m}$

So, maximum length travelled on inclined plane,
$l=\frac{7}{\sin {30}^{\circ }}$
$\Rightarrow l=\frac{7}{\frac{1}{2}}=14\text{m}$

$\begin{array}{c}\text{Why this question ?}\\ \text{Concept - Condition for pure rolling when the surface on which the body is rolling does not move is}v=wr\\ \text{and at topmost point, the body stops for an instant.}\end{array}$