Question

A solid cylinder having a radius of $0.4\text{m}$, initially rotating (at $t=0$) with angular velocity ${\omega }_0=54\text{rad/s}$ is placed on a rough inclined plane with $\theta ={37}^{\circ }$ having friction coefficient $\mu =0.5$. The time taken by the cylinder to start pure rolling is
[Take $g=10{\text{m/s}}^2$]

• A. $5.4\text{s}$
• B. $2.4\text{s}$
• C. $4.4\text{s}$
• D. $1.1\text{s}$

Answer 1

The correct option is D $1.1\text{s}$

Given:
Initial angular speed, ${\omega }_0=54\text{rad/s}$
Initial linear speed, $v_0=0$

Bottommost point of the cylinder has a tendency to slip upwards.

Hence, direction of friction is downward along the inclined plane as we can see in the figure. Friction will act such that torque due to it will decrease the angular speed and increase the linear speed.


On drawing the free body diagram, we obtain.


So, $mg\sin {37}^{\circ }+f=ma$

& $N=mg\cos {37}^{\circ }$

$\Rightarrow mg\sin {37}^{\circ }+\mu mg\cos {37}^{\circ }=ma$

$[f=\mu N]$

$\Rightarrow a=g\sin {37}^{\circ }+\mu g\cos {37}^{\circ }$

Using, $g=10{\text{m/s}}^2$ and $\mu =0.5$ we get,

$a=10[\frac{3}{5}+0.5\times \frac{4}{5}]=10{\text{m/s}}^2$

Further we have the first equation of motion, $v=u+at$

$\Rightarrow v=0+10\times t=10t......\left(1\right)$

Now, using the balance of torque, $\tau =fr=I\alpha$ [Anticlockwise]

$\Rightarrow \mu mg\cos {37}^{\circ }\times r=\frac{m{r}^2}{2}\times \alpha$

$\Rightarrow \alpha =\frac{2\mu g\cos {37}^{\circ }}{r}$
$\Rightarrow \alpha =\frac{2\times 0.5\times 10\times \frac{4}{5}}{0.4}=20{\text{rad/s}}^2$

Further, $\omega ={\omega }_0+\alpha t$

$\Rightarrow \omega =54-20\times t......\left(2\right)$
[As the acceleration is retarding]

When cylinder starts pure rolling, $v=\omega r$

$\Rightarrow 10t=(54-20\times t)\times 0.4$

$\Rightarrow 25t=50-20t$

$\Rightarrow t=1.1\text{s}$

$\begin{array}{c}\text{Why this question}?\\ \text{Concept - Direction of friction is dependent on which}\\ \text{motion (translation or rotation) is more or less.}\\ \text{If translation is less, it acts so as to increase}\\ \text{translation and decrease rotation, so that}\\ \text{the object starts pure rolling after some time.}\end{array}$