Question 34: Work, Energy and Power & Circular Motion

Question

A large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in figure.
The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is :

Accepted Answer

When the drum rotates, the mass M experiences forces :
1. Weight $(\mathrm{Mg})$ acts vertically downwards.
2. Normal reaction ( N ) acts horizontally toward the centre of the drum. This provides the necessary centripetal force.
3. Frictional force (f) acts vertically upwards, opposing the tendency of the mass to slide down due to gravity.
For the body to remain rest with respect to the cylindrical surface, the vertical forces must be balanced :
$$ \mathrm{f}=\mathrm{Mg} $$
The horizontal normal force provides the centripetal acceleration ( $\mathrm{a}_{\mathrm{c}}=\mathrm{R} \omega^2$ ) :
$$ N=M R \omega^2 $$
The frictional force (f) is a self-adjusting force up to a maximum limit (limiting friction). For the body to stay at rest relative to the wall :
$$ \mathrm{f} \leq \mathrm{f}_{\max } $$
$$\Rightarrow $$ $\mathrm{f} \leq \mu \mathrm{N}$
Substituting the values of f and N into this inequality :
$$ M g \leq \mu\left(M R \omega^2\right) $$
$$\Rightarrow $$ $g \leq \mu R \omega^2$
$$\Rightarrow $$ $\omega^2 \geq \frac{g}{\mu R}$
$$\Rightarrow $$ $$ \omega \geq \sqrt{\frac{g}{\mu R}} $$
Therefore, the minimum angular speed is $\omega_{\min }=\sqrt{\frac{\mathrm{g}}{\mu \mathrm{R}}}$.
Hence, the correct option is (D).

Answer

$\sqrt{\dfrac{2g}{\mu R}}$

Answer

$\sqrt{\dfrac{g}{2\mu R}}$

Answer

$\sqrt{\dfrac{\mu g}{R}}$

Answer

$\sqrt{\dfrac{g}{\mu R}}$

Work, Energy and Power & Circular Motion

A large drum having radius R is spinning around its axis with angular velocity \omega, as shown in figure. The minimum value of \omega so that a body of mass M remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass M as \mu, is :

Choose the correct answer: