A large drum having radius R is spinning around its axis with angular velocity ω, as shown in figure. The minimum value of ω 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 μ, is :
Q. A large drum having radius R is spinning around its axis with angular velocity ω, as shown in figure. The minimum value of ω 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 μ, is :
A. \( \sqrt{\dfrac{2g}{\mu R}} \)
B. \( \sqrt{\dfrac{g}{2\mu R}} \)
C. \( \sqrt{\dfrac{\mu g}{R}} \)
D. \( \sqrt{\dfrac{g}{\mu R}} \)
Correct Answer: \( \sqrt{\dfrac{g}{\mu R}} \)

Step 1: Centripetal Force Condition

Normal reaction provides centripetal force:

\[ N = M \omega^2 R \]

Step 2: Friction Balances Weight

For minimum ω, friction is maximum:

\[ f_{max} = \mu N \]

Equilibrium vertically:

\[ \mu N = Mg \]

Step 3: Substitute N

\[ \mu (M\omega^2 R) = Mg \]

Cancel M:

\[ \mu \omega^2 R = g \]

Step 4: Final Expression

\[ \omega^2 = \frac{g}{\mu R} \]

\[ \boxed{\omega = \sqrt{\frac{g}{\mu R}}} \]

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