Question

A rotor of radius $r$ is rotating about its own vertical axis and a person in contact with inner wall of rotor remains in equilibrium without slipping down. If $\omega$ is angular velocity of rotor and $\mu$ is minimum coefficient of friction between person and the wall of rotor then which of the following are correct?
A) $\mu \propto {\omega }^2$
B) $\mu \propto \frac{1}{r}$
C) $\mu \propto \frac{1}{{\omega }^2}$
D) $\mu \propto r$

• A. A and B are true
• B. A and D are true
• C. B and C are true
• D. C and D are true

Answer 1

Answer: (C)

B and C are true

Given rotor radius $r,$ angular velocity $=\omega ,$ coefficient of friction $=\mu$

assume the mass of person is $m$

and person is in contact with inner wall of rotor.

From the Free Body Diagram:

$\Rightarrow F_C-F_N=0..........\left(1\right)$

Centrifugal Force $F_C=m{\omega }^{2}r......\left(2\right)$

$\Rightarrow F_N=m{\omega }^{2}R$

Friction force $f$ is given by

$\Rightarrow f=\mu F_N$

Person remains in equilibrium without slipping down, it means weight of person downward will be balanced with friction force $f$ upwards

$\Rightarrow f=mg$

$\Rightarrow \mu F_N=mg\Rightarrow \mu m{\omega }^{2}R=mg\Rightarrow \mu =\frac{g}{R{\omega }^2}$

so if R is constant then $\mu \propto \frac{1}{{\omega }^2}$

and if $\omega$ is constant then $\mu \propto \frac{1}{R}$

so the option $C$ is true