Question

A hollow vertical cylindrical structure rotates about its axis. Tom is standing on a horizontal platform against inner wall of cylindrical structure. At a certain angular speed of hollow cylinder, the horizontal platform below him is removed but he remains stationary against the inner wall. If the radius of the hollow cylinder is $5\text{m}$ and the coefficient of static friction between the cylindrical wall and Tom is $0.15$, find the minimum angular speed of the cylindrical structure at which the platform may be removed.
[Take $g=10{\text{m/s}}^2$]

• A. $28.26\text{rad/s}$
• B. $3.65\text{rad/s}$
• C. $0.54\text{rad/s}$
• D. $18.26\text{rad/s}$

Answer 1

The correct option is B $3.65\text{rad/s}$

FBD of Tom:
From FBD:
Since the tom is at rest, when the floor is removed
Static frictional force is balancing the weight
$f_s=mg$ ... (1)
As tom is resting of the inner wall of hollow cylinder there is normal reaction force. This provides the necessary centripetal force.
$N=m{\omega }^{2}r$ ... (2)
Since, $f_s={\mu }_{s}N$, from equation (1) we get
${\mu }_{s}N=mg$
From equation (2)
${\mu }_s\left(m{\omega }^{2}r\right)=mg$
$\Rightarrow {\omega }^2=\frac{g}{{\mu }_{s}r}$
$\Rightarrow \omega =\sqrt{\frac{g}{{\mu }_{s}r}}$
Given $r=5\text{m},{\mu }_s=0.15;g=10{\text{m/s}}^2$
$\Rightarrow \omega =\sqrt{\frac{10}{0.15\times 5}}=3.65\text{rad/s}$