Question

A cone-shaped funnel is being rotated with constant angular velocity $\omega$. An object is placed on the inner wall of the funnel. The object can freely move along the surface of the cone, but during the rotation of the cone, it is in a state of equilibrium. Choose the correct statement(s).

• A. Radius of circular path of the body $=\frac{g\cot \theta }{{\omega }^2}$
• B. Radius of circular path of the body $=\frac{g\tan \theta }{{\omega }^2}$
• C. The equilibrium is stable.
• D. The equilibrium is unstable.

Answer 1

Answer: (A), (D)

The equilibrium is unstable.

FBD of the object in rotational frame (cone):


For equilibrium,
$N=mg\sin \theta +F_c\cos \theta$
& $F_c\sin \theta =mgcos\theta$

where $F_c=m{\omega }^{2}r$ is the centrifugal force, $r$ is the radius of the circle in which the object rotates.
$\Rightarrow m{\omega }^{2}r\sin \theta =mg\cos \theta$
or $r=\frac{g\cot \theta }{(\omega {)}^2}$

If we displace the object by a small distance, then we can see that it will tend to move away from the position of equilibrium. Hence, the equilibrium is unstable.

Option (a) and (d) are correct.