Question

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is \( F \) and the angular velocity of the tube is \( \omega \), then the value of \( \alpha \) is _______ in SI units.

Hint:

For rotational motion, remember that the centripetal force depends on the mass, the radius of rotation, and the square of the angular velocity.

Answer 1

The force \( F \) exerted by the liquid is related to the centripetal force required for the rotation. The centripetal force at a point at a distance \( r \) from the center is given by: \[ F_{\text{centripetal}} = m r \omega^2. \] The total force exerted by the liquid is the sum of the forces over the length of the tube. Since the mass of the liquid is \( 2M \), and the force exerted is proportional to the square of the angular velocity, we have: \[ F = \alpha M \omega^2. \] Thus, the value of \( \alpha \) is: \[ \boxed{\frac{F}{M \omega^2}}. \]