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.
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For rotational motion, remember that the centripetal force depends on the mass, the radius of rotation, and the square of the angular velocity.
The liquid's force \( F \) is linked to the centripetal force needed for rotation. The centripetal force at radius \( r \) from the center is defined as:\[F_{\text{centripetal}} = m r \omega^2.\]The liquid's total force is the aggregate of forces along the tube's length. Given the liquid's mass is \( 2M \) and the applied force is directly proportional to the square of the angular velocity, the relationship is:\[F = \alpha M \omega^2.\]Consequently, the coefficient \( \alpha \) is determined as:\[\boxed{\frac{F}{M \omega^2}}.\]
