4.2 Problems
143
Fig. 4.6
4.31 A tube of length L is filled with an incompressible liquid of mass M and
closed at both the ends. The tube is then rotated in a horizontal plane about
one of its ends with a uniform angular velocity ω. Show that the force exerted
by the liquid at the other end is F =
1
2 mω 2 L.
4.32 A uniform bar of length 6a and mass 8m lies on a smooth horizontal table.
Two point masses m and 2m moving in the same horizontal plane with speed
2v and v, respectively, strike the bar and stick to the bar (Fig. 4.7). The bar is
set in rotation. Show that
(a) the centre of mass velocity v c = 0
(b) the angular momentum J = 6mva
(c) the angular velocity ω = v/5a
(d) the rotational energy E = 3mv 2 /5
Fig. 4.7
4.33 A thin rod of negligible weight and of length 2d carries two point masses of m
each separated by distance d, Fig. 4.8. If the rod is released from a horizontal
position show that the speed of the lower mass when the rod is in the vertical
position will be v =
24 gd
5
.
4.34 If the radius of the earth suddenly decreases to half its present value, the mass
remaining constant, what would be the duration of day?