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4 Rotational Dynamics
4.2 Problems
4.2.1 Moment of Inertia
4.1 Calculate the moment of inertia of a solid sphere about an axis through its
centre.
4.2 Two particles of masses m 1 and m 2 are connected by a rigid massless rod of
length r to constitute a dumbbell which is free to move in a plane. Show that
the moment of inertia of the dumbbell about an axis perpendicular to the plane
passing through the centre of mass is μr 2 where μ is the reduced mass.
4.3 Show that the moment of inertia of a right circular cone of mass M, height h
and radius ‘a’ about its axis is 3Ma 2 /10.
4.4 Calculate the moment of inertia of a right circular cylinder of radius R and
length h about a line at right angles to its axis and passing through the middle
point.
4.5 Show that the radius of gyration about an axis through the centre of a hollow
cylinder of external radius ‘a’ and internal radius ‘b’ is
2
5
a 5 − b 5
a 3 − b 3
.
4.6 Calculate the moment of inertia of a thin rod (a) about an axis passing through
its centre and perpendicular to its length (b) about an end perpendicular to the
rod.
4.7 Show that the moment of inertia of a rectangular plate of mass m and sides 2a
and 2b about the diagonal is
2
3
ma 2 b 2
(a 2 + b 2 )
4.8 Lengths of sides of a right angle triangular lamina are 3, 4 and 5 cm, and the
moment of inertia of the lamina about the sides I 1 , I 2 and I 3 , respectively
(Fig. 4.1). Show that I 1 > I 2 > I 3 .
Fig. 4.1
4.9 A circular disc of radius R and thickness R/6 has moment of inertia I about
the axis perpendicular to the plane and passing through its centre. The disc is