Kinetic Energy of A Rolling Body: Consider a body that
is rolling on a horizontal plane without slipping. Its motion is composed of
two motions
a) Rotation about a horizontal axis through its centre of mass and
b) translation of the centre of mass. Let M, R and n be the mass, radius
and uniform velocity of the body. If T is the time period of rotation of the
body, it describes an angle of 2 p radians in this time and its centre of mass
is displaced by 2 p R in the same time.
Thus angular
velocity w =

2p


T


and the linear velocity of translation of the centre of mass,

n = w R
The total kinetic energy of the rolling body, E_{k}
is given by
E_{k} = K. E. of rotation + K.E. of
translation
=

1


2


Iw ^{2} +

1


2


Mn^{2}

=

1


2


MK^{2}w^{2} +

1


2


Mv^{2}

where K is the radius of gyration about a horizontal axis
through the centre of mass
\ E_{k} =

1


2


M


=

1


2


Mv^{2}


...(1)

or \ E_{k} =

1


2


M [K^{2}w^{2} + R^{2}w^{2} ] =

1


2


mw ^{2}[K^{2} + R^{2} ].

In the case of a cylinder or a disc,
K^{2} =

R^{2}


2


or

K^{2}


R^{2}


=

1


2


\ E_{k} =

3


4


Mn^{2} =

3


4


MR^{2}w^{2}

Rolling can be considered as a combination of pure rotation and
pure translation.
For a body rolling down an inclined plane AB making an angle q with the horizontal.
