**Combined motion of a body on horizontal surface**

Body in combined motion
has both translatory motion and rotatory motion and correspondingly it has both
translatory kinetic energy and rotatory kinetic energy.A rolling ball on a
flat and horizontal floor is a simple example where the body has both
translatory motion and rotatory motion. Body in translatory motion has
translatory kinetic energy and it is dependent of velocity and mass of the
body. Body in rotational motion has rotational kinetic energy and it depends on
moment of inertia and angular velocity.

We can express the
rotational kinetic energy in terms of translatory kinetic energy. We know that linear
velocity of the body a body can be expressed as the cross product of radius
vector of rotational motion and angular velocity. By substituting this we can
express rotational kinetic energy in terms of transnational kinetic energy.

We can express moment
of inertia as the product of mass of the body and square of radius of gyration.
Thus by simplifying the sum of both the energies, we can express the total
energy in terms of transnational kinetic energy as shown in the video shown
below.

**Combined motion on a inclined Plane**

Let us consider a
rolling body of known mass on a inclined plane having the base at a known
height and angle of inclination is not known. Alternatively, let us assume that
we know the length of the inclined plane also. If the body is initially on the
top of the inclined plane and starting from the state of rest, all its energy
in the form of potential energy.

As the body starts
rolling down, it has both translatory motion and rotational motion. We can
measure the impact of translatory motion in terms of translatory kinetic energy
and rotational motion impact basing on rotational kinetic energy.

We can use the concept
of conservation of energy that the energy is neither created nor destroyed and
the total energy of the system always remains constant. As it is shown in the
previous case, we can express the total energy in terms of translatory kinetic
energy and we can further find the velocity of the body sliding down on the
smooth inclined plane as shown below. It can be further found that the velocity
of the body depends on the radius of gyration and hence moment of inertia of
the body.

Using the value of the
velocity acquired by the body in the previous case and further using equation
of motion, we can find the acceleration acquired by the body during the process
of reaching the bottom of the inclined plane.

Further, by
substituting the value of this acceleration in the displacement equation from
the equation of motion list and further we can find the time taken by the body
to reach the bottom of the smooth inclined plane.

Substituting the
values of velocity, acceleration and time taken to reach the bottom of the
inclined plane and further knowing the value of ratio of radius of gyration and
radius of the body, we can find and compare different bodies like ring, disc,
hallow sphere and solid sphere, we will be knowing that who will reach the
bottom of the inclined plane first, who will come down with higher velocity and
who will have more acceleration.

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