## Monday, February 29, 2016

### Combined Motion of a body on horizontal surface and inclined surface

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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