Rotational Motion
Rigid Body: A body with a perfectly definite and unchanging shape is called a rigid body. The distances between all pairs of particles (of a rigid body) do not change. But no real body is rigid because real bodies deform under the influence of forces. But such deformations are negligible in many situations.
Motion of Rigid Bodies:
Translational Motion: In pure translational motion at any instance of time, all particle of a body have the same velocity.
The first figure shows a block sliding down an inclined plane and it has no sidewise movement. In this case, all the particles of the block are moving together. It means, at any instance of time all the particles have the same velocity. This is an example of pure translational motion.
The second figure shows a cylinder rolling down an inclined plane. In this case, different particles of the body are moving at different velocities. This is not a case of pure translational motion.
Rotation: When a body is moving around a fixed axis, its movement is called rotation. In this case, every particle moves on a circle. The circle lies in a plane that is perpendicular to the axis. The center of circle lies on the axis.
This figure shows the rotational motion of a rigid body about a fixed axis. Let us consider three arbitrarily chosen particles on this body. Radii of their circular paths are as follows:
| Particle | Radius of circular path |
|---|---|
| P1 | r1 |
| P2 | r2 |
| P3 | 0 (at rest) |
However, the axis may not be fixed in some cases of rotation, e.g. a spinning top or an oscillating table fan. The spinning top moves around the vertical through its point of contact with the ground. While doing so, it sweeps out a cone. But the point of contact of the top with ground is fixed. So, at any instant, the axis of rotation of the top passes through the point of contact with the ground.
Centre of Mass
It is the average position of all the parts of the system, weighted according to their masses. For simple rigid objects with uniform density, the center of mass is located at the centroid.
Let us consider and object and consider two particles in the object. Let us assume that the line joining the two particles is x-axis. Distances of objects from the origin O are x1 and x2 and their masses are respectively m1 and m2. The center of mass of the system of these particles is point C which is at a distance X from O. This distance is given by following equation.
`X=(m_1x_1+m_2x_2)/(m_1+m_2)`
In this equation, X can be considered as the mass-weighted mean of x1 and x2. If the two particles have equal mass, i.e. m1 = m2 then
`X=(mx_1+mx_2)/(2m)=(x_1+x_2)/2`
This equation shows that for two particles of equal mass, the center of mass lies exactly midway between them.
For n number of particles, the center of mass is given by following equation:
`X=(Σm_1\x_i)/(Σm_i)`
Let us assume that there are three particles which are not lying on a straight line.
Coordinates of three particles are: (x1, y1), (x1, y2) and (x3, y3)
Masses of three particles are: m1, m2 and m3
Coordinates of center of mass of three particles can be given as follows:
`X=(m_1x_1+m_2x_2+m_3x_3)/(m_1+m_2+m_3)`
`Y=(m_1y_1+m_2y_2+m_3y_3)/(m_1+m_2+m_3)`
For three particles of equal masses (m1 = m2 = m3), coordinates center of mass is given by following equation.
`X=(m(x_1+x_2+x_3))/(3m)=(x_1+x_2+x_3)/3`
`Y=(m(y_1+y_2+y_3))/(3m)=(x_1+x_2+x_3)/3`
So, if the particles form a triangle then the centroid of the triangle gives the center of mass of a system of these three particles.