When a rigid body such as a merry-go-round rotates around an axis, each particle in the body moves in its own circle around that axis. Since the body is rigid, all particles make one revolution in the same amount of time. i.e., they all have the same angular speed w.
However, far a particle is from the axis, greater the circumference of its circular trajectory, the greater its linear speed. Let us try to relate the linear variables 'S', 'v' and 'a' for a particular point in a rotating body to the angular variables q, w of the body. The two sets of variables are related by r, the perpendicular distance of the point from the rotation axis. It is also the radius 'r' of the circle traveled by the point around the axis of rotation.
The position
If a reference line on a rigid body rotates through an angle , a point within the body at a position 'r' from the rotation axis moves a distance 's' along a circular arc where 's' is given by
The speed
Differentiating equation (1)
The acceleration
Differentiating equation (2)
We get represents only the part of
the linear acceleration that is responsible for changes in the magnitude 'v' of the
linear velocity . Let us call it tangential component at.
Combining all the above equations, a table can be framed that compares the translation and rotational motion parameters.
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, a point within the body at a position 'r' from the rotation axis moves a distance 's' along a circular arc where 's' is given by 
represents only the part of
the linear acceleration that is responsible for changes in the magnitude 'v' of the
linear velocity 
. Let us call it tangential component at.
