Comparison of Linear and Rotational Motion

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.

Sub Topics

The position
The speed
The acceleration
Combining all the above equations, a table can be framed that compares the translation and rotational motion parameters.

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 a_t.

Combining all the above equations, a table can be framed that compares the translation and rotational motion parameters.

Pure translation (fixed direction)	Pure rotation (fixed axis)
Position x	Angular position θ
Velocity v = dx/dt	Angular velocity ω= dθ/dt
Acceleration a = dv/dt	Angular acceleration α= dω/dt
Mass m	Rotational inertia I
Newton’s second law F_net = ma	Newton’s second law t_net = I α
Work w=∫ Fdx	Work w= ∫ tdθ
Kinetic energy K = ½ mv²	Kinetic energy K = ½ I ω²
Power (constant force) P = Fv	Power (constant torque) P = t ω
Work — kinetic energy theorem W = Δk	Work — kinetic energy theorem W = Δk
Equations of translational motion v=u+at S=ut+½ at ² v²=u²+2as	Equation of rotational motion ω = ω ₀+ αt θ= ω₀ t + ½ α t² ω²= ω₀² + 2 αθ