# Kinematics of Rigid Bodies

A rigid body is a system of particles in which the distance between the particles remains unchanged. It executes plane motion if all parts of the body moves in parallel planes.

A plane motion in which every line in the body remains parallel to its original position is called translation.

## Fixed Axis Rotation

A plane motion in which all particles of the body moves in circular paths about an axis is called fixed axis rotation.

In a fixed axis rotation, all lines in the body which are perpendicular to the axis of rotation rotates through the same angle in same time. Thus, all lines on a body in its plane of motion have the same angular displacement ($\theta$), the same angular velocity ($\omega=\mathrm{d}\theta/\mathrm{d}t$), and the same angular acceleration ($\alpha=\mathrm{d}\omega/\mathrm{d}t$).

The angular velocity and the angular displacement of a body rotating in a plane with a constant angular acceleration $\alpha$ are given by \begin{align} &\omega=\omega_0+\alpha t, \nonumber\\ &\theta=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2, \nonumber\\ &\omega^2=\omega_0^2+2\alpha t, \nonumber \end{align} where $\theta_0$ and $\omega_0$ are angular displacement and angular velocity at time $t=0$.

## General Plane Rotation

In plane motion of a rigid body, the directions of angular velocity $\vec\omega$ and angular acceleration $\vec\alpha$ remains fixed (these are normal to the plane of rotation).

The velocities and accelerations of two points O and P on the rigid body are related by \begin{align} &\vec{v}_P=\vec{v}_O+\vec{\omega}\times{\vec{\mathrm{OP}}}=\vec{v}_O+\vec{\omega}\times\vec{r}, \nonumber\\ &\vec{a}_P=\vec{a}_O+\vec{\omega}\times\vec{\omega}\times\vec{r}+\vec{\alpha}\times\vec{r}, \nonumber \end{align} where $\vec{r}={\vec{\mathrm{OP}}}$ is the position vector from O to P. The acceleration term $\vec{\omega}\times\vec{\omega}\times\vec{r}$ is called centripetal acceleration and $\vec{\alpha}\times\vec{r}$ is called tangential acceleration.

If the axis of rotation is fixed then $\vec{v}_O=\vec{a}_O=\vec{0}$ for point O lying on the axis. This type of motion is called fixed axis rotation. In this case, the velocity of a point at a perpendicular distance $r$ from the axis is $v=\omega r$ (tangential). The centripetal acceleration of this point is $\omega^2 r$ and its tangential acceleration is $\alpha r$.

## Instantaneous Axis of Rotation

An axis perpendicular to the plane of rotation that passes through the point of zero instantaneous velocity (this point may or may not lie on the body) is called instantaneous axis of rotation.

If two points on a rigid body are instantaneously at rest then all points on the line joining these points is at rest (instantaneous axis of rotation). The angular velocity of the body at that instant is along this line.

## In-extensible String Constraint

The tangential component of velocity (i.e., component along the string) of any point on the string is same. Same is true for the acceleration.

## Rolling without Slipping Constraint

Consider motion of a ball on a surface (fixed or moving). At the contact point, the ball may or may not slide on the surface. If tangential velocity of the ball at the contact point is equal to the velocity of the surface at this point then there is no slipping. Same is true for the acceleration.