Linear and Rotary Motion

1. Introduction
Many dynamic systems involve both linear translation and rotary motion. These problems are commonly solved by applying Newton's law, where components are undergoing translation only then the equation of linear motion is used:

Force = mass x acceleration
Where a body is rotating about a fixed pivot point then the equation:

Torque = Moment of Inertia about the Pivot point x angular acceleration

can be used.
NB Where a body is translating and rotating, the best approach is to consider the linear motion of the centre of mass and the rotation about the centre of mass. There will also be at least 1 equation linking the rotation and the translation. This is shown in the following example where the rotation of the wheels on a vehicle need to be related to the translation of the entire vehicle. In this example it is required to determine an expression for it's acceleration in terms of the torque applied to the driving wheels, their radius, mass and moment of inertia and the mass of the vehicle body. It is assumed that the wheels do not skid.

The approach used is to draw a layout diagram, then the free body diagrams (FBD). As the two rear wheels are identical, only 1 FBD is shown. The same applies for the front wheels. It is assumed that the vehicle is travelling on horizontal ground so only forces and motion in the horizontal direction are considered. Having drawn the FBDs the 5 system equations are can be written down - translation for the vehicle body (eqn. 2) rear wheel (eqn. 1) front wheel (eqn. 3) and rotation of the front (eqn. 5) and rear (eqn. 4) wheels.

David J Grieve. Modified: 10th March 2011, original: 14th November 2001.