Introduction: Rigid Bodies and the Two Kinds of Motion

Every body studied so far in mechanics -- a block, a ball, a car -- was treated as though it were a

single point, with all of its mass concentrated at one location. That approximation works well while a

body only translates from place to place, but it breaks down completely the moment a body's own spin or

turning matters: a spinning top, a rolling wheel, a swinging door, or a somersaulting diver cannot be

described by a single point alone, because different parts of the SAME body are moving in different

directions at the same instant. WBCHSE's Unit 5 develops exactly the mechanics needed for such an

extended body.

A rigid body is an idealised extended body in which the distance between any two of its constituent

particles stays exactly fixed, however the body as a whole moves -- no bending, no stretching, no internal

rearrangement is allowed. The most general possible motion of a rigid body can always be split into two

independent parts:

• Translational motion, in which every particle of the body undergoes exactly the same displacement in the same time interval, so the body's orientation in space never changes (a block sliding down a frictionless incline without tipping).
• Rotational motion, in which every particle moves in its own circle about a common straight line called the axis of rotation, so the body's orientation is continuously changing even though (for rotation about a fixed axis) the body's overall location may not be.

A rolling wheel is the everyday example of both happening together: its centre translates forward in a

straight line while the wheel itself simultaneously rotates about an axis through that centre.

This chapter builds up the tools needed for such motion in a fixed order. It first extends the idea of

"centre of mass" from a single particle to a genuine, well-defined point for any system of particles or

rigid body (Sections 5.2-5.3), and shows that this one point moves exactly as Newton's second law would

predict for the whole system's total mass, regardless of any internal forces at work (Section 5.4). It

then develops the rotational analogues of force and momentum -- torque and angular momentum (Sections

5.5-5.7) -- states the full conditions for a rigid body to be in equilibrium (Section 5.8), and

distinguishes centre of gravity from centre of mass (Section 5.9). Finally it introduces moment of

inertia, the rotational analogue of mass, together with its standard values for simple bodies, the radius

of gyration, and the two axis theorems that connect a moment of inertia about one axis to another

(Sections 5.10-5.13), before closing with the equations of rotational kinematics and dynamics and a

complete side-by-side comparison with straight-line motion (Sections 5.14-5.15).