Rigid Bodies and Centre of Mass

Rigid bodies and types of motion

Why size now matters

A point-particle model can describe a car travelling between towns, but it cannot describe how the wheels turn or why a door opens more easily when pushed at its edge. An extended body contains many particles at different positions. Their individual motions need not be identical. We therefore describe the motion of the body as a whole and the motion of its parts relative to one another.

An ideal rigid body maintains the distance between every pair of its particles. It can move and change orientation, but cannot bend, stretch or vibrate. Real bodies deform, so rigidity is an approximation. It works when those deformations have negligible influence on the question. A steel beam in a balance problem may be treated as rigid; its bending under a heavy load requires a more detailed model.

Translation preserves orientation

In pure translation, every point of the body has the same velocity at any given instant. A block sliding down an incline without turning is an example. The shared velocity may change with time, so translation need not mean uniform motion. A marked line on the body remains parallel to its original direction. The paths of different points are translated copies of one another; translation can also follow a curved route while orientation remains fixed.

Rotation gives circular paths about an axis

For rotation about a fixed axis, each point away from the axis moves in a circle. The circle lies in a plane perpendicular to the axis and its centre lies on the axis. The circle’s radius is the perpendicular distance to the axis. Points on the axis remain stationary. All particles share the same angular motion, although points farther from the axis travel faster along their larger circles. A ceiling fan and a potter’s wheel illustrate this model.

A spinning top can have a fixed contact point while its axis changes direction, sweeping a cone. This change in the axis direction is called precession. An oscillating table fan combines spinning blades with a side-to-side change of the blade axis. These show why fixing one point is different from fixing an entire axis. Most calculations in this chapter use the simpler fixed-axis model.

Translation and rotation can occur together

A rolling cylinder moves down an incline while its orientation changes. The centre translates, and other points rotate relative to it. Different points then have different ground velocities. For rolling without slipping on stationary ground, the instantaneous contact-point velocity is zero while the top point moves faster than the centre. That point is not permanently at rest: a different part of the rim becomes the contact point as rolling continues. Here rolling illustrates combined motion; detailed rolling dynamics is beyond the supplied chapter’s treatment.

Centre of mass of particle systems

A mass-weighted location

The centre of mass, or CM, is the position obtained by weighting each particle’s location by its mass. Heavy particles influence it more strongly. For two equal masses it lies midway between them. For unequal masses it lies nearer the heavier mass. It is a geometric property of the mass distribution and can be defined even where there is no gravity.

To see the division of the joining segment, put the first particle at zero and the second at distance d. Then X = m₂d/(m₁ + m₂). The distances from the CM satisfy m₁X = m₂(d − X). Thus the heavier mass has the shorter lever arm. This also shows that with positive masses the CM lies between the two particle positions.

Changing the origin changes every coordinate and the CM coordinate consistently; it does not move the physical CM relative to the particles. If the origin is placed at the CM, the mass-weighted vector sum Σmᵢrᵢ is zero. This does not mean the particles are all at zero. Their positive and negative weighted coordinates cancel.

Masses at the corners of a triangle

For three equal particles at triangle vertices, the CM coordinates are the arithmetic averages of the vertex coordinates. That point is the triangle’s centroid. Unequal vertex masses generally shift the CM away from the geometric centroid. Do not replace the mass-weighted formula by a simple average unless the masses really are equal.

Continuous and composite bodies

From particles to a continuous mass distribution

A macroscopic rod contains too many atoms to sum individually. Divide it into small pieces with masses Δm and known approximate positions. As the pieces become smaller, the sum approaches an integral. The element dm means a small amount of mass; it is not automatically a length or an area. For a uniform thin rod dm = λ dx, where λ is mass per length. For a uniform lamina dm = σ dA, and for a volume distribution dm = ρ dV.

For a uniform rod from x = 0 to L, its constant linear density is M/L. Integrating x(M/L) dx from 0 to L and dividing by M gives X = L/2. This calculation shows explicitly how the integral reproduces the familiar midpoint. If density varies along the rod, it must stay inside the integral and the midpoint need not be the CM.

Symmetry removes unnecessary calculation

For a uniform rod centred at zero, each mass element at x has an equal partner at −x, so their weighted positions cancel. The same reflection argument places the CM at the geometric centre of a uniform sphere, cylinder, ring or cube. Symmetry of shape alone is insufficient if density is nonuniform. A ring’s CM lies at its empty centre: the CM need not lie in the material of the body.

For a uniform triangular lamina, divide the area into narrow strips parallel to one side. Each strip has its CM at its midpoint. These midpoints lie on the median from the opposite vertex. The whole lamina’s CM must therefore lie on that median. Repeating with the other two directions places it at the intersection of the medians, the centroid. This is an argument about a continuously filled triangle, distinct from three unequal vertex masses.

Replace whole components by mass points

Subtracting a hole

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