System of Particles and Rotational Motion · Lesson 1 of 15
Rigid Bodies, Translation and Rotation
“Distinguish whole-body translation from rotation and identify the assumptions behind a rigid-body model.”
• Explain a rigid body as an idealised system of particles. • Distinguish translation, fixed-axis rotation and combined motion. • Describe fixed and moving axes. • Recognise the limits of a point-particle model.
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.
Problem
Slide a book across a table, then move it while turning it. What distinguishes the motions?
- 1.In the first motion, keep a marked edge parallel to itself. Every book point has the same instantaneous velocity: pure translation.
- 2.In the second, the marked edge changes direction. A point near an edge has an additional velocity relative to the centre, so translation and rotation coexist.
- 3.The centre can follow the same path in both demonstrations. The centre path alone does not reveal the complete motion.
Quiz
Which statement defines a rigid body?
Rigidity constrains internal distances, not the body’s overall motion.
In pure translation, what is shared by every point at an instant?
Each point has the same velocity vector, even when that common velocity changes.
A point on a fixed rotation axis has what speed?
Its perpendicular distance from the axis is zero.
A rolling cylinder generally exhibits what motion?
The centre moves while the cylinder changes orientation.
A top’s axis sweeping a cone is called what?
Precession describes a changing orientation of the rotation axis.
Key Takeaways
• Rigidity is a useful approximation of unchanging internal distances. • Translation gives all points equal instantaneous velocities. • Fixed-axis rotation gives circular paths with a common angular motion. • A body can translate and rotate simultaneously.
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Centre of Mass of Particle Systems