System of Particles and Rotational MotionIntroduction
After the evening match Sina tosses her cricket bat to Kabir. It does not sail across like a ball: it tumbles end over end, the handle and the blade chasing each other round. Kabir catches it anyway, and shouts that the bat “went on a parabola like a ball”. Sina says that is nonsense — it was spinning all over the place. They are both right, and this chapter shows why.
So far we have treated a car, a ball or a person as a single point, a particle. That works when only the overall motion matters. A tumbling bat, a turning wheel or an opening door cannot be squeezed into one point: different parts of the body move differently. The figure shows the bat at seven moments of its flight, one sixth of the flight apart. Tap a dot on the bat below the picture to trace that point through the whole flip.
Every point you traced looped round except one, which followed the purple parabola exactly like a thrown ball. That point is the centre of mass of the bat, and it sits nearer the heavy blade than the light handle. The whole tumble is just two simple motions added together: the centre of mass flies like a particle, and the bat spins about it.
Rigid Bodies
An extended body — one with a real size — is a system of particles: a huge number of atoms held together. We will first look at the motion of the system as a whole, and the centre of mass will be the key idea there.
Many problems become much easier if we treat the body as a rigid body: a body whose shape never changes, so the distance between every pair of its particles stays fixed. No real body is perfectly rigid, since every body bends, twists or squashes a little under a force. But for wheels, tops, steel beams, molecules and even planets, those changes are usually so small that we can ignore them.
Which of these behaves most like a rigid body?
Right. Pressing a steel ball bearing changes the distances between its atoms by a tiny fraction of a millimetre, far too little to matter for its motion.
What Kind of Motion Can a Rigid Body Have?
Think of three things you see every day. A block sliding down a smooth slope (Fig. 6.1). A steel cylinder rolling down the same slope (Fig. 6.2). A ceiling fan turning on its fixed rod (Fig. 6.3, Fig. 6.4). Choose each one and tap the points P₁ to P₄ to see their velocities at one instant.
So there are three cases.
- Pure translation. At any instant every particle of the body has the same velocity, like the sliding block. The body can even move along a curved path and still be in pure translation, as long as it does not turn.
- Rotation about a fixed axis. If a body is fixed along a line, it can only rotate about that line, its axis of rotation: a ceiling fan, a potter’s wheel, a giant wheel at a mela. Every particle moves in a circle that lies in a plane perpendicular to the axis and has its centre on the axis. Particles on the axis stay put.
- Translation and rotation together. The rolling cylinder moves forward and turns at the same time. That is the “something else” in rolling: a rotation added to a translation, just like the flipped bat.
The axis does not always stay fixed. A spinning top keeps only one point fixed, its tip on the ground, while its axis sweeps out a cone around the vertical; this movement is called precession (Fig. 6.5(a)). The axis of an oscillating table fan’s blades swings from side to side about its pivot (Fig. 6.5(b)). In this chapter we will deal only with rotation about a fixed axis.
In short: a body that is fixed along a line can only
Speed Round: What Kind of Motion?
Ask of each body: does every point have the same velocity, is some line of it fixed, or both is it moving and turning? Get five right to move on.