At the Dussehra mela a giant wheel carries twelve cabins round and round. Each cabin hangs from a pin on the rim. A child in a cabin rises, passes over the top and descends again, on a large circle. Yet her ice cream never spills, and she faces the same way throughout, towards the crowd. Is the cabin rotating or not?
Translation
A rigid body is in translation when every one of its particles has the same velocity at each instant. Two things follow at once. First, a line painted on the body stays parallel to itself, so the orientation of the body never changes. Second, all its particles trace paths of the same shape and size, each one shifted from the others.
The path need not be straight. A drawer sliding out of a desk translates along a line. A cabin on a giant wheel translates along a circle. What decides translation is whether the body turns, not the shape of the path.
In Fig 2.1 the wheel and the cabin move together but in different ways. A spoke of the wheel points up, then left, then down; the wheel rotates. The arrow on the cabin points right the whole time. Its two corners trace circles of the same radius, \(2.2\,\text{m}\), with centres \(0.9\,\text{m}\) apart. So at every instant the two corners have the same velocity, and the cabin translates on a circle.
A giant-wheel cabin goes round the wheel while its floor stays horizontal. Its motion is
Rotation about a fixed axis
A rigid body is in rotation about a fixed axis when every particle moves on a circle centred on one fixed line. That line is the axis of rotation. The radius of each circle is the perpendicular distance of that particle from the axis. Particles on the axis do not move at all. A ceiling fan, a door on its hinges and a potter's wheel all rotate about fixed axes.
Because the body is rigid, every particle turns through the same angle in the same time. So the whole body has one angular position \(\theta\), one angular velocity \(\omega = d\theta/dt\) and one angular acceleration \(\alpha = d\omega/dt\). You met these in Kinematics of Circular Motion, for a single particle on a circle. For a rigid body they belong to the body as a whole.
What differs from particle to particle is the linear motion. A particle at distance \(r\) from the axis has speed \(v = \omega r\), a tangential acceleration \(a_t = \alpha r\) and a centripetal acceleration \(a_c = \omega^2 r\) towards the axis. These are the links of Kinematics of Circular Motion, now read for every particle of one body at once.
A rigid body has one \(\omega\), but not one speed. The tip of a fan blade and a point near its hub share \(\omega\), while the tip moves many times faster. Never give a rigid body "a speed" in rotation; give its \(\omega\), or the speed of a named point.
Signs and the right-hand rule
For a rotation in one plane we choose a positive sense, usually anticlockwise as seen with the axis pointing at us. Then \(\omega\) and \(\alpha\) carry signs. By the right-hand rule, a positive \(\omega\) points along the axis towards the viewer. The sign of \(\alpha\) alone does not say whether the body gains or loses speed. The body turns faster when \(\omega\) and \(\alpha\) have the same sign, and slower when their signs differ.
With one \(\omega\) and one \(\alpha\) for the whole body, the equations for a constant angular acceleration from Kinematics of Circular Motion apply unchanged. The example below recalls them.
A ceiling fan in a Chennai classroom starts from rest. Seen from below it turns anticlockwise, which we take as positive, with a constant angular acceleration of \(4\,\text{rad/s}^2\). Find its angular velocity after \(3\,\text{s}\), and the angle it turns through during the third second.
\(\omega_0 = 0\), \(\alpha = +4\,\text{rad/s}^2\). By the right-hand rule \(\vec\omega\) points down from the ceiling towards the students, and \(\vec\alpha\) points the same way.
\(\omega = \omega_0 + \alpha t\) and \(\theta = \omega_0 t + \tfrac12\alpha t^2\). The third second runs from \(t = 2\,\text{s}\) to \(t = 3\,\text{s}\).
\(\omega(3) = 4 \times 3 = 12\,\text{rad/s}\). \(\theta(3) = \tfrac12 \times 4 \times 9 = 18\,\text{rad}\) and \(\theta(2) = \tfrac12 \times 4 \times 4 = 8\,\text{rad}\), so the third second gives \(10\,\text{rad}\).
After \(3\,\text{s}\) the fan turns at \(12\,\text{rad/s}\), and in the third second it turns through \(10\,\text{rad}\), about \(1.6\) turns. Every blade, and every point of every blade, shares these values.
A potter's wheel turns anticlockwise, seen from above, and is slowing at a steady rate. Taking anticlockwise as positive, what are the signs of \(\omega\) and \(\alpha\)?
A disc turns about a fixed axis. At one instant \(\omega = 2\,\text{rad/s}\) and \(\alpha = 3\,\text{rad/s}^2\). What is the magnitude of the acceleration of a point \(0.5\,\text{m}\) from the axis, in m/s²?
General motion: both at once
Most real motions are neither pure translation nor pure rotation. A thrown baton, a rolling wheel and a falling ladder all move and turn at once. Such general motion is simpler than it looks. It is always a translation of one point of the body together with a rotation about an axis through that point.
The centre of mass is the natural point to choose, because its motion follows from the net force alone. In Fig 2.2 the translation carries C along the parabola, and the rotation turns the baton about C at a steady rate. Add the two chips, and the loops of the end P appear. Neither motion alone could make them.
You may choose any other point instead. The translation part then changes, because a different point moves differently. The rotation part does not change: the body turns through the same angle in the same time about every point of it. So a rigid body has one angular velocity, whatever point we describe it from.
For a rolling wheel there is a special choice. The point of the wheel touching the road is at rest for an instant. At that instant the whole wheel is in pure rotation about it. This point marks the instantaneous axis of rotation. The chapter on Rolling Motion builds on it.
Which of these motions are pure translation?
A spinning baton flies through the air. Seen by an observer moving with its centre of mass, the baton
In 1830 the French geometer Michel Chasles proved a striking result about rigid bodies. Any change of position of a rigid body, however it was produced, can be made by one translation and one rotation. He went further: the translation can be chosen along the axis of the rotation. So every displacement is a twist, like a screw driven into wood. Engineers who program robot arms still use his result to plan the motion of each part.
A welding robot in a car factory near Pune must carry a welding gun from one joint of a car body to the next. The controller describes each move as the translation of the gun's tip plus a rotation about that tip. The translation brings the tip to the joint, and the rotation turns the gun to the right angle. Splitting the move this way keeps the tip on a safe path while the gun turns.
The cabin of the giant wheel can now be settled. Its painted arrow never turns, so it has no rotation at all. It translates on a circle, while the wheel that carries it rotates about its axle.
- Calling every motion on a circular path a rotation.Look at the orientation. If a line on the body keeps its direction, the body translates, even on a circle.
- Giving different angular velocities to different points of one rigid body.Every point of a rigid body has the same \(\omega\) and the same \(\alpha\). Only \(v\), \(a_t\) and \(a_c\) change with \(r\).
- Reading a negative \(\alpha\) as a loss of speed.A body turns more slowly only when \(\omega\) and \(\alpha\) have opposite signs.