At a mela, two children ride the same giant wheel. One sits in a cabin on the rim. The other, too small for a cabin, sits with her father on a platform halfway along a spoke. Both go round once in the same time. Yet the child on the rim covers a much longer path, and the wind in her face is much stronger.
One number for a position on a circle
A body on a circle of fixed radius needs only one number to fix its position. Draw a reference line from the centre, usually along the \(+x\) axis. Join the centre to the body. The angle \(\theta\) between the reference line and this radius is the angular position of the body. By the usual convention, \(\theta\) is measured anticlockwise from the reference line.
Degrees are not the natural unit for this angle. Physics uses the radian. The angle in radians is the length of the arc divided by the radius. An arc as long as the radius makes an angle of one radian at the centre. A full circle has an arc of \(2\pi r\), so it makes an angle of \(2\pi\) radians.
The angle subtended at the centre of a circle by an arc equal in length to the radius is one radian. An angle \(\theta\) in radians is \(\theta = s/r\), where \(s\) is the arc length and \(r\) the radius. A full turn is \(2\pi\,\text{rad} = 360°\), so \(1\,\text{rad} \approx 57.3°\).
The radian is a ratio of two lengths, so it has no dimension. You may write \(\text{rad}\) after a number as a reminder, or leave it out. Because of this, \(s = r\theta\) needs no conversion factor, and neither will the formulas built on it. That is the whole reason for using radians.
A body moves \(1.5\,\text{m}\) along the rim of a circle of radius \(0.5\,\text{m}\). Through what angle has it turned about the centre?
Angular displacement
When a body moves from angular position \(\theta_1\) to \(\theta_2\), its angular displacement is \(\Delta\theta = \theta_2 - \theta_1\). It is positive for an anticlockwise turn and negative for a clockwise turn. This is the twin of the displacement \(\Delta x = x_2 - x_1\) on a line. The arc covered is \(r\,|\Delta\theta|\) when the body does not turn back.
Is angular displacement a vector? It has a size and a sense of turning, which suggests a direction. But a vector must also obey the rule of addition, and finite rotations do not. Take a book lying flat on a table. Turn it by \(90°\) about a vertical axis, then by \(90°\) about an axis pointing east. Now start again and do the same two turns in the other order. The book finishes in a different position. So the order of two finite rotations matters, while the order of two vectors in a sum never does.
Very small rotations do add in either order: the difference between the two orders shrinks faster than the angles themselves. So an infinitesimal angular displacement \(d\vec\theta\) is a vector, directed along the axis of rotation. This is why angular velocity, which is \(d\vec\theta/dt\), is a true vector even though a finite \(\Delta\theta\) is not.
Angular velocity
The average angular velocity over a time \(\Delta t\) is the angular displacement divided by that time, \(\omega_{\text{avg}} = \Delta\theta/\Delta t\). Shrink \(\Delta t\) towards zero, as in Motion in One Dimension. The limit is the instantaneous angular velocity, the rate at which the angular position changes.
Machines give their speed of turning in revolutions per minute. To convert, multiply the number of revolutions by \(2\pi\) and divide the minutes by \(60\). A ceiling fan at \(300\,\text{rpm}\) turns at \(300 \times 2\pi/60 = 10\pi \approx 31.4\,\text{rad/s}\).
The direction of angular velocity
A rotation happens about an axis, so the direction of \(\vec\omega\) is taken along that axis. The right-hand rule picks one of the two senses along the axis. Curl the fingers of your right hand in the sense of the turning. Your outstretched thumb then points along \(\vec\omega\). For a body turning anticlockwise in the page, \(\vec\omega\) points out of the page, towards you.
The direction of \(\vec\omega\) is not the direction in which anything moves. Every point of a turning wheel moves in the plane of the wheel, while \(\vec\omega\) points along the axle. The axis direction is the only direction that is the same for every point of the wheel, and that is why it is chosen.
You face a wall clock. In which direction does the angular velocity of its second hand point?
Period, frequency and speed
For a body turning at a steady \(\omega\), the time period \(T\) is the time for one full revolution, \(2\pi\) radians. So \(T = 2\pi/\omega\). The frequency \(f\) is the number of revolutions in one second, \(f = 1/T\). Its unit is the hertz, \(1\,\text{Hz} = 1\,\text{s}^{-1}\). Put together, \(\omega = 2\pi f\).
Now return to the arc length, \(s = r\theta\). For a body on a circle, \(r\) does not change. Differentiate both sides with respect to time. The rate of change of \(s\) is the speed \(v\), and the rate of change of \(\theta\) is \(\omega\). So \(v = r\omega\). This is the link between the linear world and the angular world.
The two riders of the giant wheel are now explained. Every point of a rigid body turns through the same angle in the same time, so both riders share one \(\omega\). But \(v = \omega r\). The rider on the rim is twice as far from the axle, so she moves twice as fast. On a turning body, \(\omega\) is the one number for the whole body, and the speed grows in step with the distance from the axis.
A ceiling fan turns steadily at \(300\,\text{rpm}\). The tips of its blades are \(0.6\,\text{m}\) from the axis. Find its angular velocity, its period, its frequency and the speed of a blade tip.
\(\omega = 300 \times 2\pi/60 = 10\pi \approx 31.4\,\text{rad/s}\).
\(T = 2\pi/\omega = 2\pi/10\pi = 0.2\,\text{s}\), and \(f = 1/T = 5\,\text{Hz}\). The fan makes five turns every second.
\(v = \omega r = 10\pi \times 0.6 = 6\pi \approx 18.8\,\text{m/s}\), about \(68\,\text{km/h}\). A point halfway along the blade moves at half this speed.
A motorcycle wheel of radius \(0.35\,\text{m}\) turns at \(600\,\text{rpm}\). What is the speed of a point on its rim relative to the axle, in m/s?
Rank these by angular velocity, fastest first.
Two children sit on a merry-go-round turning steadily, one at \(1\,\text{m}\) from the centre and one at \(3\,\text{m}\). Which statements are true?
The Earth turns once a day about its axis, and its radius is about \(6400\,\text{km}\). Estimate the speed of a point on the equator due to this spin, in m/s.
Roger Cotes was a young mathematician who edited the second edition of Newton's Principia. In 1714 he saw that the natural measure of an angle is the arc it cuts off on a circle of unit radius. Mathematicians used the idea for more than a century without giving the unit a name. The word "radian" first appeared in print in 1873, in examination questions set by James Thomson at Queen's College, Belfast. Thomson was the elder brother of Lord Kelvin. The unit is still the one that makes \(s = r\theta\) and \(v = \omega r\) true without a conversion factor.
A bicycle speedometer does not measure speed directly. A small magnet on a spoke passes a sensor on the fork once per turn of the wheel. The meter counts these turns and times them. That gives the wheel's \(\omega\). The meter then multiplies by the wheel's radius, set by the rider, to show \(v = \omega r\). If the rider enters too large a wheel size, every reading comes out too high by the same factor.
- Using \(s = r\theta\) or \(v = \omega r\) with the angle in degrees or \(\omega\) in rpm.Convert to radians and rad/s first. \(1\,\text{rpm} = \pi/30\,\text{rad/s}\).
- Thinking that points farther from the axis have a larger angular velocity.Every point of a rigid body has the same \(\omega\). Points farther out have a larger speed, \(v = \omega r\).
- Drawing \(\vec\omega\) along the velocity of a point or along a radius.\(\vec\omega\) lies along the axis of rotation, by the right-hand rule, at right angles to the plane of turning.
- Adding two finite rotations about different axes as vectors.Finite rotations depend on their order, so they are not vectors. Only infinitesimal rotations, and so \(\vec\omega\), are.