A ceiling fan in a classroom turns steadily at \(300\) revolutions per minute. The centre of the fan hardly seems to move. The tips of the blades, \(0.6\,\text{m}\) from the centre, cut the air fast enough to hum. Every point of a blade turns through the same angle in the same time. Yet the tip covers a much longer path than a point near the centre. How fast is the tip moving?
An angle measured by an arc
Degrees divide one turn into \(360\) equal parts. The number 360 is a choice made by ancient astronomers; nothing in geometry forces it. Physics uses a measure that comes from the circle itself. Draw a circle of radius \(r\) with its centre at the corner of the angle. The two arms of the angle cut off an arc of length \(s\).
A larger angle cuts off a longer arc. On a circle twice as large, the same angle cuts off an arc twice as long. So the ratio \(s/r\) depends on the angle alone, not on the size of the circle. This ratio is the angle measured in radians.
The angle in radians at the centre of a circle is the length of the arc it cuts off divided by the radius, \(\theta = s/r\). One radian is the angle whose arc is exactly as long as the radius. The radian is a ratio of two lengths, so it is a pure number. The word "rad" only reminds the reader how the angle was measured.
A full turn is \(2\pi\) radians
The whole circle has a circumference of \(2\pi r\). So a full turn, measured in radians, is \(2\pi r/r = 2\pi\). This gives the one conversion that matters: \(360° = 2\pi\) rad, or \(180° = \pi\) rad. One radian is therefore \(180°/\pi = 57.3°\), a little less than the angle of an equilateral triangle.
Converting in both directions
Every conversion uses \(\pi\) rad \(= 180°\). To turn degrees into radians, multiply by \(\pi/180\). To turn radians into degrees, multiply by \(180/\pi\). Physics usually leaves \(\pi\) in the answer, because \(\pi/3\) is exact and \(1.047\) is rounded.
The angles of the last lesson, in radians:
- \(30° = \pi/6 = 0.524\) rad and \(45° = \pi/4 = 0.785\) rad.
- \(60° = \pi/3 = 1.047\) rad and \(90° = \pi/2 = 1.571\) rad.
- \(180° = \pi = 3.142\) rad and \(270° = 3\pi/2 = 4.712\) rad.
- \(360° = 2\pi = 6.283\) rad.
Convert (a) \(270°\) and (b) \(150°\) into radians, and (c) \(1.5\) rad into degrees.
(a) \(270 \times \pi/180 = 3\pi/2 = 4.71\) rad.
(b) \(150 \times \pi/180 = 5\pi/6 = 2.62\) rad.
(c) \(1.5 \times 180/\pi = 85.9°\), a little less than a right angle.
A bead moves along a circular wire of radius \(2\,\text{m}\) and covers an arc of \(3\,\text{m}\). Through what angle has it turned about the centre?
One angle, many arcs
A turning body carries every point through the same angle. A point twice as far from the centre moves along an arc twice as long. This is \(s = r\theta\) read the other way: for a fixed \(\theta\), the arc grows in proportion to the radius. On a giant wheel at a mela, a rider on the rim and a child halfway along a spoke turn together. The rider on the rim travels twice as far.
Arrange these angles from the smallest to the largest.
Radians per second
A body turning at a steady rate sweeps equal angles in equal times. The angle swept in each second is the angular speed, \(\omega = \theta/t\). In radians its unit is rad/s. One revolution is \(2\pi\) rad, so a body making \(n\) revolutions per second turns at \(\omega = 2\pi n\). A speed in revolutions per minute (rpm) is divided by 60 first: \(1\) rpm \(= 2\pi/60\) rad/s.
Now divide \(s = r\theta\) by the time. A point at distance \(r\) from the axis covers the arc \(r\theta\) in the time \(t\). Its speed along the circle is therefore \(v = s/t = r\theta/t = \omega r\). The same angle per second gives every point a speed in proportion to its distance from the axis.
The rules \(s = r\theta\), \(v = \omega r\) and \(\sin\theta \approx \theta\) hold only with the angle in radians. With the angle in degrees, the answer is \(57.3\) times too large. A wheel of radius \(0.5\,\text{m}\) turning at \(360°\) per second does not move its rim at \(360 \times 0.5 = 180\,\text{m/s}\). It turns at \(2\pi\) rad/s, so its rim moves at \(3.14\,\text{m/s}\). Convert to radians before using any of the three.
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.
One revolution is \(2\pi\) rad, and one minute is \(60\,\text{s}\). The tip is at \(r = 0.6\,\text{m}\).
\(\omega = 300 \times 2\pi/60\), \(T = 2\pi/\omega\), \(f = 1/T\) and \(v = \omega r\).
\(\omega = 10\pi \approx 31.4\,\text{rad/s}\). \(T = 2\pi/10\pi = 0.2\,\text{s}\), so \(f = 5\,\text{Hz}\). \(v = 10\pi \times 0.6 = 6\pi \approx 18.8\,\text{m/s}\).
The fan turns at \(31.4\,\text{rad/s}\), five turns every second. The blade tip moves at \(18.8\,\text{m/s}\), about \(68\,\text{km/h}\). A point halfway along the blade moves at half this speed.
The fan's question has its answer. Every point of a blade turns at the same \(31.4\,\text{rad/s}\), because the whole blade sweeps one angle. The tip, \(0.6\,\text{m}\) out, moves at \(18.8\,\text{m/s}\). A point \(0.1\,\text{m}\) from the centre moves at only \(3.1\,\text{m/s}\). That is why the tips hum and the centre seems still.
A giant wheel of radius \(2.5\,\text{m}\) turns once in \(10\,\text{s}\). Find the speed of a rider on its rim, in m/s.
A potter's wheel of radius \(0.2\,\text{m}\) turns at \(180°\) per second. A student writes the speed of its rim as \(v = 180 \times 0.2 = 36\,\text{m/s}\). What is the true speed?
Kinematics of Circular Motion gives a body on a circle its angular position \(\theta = s/r\), its angular velocity in rad/s and its speed \(v = \omega r\). Rolling Motion rolls a wheel through a distance \(R\theta\) for a turn of \(\theta\). Kinematics of SHM measures the phase of an oscillation in radians. The chapters on the magnetic field find the field of an arc of wire from the angle it spans. The optics chapters measure three things in radians. They are the apparent size of the Moon, the spread of a diffraction pattern and the limit of a telescope.
- Using degrees in \(s = r\theta\) or \(v = \omega r\).Convert to radians first. Both rules come from \(\theta = s/r\), which defines the radian.
- Taking one radian as one degree.One radian is \(57.3°\). A full turn is \(2\pi\) rad, only about 6.28 radians.
- Converting rpm to rad/s by multiplying by \(2\pi\) alone.Divide by 60 as well. rpm counts turns per minute; rad/s counts radians per second.