Angular Displacement
When a body moves a long a straight line, we measure the displacement using distance. If it rotates or move along a circular path, we can measure the displacement using angle.
Consider a small body moving along a circle. Imagine a line joining the centre of the circle to the body. This line would rotate round with the object, like the hand of a clock (the old type, not the digital one).
Think of the line at the start, and think of it some time later. The two line positions form an angle in between. This angle is called the angular displacement of the body. There are two ways to measure this. One is in degrees. The other is in radians.
Suppose the angle is θ, the radius of the circle is r, and the distance the body moves along the circle is s. Then the angle in radians is given by θ = s/r.
Radian is a rather strange unit. It really has no dimension - since s and r are both in metres, so the metres cancel in s/r. Because of this, the unit may sometimes be left out. For example, an angle of 1 radian may be written as 1 rad, or just 1. Of course, writing 1 rad would be much clearer.
It is useful to know how to convert between radians and degrees. To work out how, note that if the body moves round the circle once, it is 360o, and s is the circumference 2πr. So in radians, this is s/r = 2πr/r = 2π.
So the conversion is 360o = 2π rad, or 1o = π/180 rad.
It is also useful to remember some special cases: 2π rad is 360o, π rad is 180o, π/2 rad is 90o, π/3 rad is 60o, and π/6 rad is 30o.
Note that although we have developed the idea of angular
displacement using a small body on a circular path,
it also applies to a large body rotating about an axis
inside the body. So if you turn about face on the spot,
then your angular displacement is π rad, or 180o.
Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 21 September 2010.