Before you read this, it might be useful to read post 16.7.
The most common way to measure angles is to divide a complete revolution (for example, of a wheel) into 360 degrees, usually written as 360o. Then a right-angle is a quarter of a revolution, 360o/4 = 90o.
Another way is to divide a complete revolution by 2π radians, often written as 2π rad; 2π is the number of times that the radius of a circle fits into its circumference. So, as well as being 90o, a right-angle is also 2π/4 = π/2 rad. Because the radius is a straight line and the circumference is a circle, we have to bend the radius to make it fit. As a result, 2π is not a whole number; another way of saying this is that it is not an integer. Also, π is an irrational number – it can’t be written as one integer divided by another. The numbers 3 (= 3/1) and 1.5 (= 3/2) are rational numbers but π isn’t. So, if you have been taught that π = 22/7, this isn’t true. But 22/7 = 3.143 is a reasonable approximation to the value of π.
We can calculate a better value for π from the series
π = 3 + 4/(2 ×3 ×4) – 4/(4 × 5 ×6) + 4/(6 ×7 ×8) – 4/(8 ×9 ×10) + …
= 3 + (4/24) – (4/120) + (4/336) – (4/720) + …
This series goes on for ever – it’s an infinite series. But each term is smaller than the one before. So if we add a lot of terms, the later ones have a progressively smaller effect on the result. A series like this is called a convergent series. As a result, we can use this series to calculate π with the precision (see post 16.7) that we require.
|Term number||Term value||Cumulative sum|
The table above shows the values of the first 30 terms of the series, together with their cumulative sums. What is a cumulative sum? For the fifth term, the cumulative sum is the sum of the first five terms; for the ninth term, the cumulative sum is the sum of the first nine terms. And so on.
If we want the value of π that is precise to within 2 decimal places, we can we can see that the series has converged to a value of 3.14 by the sixth term. If we want the value of π that is precise to within 3 decimal places, we can we can see that the series has converged to a value of 3.142 by the 15th term. If we want the value of π that is precise to within 4 decimal places, we can we can see that the series has converged to a value of 3.1416 by the 19th term. If the reasoning behind this is not clear, see post 16.7.
How can we convert between the two methods for measuring angles? If an angle has a value of θ rad, it represents θ/2π of a revolution. If the same angle has a value of β degrees, it also represents β/360 of a revolution. So θ/2π = β/360; multiplying both sides of this equation by 2 gives θ/π = β/360. Since we can calculate a value for π we can convert an angles measured by one method to angles measured by the other method.
Now let’s think about a point P moving around a circular path. After an interval of time, it has moved from P to P’. If O is the centre of the circular path, we can denote the angle swept out by OP by θ. Conventionally a positive value of θ denotes an anti-clockwise movement of P to P’; a negative value denotes a clockwise movement. This convention is consistent with the definition of positive and negative torques (post 17.10); a positive torque induces a change in angle in the positive direction and a negative torque induces a change of angle in the negative direction.
What is the length of the curved line joining P to P’? From the picture, you can see that it is part of the circumference of the circle (called an arc of the circle) and its length is greater than the length of the straight line between the two points. If P did a complete revolution, of a circle of radius r, the length of its path would be 2πr, the circumference of the circle (from the definition of π- see above). But in moving to P’, P has moved only a fraction (θ/2π) of the way round. So the length of the arc is 2πr × (θ/2π) = rθ.
Remember how distance (a scalar quantity) was related to displacement (a vector quantity) – see post 17.4). We can also define a vector, θ, called angular displacement, where θ is the modulus of θ (see post 17.2). The direction of θ is defined to be the axis about which P rotates in an anti-clockwise direction – so it passes through O. It also points towards you – out of the plane of the picture. You can find this direction by sticking the thumb of your right hand up and curling your fingers in the direction in which P moves; your thumb then points in the direction of the vector θ. This is exactly the same idea that we used to find the direction for T, the vector definition of torque (see post 17.10).
So, there are two different ways of measuring angles and we can use the concept of angle to define a vector quantity called angular displacement. Angular displacement has the same relationship to angle as displacement has to distance.