The speed of something going round a circle can be measured by the distance it covers. But then the length of one lap changes with the radius, which makes the inside hard to compare with the outside. For something that turns there is a more natural ruler: the angle it has turned through.
How many radians it turns per second is called the angular velocity, written . If it turns through an angle in a time , then . Whatever the radius, one lap is .
The distance can be written with the angle as well. The arc of a circle of radius subtending an angle has length . Divide by the time and that is the speed, so . At the same , the farther out you are the faster you go, and the equation says so.
Divide one lap of by and out comes the time for one lap, the period . Fix the radius and any one of , and , and the other two are fixed with it.
It goes by the name of uniform circular motion, but the only thing that is constant is the speed. The velocity keeps changing, because a velocity has both a size and a direction, and while the body goes round the circle its direction never holds still for an instant.
Acceleration is the rate at which the velocity changes. It does not refer to a change of speed alone. So uniform circular motion, in which the speed does not change, has an acceleration too.
The figure shows which way it points. The velocity a moment ago and the velocity now have been carried to the centre of the circle so that they start from the same point. They are the same length but differ in direction, so a gap opens between their tips. The arrow filling that gap is the change in velocity, .
Since and are the same length, the three arrows form an isosceles triangle. The that forms its base always points toward the centre of the circle. The velocity is being turned toward the centre, and so the acceleration points at the centre as well.
Now that the direction is known, find the size. The velocity arrow kept its length and turned through an angle . The tip of an arrow of length swinging through travels a distance , and that is the size of .
All that is left is to divide by the time: . Put in and it is ; put in and it is . These are one quantity written three ways, and you may take whichever form is convenient.
The two arrows in the figure always meet at a right angle, because the velocity lies along the tangent and the acceleration along the radius.
This is also why the motion can go on turning at constant speed. An acceleration at right angles to the direction of travel cannot change the speed. It changes only the direction, and bends the path accordingly. Were the acceleration to have any component along the tangent, the speed would change by that much and the motion would no longer be uniform.
The in sits in the denominator. At the same speed, a smaller radius means a larger acceleration.
The two bodies in the figure go round the same centre at the same speed, the inner radius being half the outer. To follow a circle of half the size at the same speed, the direction must be changed twice as fast, so the acceleration is twice as large. The arrow is drawn at exactly twice the length.
The angular velocity is doubled too. In the is the same, so halving doubles . In the figure the inner body completes two laps while the outer one completes a single lap.
This is why a tight bend in a car is hard on the body. At the same speed, the sharper the turn the smaller is and the greater the acceleration felt. Even on the same road, the inner lane feels harder.