Further Mechanics

Radians

One radian is the angle created when the radius of a circle is curved around its circumference. 1 radian is approximately 57 degrees.

Converting between radians and degrees

Worked Example

Convert 180 degrees into radians.

Solution

180 / 360 = 1/2

1/2 x 2 pi = pi

360 degrees is equal to 2 pi radians.

Variables in circular motion

Radius, r / m

Mass, m / kg

Time Period, T / s

Frequency, f / Hz

Linear speed/velocity, v / ms-1

angular velocity, w / rad s-1

Speed

To calculate the speed at which an object revolves, the simple equation of speed can be used:

v = d / t

The distance travelled around the circumference of the circle is equal to 2πr. The time taken to make one revolution is the time period, t, which can be calculated if the frequency is known from the equation:

T = 1 / f

These can be substituted into the equation to give;

v = 2πr/T

This equation can explain the variation in speeds at different radii:

Take the Earth as an example. Looking from above the North Pole, a polar bear 1m from the North Pole (on the spin axis of the Earth) has a smaller speed than a Lion at the equator. This is because they both take the same amount of time to make one revolution (24 hrs ) but the Lion…