What's the connection between period and radius in circular motion?

The period of circular motion is directly related to the radius of the circle, as per the formula for centripetal acceleration.

In circular motion, an object moves along the circumference of a circle. The period of this motion, often denoted as T, is the time it takes for the object to complete one full revolution around the circle. The radius, r, is the distance from the centre of the circle to the point where the object is located.

The connection between the period and the radius can be understood through the concept of centripetal acceleration. Centripetal acceleration is the rate of change of tangential velocity and always points towards the centre of the circle. The formula for centripetal acceleration is a = v²/r, where v is the tangential velocity.

However, tangential velocity can also be expressed in terms of the period and the radius, using the formula v = 2πr/T. Substituting this into the formula for centripetal acceleration gives a = 4π²r/T². This shows that the period of circular motion is inversely proportional to the square root of the radius - if the radius is quadrupled, the period will be doubled, assuming a constant centripetal acceleration.

This relationship is fundamental in many areas of physics. For example, in planetary motion, the period of a planet's orbit around the sun (a form of circular motion) is determined by the radius of the orbit. This is known as Kepler's Third Law.

IB Physics Tutor Summary: In simple terms, the period of circular motion (how long it takes to go around a circle) and the circle's radius (distance from centre to edge) are linked through centripetal acceleration. Specifically, if you increase the radius, the period changes because the speed needed for circular motion alters. This principle helps us understand movements in space, like planets orbiting the sun.