Topic 6: Circular Motion and Gravitation

6.1 Circular motion

Period — Time taken for one complete revolution

Frequency — Number of complete revolutions per unit time, 1/T.

Angular displacement — Angle, in radians, that a body has travelled around the centre of circular motion.

Angular velocity — How fast an object revolves around a particular point.

For the purposes of this syllabus, the centripetal force is the net force, which acts in the direction of the centre of the circular motion (perpendicular to the velocity). The centripetal force is provided by a sum of all the forces acting on the body undergoing circular motion. The centripetal acceleration is the acceleration of a body undergoing such a motion. Such a body has a constant speed, but the direction of its velocity is constantly changing. Hence, the centripetal acceleration is perpendicular to the velocity. By considering the velocity at time t, and the velocity at an infinitesimal time step later, one can show that the centripetal acceleration is

a = ω2r = v2/r,

where ω is the angular velocity, r is the distance between the object and the centre of rotation, and v is its velocity.

6.2 Gravitation

Newton’s Law of Gravitation states that every body attracts every other body with a force proportional to the product of their masses and inversely proportional to the distance between them.

F = G m1 m2 r-2.

Here G is the constant of proportionality, the Gravitational Constant.

The gravitational field strength is

g = G m1 r-2.

This can be expressed in force per unit mass (F m-1) or in terms of acceleration, also called the acceleration due to gravity. On earth, this is 9.81 ms-2. The gravitational potential energy of an object in a gravitational field is the work done by an external agent to bring that object from infinity to that point.

U = – G m1 m2 r-1.

The gravitational potential at a particular point is the work per unit mass that an external agent would have to do to bring a mass from infinity to that point.

V =- G m1 r-1.

Maybe give some examples. Kepler’s 3rd and geostationary orbits