Circular motion Objectives: understand that acceleration is present when the magnitude of the velocity, or its direction, or both change; understand that in motion on a circle with constant speed there is centripetal acceleration of constant magnitude, directed at the centre of the circle; recognize situations in which a force is acting in a direction toward the centre of a circle; solve problems involving applications of Newton’s second law to motion on a circle.

Uniform circular motion Let T be the time to complete one full revolution. We call T the period of the motion. Since the speed is constant and the object covers a distance of 2πR in a time T, it follows that: v = 2πR/T

Angular speed The object sweeps out an angle of 2π radians in a time equal to the period. So we define the angular speed as angular speed = angle swept/ time taken ω = 2π/T

Tangential velocity It is important to note that the speed may be constant but the velocity is not. It keeps changing direction. It is a general result of the velocity vector is always tangent to the path. Since the velocity changes, we have acceleration!!!

Centripetal acceleration A body moving along a circle of radius R with speed v experiences centripetal acceleration that has mangitude given by a c = v 2 /R and is directed toward the centre of the circle.

Tangential acceleration If the magnitude of the velocity changes, we have tangential acceleration. This is a vector directed along the velocity vector if the speed is increasing and opposite to the velocity vector if the speed is decreasing. The magnitude of the tangential acceleration is given by a t = Δv/Δt When the velocity direction and magnitude are changing, we have both centripetal acceleration and tangential acceleration. The total acceleration is then the vector sum of the vectors representing these accelerations.

Example questions

Centripetal forces If a body moves in a circle a net force must be acting on the body. If the speed is constant, the direction of the acceleration is towards the centre of the circle, and therefore the net force is centripetal. F net = ma C

Example questions

Work done by a centripetal force Since a centripetal force is at right angles to the direction of motion, the work done by the force is zero.

Exercises