by the Dutch scientist Christiaan Huygens ( - ) but it was probably known to Newton also some years earlier. “Centripetal” comes from a Greek term which means ‘centre-seeking’. Since v and R are constant, the magnitude of the centripetal acceleration is also constant. However, the direction changes — pointing always towards the centre.
Therefore, a centripetal acceleration is not a constant vector. We have another way of describing the velocity and the acceleration of an object in uniform circular motion. As the object moves from P to P ′ in time ∆ t (= t ′ – t ), the line CP (Fig. .
) turns through an angle ∆ θ as shown in the figure. ∆ θ is called angular distance. We define the angular speed ω (Greek letter omega) as the time rate of change of angular displacement : ω ∆ t ( . ) Now, if the distance travelled by the object during the time ∆ t is ∆ s, i.e.
PP ′ is ∆ s , then : s = ∆ but ∆ s = R ∆ θ . Therefore : ω v = R ω ( . ) We can express centripetal acceleration a c in terms of angular speed : c = ω ω c = ω ( . ) The time taken by an object to make one revolution is known as its time period T and the number of revolution made in one second is called its frequency ν (= / T ).
However, during this time the distance moved by the object is s = π R . Therefore, v = π R / T = π R ν ( . ) In terms of frequency ν , we have ω = πν v = π R ν a c = π ν R ( . ) Example .
An insect trapped in a circular groove of radius cm moves along the groove steadily and completes revolutions in s. (a) What is the angular speed, and the linear speed of the motion? (b) Is the acceleration vector a constant vector ? What