6-Motion in a circle notes

6. Motion in a circle

Angular displacement θ is defined as:



θ = S/r r θ s
Where θ is the angular displacement, in radian
s is the arc length, in metre
r is the radius of arc, in metre
Angular velocity ω is the rate of change of angular displacement.
ω = θ /t = 2π / T
ω is in rad s-1

Period is the time taken for the object to complete one revolution.

Frequency - the number of revolutions per unit time.
Linear velocity v = r ω
A body moving with constant speed in a circle must experience a force (or an acceleration) towards the centre of the circle. This force is called the centripetal force F.
F = ma = mv2/r = mrω2; centripetal acceleration a = v2/r or rω2

Note:
Centripetal force is always perpendicular to the direction of motion or the linear velocity v.

Centripetal force is not a new kind of force. It is the resultant force acting on the particle.

E.g. A particle tied to a string moving in a horizontal circle with constant speed v. Tension provides the centripetal force. T = mv2/r



A particle tied to a string moving with constant speed v in a vertical circle. Weight of the particle provides part of the ω centripetal force.



A: Ta – mg = mv2/r

B: Tb = mv /r

C: Tc + mg = mv2/r

Ta > T b > T c