# CIRCULAR MOTION - revision notes and calculations

Circular Motion

Let a body moves in a circle as shown in the figure below:

First of all, we define several terms: -

(1)        The angular position θ: - This is the angle moved from the initial position.

(2)        The angular velocity ϖ: - This is the angle rotated per unit time.

Unit: rad s-1. i.e. ϖ = Δθ/Δt

(3)        Period T: - is the time required for the body to move one revolution. i.e. T = 2π/ϖ.

(4)        Speed V: - is the distance moved per unit time. i.e. V = s/Δt, but in circle relationship  S = r Δθ ,

Therefore,     V = r Δθ/Δt = rω, or  V = rω .

(5)        Angular acceleration of a circle :-In circular motion, velocity always changes even the speed is keep constant, therefore, there should be an acceleration, a. This value of a can be deduced from the change of velocity.

Consider a particle moves in a circular path with constant speed V from A to B in a short time interval Δt as shown below: -

The change of velocity from A to B is denoted in the vector diagram as ΔV such that:    ΔV = VB  -  VA

Since,   the magnitude of  VA  = magnitude of VB = VΔθ   (if Δθ is very small)

Therefore, a = change of velocity/time taken = ΔV/Δt = VΔθ/Δt = Vϖ

i.e.  a = Vϖ = V2/r = rω2  (in magnitude)

Note:         (I)        For direction, if Δt is made so small that A and B almost coincide, then, the vector ΔV is perpendicular to VA (or VB) i.e. the direction is pointing to the centre O.

(II)        The acceleration is pointing to the centre, which is therefore known as centripetal acceleration.

(III)        According to Newton's 2nd law, acceleration is due to force, namely the resultant force

Therefore,  F = ma = mV2/r = mrω2

This is known as the centripetal force

(IV)        We must be clear of the cause -result relationship. i.e. we must have centripetal force first, this produces a centripetal acceleration which couples with a tangential velocity to start the circular motion.

i.e.        Cause        Result

Centripetal acceleration  +        imply        Circular motion

Tangential Velocity

(V)        In our usual calculation, we must consider the net force acting on the "free body diagram' of the rotating object  being the centripetal force.

(6)        Examples of Circular Motion

(i)        Motion of Bicycle Rider Round Circular Track

The centripetal force is provided by frictional force F

Therefore,         F = mV2/r        (1)

This frictional force produce a clockwise moment about G, which must be balanced by the anticlockwise moment produced by R.

i.e.        Fh = Ra = mga        (2)

Therefore,        a/h = F/mg = tanθ

Where θ is the angle of inclination to the vertical?

tanθ = mV2/rmg = V2/rg

i.e.        tanθ ∝ V2 and  tanθ ∝ 1/r

This means that the quicker the velocity to turn a sharp angle the larger is θ (bend dipper) if the ...