# Motion in a Circle

#### Kinematics of uniform circular motion

Radian (rad) is the S.I. unit for angle, *θ* and it can be related to degrees in the following way. In one complete revolution, an object rotates through 360° , or 2π rad.

As the object moves through an angle *θ*, with respect to the centre of rotation, this angle θ is known as the angular displacement.

Angular velocity (ω) of the object is the rate of change of angular displacement with respect to time.

ω = θ / t = 2π / T (for one complete revolution)

Linear velocity, v, of an object is its instantaneous velocity at any point in its circular path.

v = arc length / time taken = rθ / t = rω

- The direction of the linear velocity is at a
*tangent*to the circle described at that point. Hence it is sometimes referred to as the*tangential velocity* - ω is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis.

A body moving in a circle at a __constant speed__ changes velocity {since its direction changes}. Thus, it *always* experiences an acceleration, a force and a change in momentum.

#### Centripetal acceleration

a = rω^{2} = v^{2} / r {in magnitude}

#### Centripetal force

Centripetal force is the ** resultant** of all the forces that act on a system in circular motion.

{It is not a particular force; “centripetal” means “centre-seeking”. Also, when asked to draw a diagram showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as “centripetal force”. }

Centripetal force, **F** = m r ω ^{2} = mv^{2} / r {in magnitude}

A person in a satellite orbiting the Earth experiences “**weightlessness**” although the gravi field strength at that height is not zero because the person and the satellite would both have the __same acceleration__; hence the contact force between man &

satellite / __ normal reaction on the person is zero__ {Not because the field strength is negligible}.