Motion in a Circle

 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 = rω² = v² / 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 = mv² / 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}.