7.0 MOTION IN A CIRCLE
Angular displacement
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.


The radian is the angle subtended at the centre of a circle by an arc length of
the circumference equal to the radius of the circle.

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

Angular velocity (ω) of the object is the rate of change of angular
displacement with respect to time.
𝜃
𝜔 =
𝑡
θ – angular displacement (rad)
ω – angular velocity of particle (rad s-1)
t – time taken (s)

The period T, of rotational motion is the time taken to complete one
revolution.
𝜔=

2𝜋
𝑇
Linear velocity, v, of an object is its instantaneous velocity at any point
in its circular path.
𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ(𝐴𝑃)
𝑟𝜃
𝑣 =
=
𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛
𝑡
∴ 𝑣 = 𝑟𝜔

Note:
(i)
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.
(ii)
ω is the same for every point in the rotating object, but the
linear velocity v is greater for points further from the axis.
Centripetal force
 A body moving in a circle at a constant speed changes velocity {since its
direction changes}
 In accordance with Newtons 1st law, a body which is moving in a circle
must have a resultant force acting on it.
 For a body moving with constant speed, there is no component of this
force which acts in direction of motion.
 The force must therefore be perpendicular to the motion of the body, ie
directed towards the centre.
 It is known as a centripetal force.
 Centripetal force is the resultant of all the forces that act on a system
in circular motion.
Centripetal acceleration
 By Newtons 2nd law, a body acted upon by a resultant force must have an
acceleration.
 The acceleration is in the same direction as the force (towards the
centre)
 It is known as a centripetal acceleration
 A body moving with constant angular velocity ω, along a path of radius r,
𝑣2
centripetal acceleration a is given by 𝑎 = 𝜔2 𝑟 = 𝑟

Centripetal force will be given by 𝐹 = 𝑚𝑎 = 𝑚𝜔2 𝑟 =
𝑚𝑣 2
𝑟
Banked roads
 ‘Banking’ roads removes the reliance on friction having to provide
centripetal force for a vehicle going round a bend.


The normal reaction, R acquires a horizontal component (R sin θ).
Consider car of mass m moving with constant speed v round a bend of
radius r.

Centripetal force needs to provide an acceleration of 𝑟
𝑚𝑣 2
𝑅 𝑠𝑖𝑛 𝜃 =
𝑟
Since no vertical acceleration 𝑅 𝑐𝑜𝑠 𝜃 = 𝑚𝑔
𝑣2
𝑣2
Dividing equations results 𝑡𝑎𝑛 𝜃 = 𝑟𝑔
Example
A satellite is moving at 2000 ms-1 in a circular orbit around a distant moon.
If the radius of the circle followed by the satellite is 1000 km, find:
i) the acceleration of the satellite
ii) the time for the satellite to complete one full orbit of the moon in
minutes(2d.p.).