Circular motion refers to the motion of an object moving in a circular path at a constant speed but changing velocity due to changing direction. Centripetal force is the inward force required to cause an object to travel in a circular path, and is equal to mass times centripetal acceleration. Centrifugal force is an equal but opposite reaction force to centripetal force. Examples of circular motion include a stone on a string, satellites orbiting Earth, and the moon orbiting Earth.

1. 3. CIRCULAR MOTION
Definition and Examples Of Circular Motion:
Circular motion is the motion of an object in a circular
path or round a circle with constant speed but
changing velocity due to change in the direction of
travel of the object as the object moves round the
circle. Examples include:
1) A stone tied to a string and whirled in a horizontal or
vertical circle.
2) A satellite circling round the earth.
3) The moon circling the earth etc.

2. YOU ARE NOT EXPECTED
TO DRAW THIS DIAGRAM

3. Centripetal force and acceleration
For a body undergoing circular motion, the
speed of the body is constant in magnitude
but its direction changes in a uniform manner
hence causing the body to accelerate. This
acceleration is directed towards the centre of
the circular path and is known as the
centripetal acceleration and denoted as a. This
centripetal acceleration is the result of an
inwardly directed force referred to as the
centripetal force.

5. The magnitude of this acceleration is given as
Where a = Centripetal acceleration
v = Uniform speed
r = radius of the circular path.
The velocity v is always directed along the tangent to
the circle at any point of the path, while a is directed
towards the centre of the circle. We define centripetal
force as the inward force required to keep an object
moving with a constant speed in a circular path.

6. The magnitude of this centripetal force is given
as
Where;
and m = mass of the object.

7. Centrifugal force
According to Newton’s 3rd law, a force must
balance the effect of the centripetal force, and
this force is termed centrifugal force. Centrifugal
force is the force which acts opposite to the force
applied to keep a body moving in a circular path
i.e. centripetal force. Centrifugal force has the
same magnitude and dimensions as
the force that keeps the particle in its circular
path (the centripetal force) but points in the
opposite direction.

8. Applications of Centripetal force
1. Banking of curves at roundabouts, car race
tracks and bends. A banked turn is a turn or
change of direction in which the vehicle
banks or inclines, usually towards the inside
of the turn.
2. Motion on a level circular road is governed by
moderation of the speeds to levels equal to
or less than the square root of µrg.

9. 3. Bending of a cyclist round a curve
When the cyclist negotiates the curve, he bends inwards
from the vertical, by an angle θ. Let R be the reaction of
the ground on the cyclist.

10. These may be resolved into two components:
The implication of equation 3 is that, for the cyclist to bend
less, (i.e. for θ to be small), the velocity v should be smaller
and radius r should be larger.

11. ANGULAR SPEED AND VELOCITY.
When a stone is tied to one end of a rope and
whirled along a horizontal circle, the stone moves
in a circular path as shown in the figure below. As
the stone is whirled, it sweeps through an angle θ
and travels a circular distance S with constant
speed V. Its velocity changes because the stone
changes its direction as it travels round the circle.
As the stone moves round the circular path, and
sweep through angle θ, the stone moves with
angular velocity, w.

13. V = rw represents the relationship between
linear velocity V, and angular velocity w.

14. Calculations on circular motion
Example 1: A body moves with a constant
speed of 5.0 m/s along a circular path. If the
angular speed of the body is 2.3rad/s,
Calculate its acceleration toward the centre of
the circle.
Example 2: A body of mass 500g moves with a
speed of 4m/s, along a circular path of radius
5m. Estimate the centripetal force on the
body.

15. Example 3: A stone tied to the end of a string
is whirled in a horizontal circle of radius 8m
with an angular speed 3 rad/s. Estimate the
centripetal force acting on the stone and
hence predict the magnitude of tangential
velocity with which the stone will move off if
the string breaks in terms of mass M.
Example 4: What is the centripetal force acting
on a particle with radial acceleration of 4× 107
m/s2 if it has mass of 2.5 × 10-6kg?