1. Topic 2.4 Uniform Circular Motion
Mechanics
Topic 2.4 Uniform Circular Motion

2. Uniform Circular Motion
An object moving in a circle requires a force to be acting towards the centre.
Imagine what would happen if the string broke?
Ball flies off at a tangent

3. Circular Forces
When the force responsible for the circular motion disappears, e.g. if the string breaks, the motion will become linear.
The force acting towards the centre is called the “Centripetal Force”
How does this explain what we feel in a car cornering a tight bend quickly? We feel as if we are being forced outwards.

4. Cornering
Doh!
Consider a car moving quickly round a corner. The driver has a carton of eggs on the seat and a coffee cup on the dashboard!
He could use the "centrifugal" force to explain why his coffee cup and the carton of eggs he has on the seat beside him tend to slide sideways.
The truth is, the friction of the seat or dashboard is not sufficient to force these objects on their curved path.

5. Take a birds eye view!
A person in a hovering helicopter above the car could describe the movement of the cup and the egg carton as just going straight.
The cup and egg carton obey Newton 1st Law. They continue in a straight line while the car travels in a curved path.
If the cup and carton are to follow the same curve as the car, a force needs to be applied to them.

6. Centrifugal Force – A Figment of Imagination
As you can see in the previous example it is the absence of any force that causes the cup and carton to slide not because of the presence of a force.
The idea that there is a force, causing the carton and cup to slide gives rise to the idea of the centrifugal force. This “force” however is completely a figment of imagination.
In fact only the presence of a centripetal force can cause objects to move in a circle

7. Circular Motion
A body moving with uniform speed in a circle is changing velocity as the direction changes.
This change of velocity, and the acceleration is directed towards the centre of the circle
This acceleration is called the centripetal acceleration

8. Vector Diagrams velocity at a tangent uniform speed Acceleration and
force directed
towards the centre

9. Proof that a centripetal force acts towards the centre of a circle.
Consider a body that rotates from A to B in a time Δt.
The velocity at A is V1 and at B is V2.
(diagram a)
The change in velocity is V2-V1 (diagram b).
Δv is towards the centre.
Velocity Vectors

10. Centripetal Acceleration
The expression for centripetal acceleration is
a = v2 = 42r r T2
Where v is the velocity at any instant (i.e.the constant speed)
And r is the radius of the circle
T is the time period of one complete circle

11. Centripetal Force The expression for centripetal force is F = mv2 r
Where v is the velocity at any instant (i.e.the constant speed)
m is the mass of the object
And r is the radius of the circle

12. Examples of Centripetal Force include
the gravitational force keeping the moon in its orbit
the friction acting sideways on the tyres of a car turning a corner
the tension in a rope, when a bucket of water is swung around your head

13. Motion in a Vertical Circle
Consider a body moving in a vertical circle.
At the top:
At the top, the resultant force of the tension and the weight provides the centripetal force. mg T1

14. Motion in a Vertical Circle
At the bottom:
At the bottom, the resultant force of the tension and the weight provides the centripetal force. T2 mg