Circular motion involves constant speed but changing direction, so it is accelerated motion. Acceleration is defined as the rate of change of velocity, which is always directed toward the center of the circle during circular motion due to vector subtraction of velocity vectors at different points. This centripetal acceleration must result from a force, known as the centripetal force, which can be caused by tension, friction, or gravity. The formula for centripetal force is F = mv2/r, where m is mass, v is speed, and r is the radius of the circle.

1. Tight Corners
Physics in Sport – Lesson 1

2. Learning intentions
 Explain why circular motion even at uniform speed is accelerated motion
 Use vector subtraction to show this
 Use formulas to solve problems

4. STARTER – give some examples of motion in a circle
 In which do you think the speed is uniform?

5. Measuring angles and arcs
 Circumference of a circle
s  2r
Where does this come from?
• Measuring angles in radians
• One radian is the angle
Angle in radians = length of arc
radius
s
 
r
s
2r
  
 2
r
r
• 360 degrees = 2π radians

6. Motion in a circle

If an object is moving in a circle what is it’s
a) Distance travelled
b) Displacement
c) Linear speed
•
2r
v
T
Theoretically what is it’s average velocity?

7. Why is it accelerated motion?
v1
 What is the definition of velocity?
 It is a vector quantity
 What is the definition of acceleration?
v4
v2
Q.
v3
 It is a vector quantity
In your own words explain using the definitions why
circular motion is accelerated motion

8. Why is it accelerated motion?
v1
 To find the change in velocity (Δv) we
need to subtract the velocity vectors.
v2
 v2-v1= v2+(-v1 )=Δv
v2
Δv Δv
-v1

9. Centripetal Acceleration
v1
v2
Δv
 To find the change in velocity any 2
short time intervals is always towards
the centre of the circle
v2
-v1
If the change in velocity is
towards the centre then so
is the acceleration

10. Centripetal Acceleration
 Acceleration is the rate of change of
velocity.
v1
v2
Δv
a= Δv
t
 It towards the centre of the circle.
 It is given by
2
v
a
r

11. Forces cause accelerations
 If there is such a thing as centripetal
acceleration then these must be
caused by forces (that’s what forces
do)
F  ma
2
v
a
r
 There are different centripetal forces
2
v
F m
r

12. Causes of Centripetal Force

13. Causes of Centripetal Force

14. Causes of Centripetal Force

15. Worked Example: Centripetal Force
A stone of mass 0.5 kg is swung round in a horizontal circle (on a frictionless surface) of radius
0.75 m with a steady speed of 4 m s-1.
 Calculate:
(a) the centripetal acceleration of the stone
acceleration = v2/r = 42 / 0.75 = 21.4 m s-2
(b) the centripetal force acting on the stone.
F = ma = mv2/r = [0.5 ´ 42] / 0.75 = 10.7 N
 Notice that this is a linear acceleration and not an angular acceleration. The angular velocity
of the stone is constant and so there is no angular acceleration.