(a) If speed doubles, centripetal force must quadruple. Radius must halve to maintain same centripetal force, so smallest radius would be r/2. (b) If mass doubles, centripetal force must double to provide the same centripetal acceleration. Radius must halve again to maintain the doubled force, so smallest radius would be r/4.

1. Uniform Circular Motion
v is constant

v is constantly changing r
1

2. Examples of UCM
http://www.youtube.com/watch?v=a3N9BanDc6E&list=PL6413B12522DDEA5D
2

3. Tangential speed
The distance travelled in one
rotation is the circumference
s  2r r
The time to travel around the
circle is T.
s 2r
v 
t T 2r
v
T 3

4. Example : Tangential Speed
A merry-go-round does 5 complete revolutions in a
minute. It has a radius of 8 metres.
Calculate the tangential speed.
4

5. 2 r
v
T
5

6. 2 r
v
T
6

7. 2 r
v
T
7

8. ∆
Centripetal Acceleration
To find the instantaneous acceleration you must make t
(the time interval) very small v
a
t
 
v  vf - vi

vf
v

- vi
t is small, approaching zero
8

9. ∆
Centripetal Acceleration
At any instant the centripetal acceleration is towards the
centre of the circle.
SO- Using vector subtraction show centripetal
acceleration is directed towards the centre of a
circle over a small time interval.
9

10. Centripetal Acceleration
Magnitude calculated by :
2
v
ac 
r
10

11. Centripetal Acceleration
Magnitude calculated by :
2
v
ac 
r
11

12. Note 2
v
if ac  then
r
ac  v 2
If you double the tangential speed, you
quadruple the centripetal acceleration
12

13. 1
also ac 
r
If you double the radius, you halve the
centripetal acceleration
13

14. Example : The Effect of Radius on
Centripetal Acceleration
The bobsled track at the 1994
Olympics in Lillehammer, Norway,
contained turns with radii of 33
m and 24 m.
Find the centripetal acceleration
at each turn for a speed of 34
m/s, a speed that was achieved
in the two-man event.
14

15. 2
v
ac 
r
15

16. v2
ac 
r
16

17. v2
ac 
r
17

18. Centripetal Force - FC
Both FC and aC are at right angles to the
velocity, directed towards the centre of the
circle.
F  ma
2
v
a 
c
r
2 Quantity: force , F
mv
FC  Units: newtons, N
r
18

19. Example: Centripetal Force
A model airplane has a mass of 0.90 kg and
moves at a constant speed on a circle that is
parallel to the ground.
Find the tension
in the 17.0m long string
if the plane circles
once every 3 seconds.
19

20. mv 2
FC 
r
2 r
v
T
20

21. aC provided by the tension force aC provided by the normal force
Describe situations in which the
centripetal acceleration is caused by a
tension force, a frictional force, a
gravitational force, or a normal force
aC provided by the frictional force aC provided by the gravitational force
21

22. aC provided by the tension force aC provided by the normal force
A bob on a string being whirled A bobsled travelling around an
around . Olympic bobsled track.
aC provided by the frictional force aC provided by the gravitational force
A car driving around a flat The Moon orbiting the Earth.
roundabout.
22

23. Free body diagrams

Box on a table FN

   Fg
FNET  Fg  FN  0N
• Draw a dot at the centre of mass
• draw labelled force vectors from the centre of mass
• If body is not accelerating the forces are balanced 23

24. Box on a slope, not moving

 FN
Ffriction
 
FNET  0 Fg
• Normal force is perpendicular to the surface of
contact
24

25. Hanging bob
25

26. Car stationary
26

27. Car constant v
27

28. Car accelerating
28

29. Draw free body diagrams for a fridge
• Pushing against a fridge (not moving)
• Pushing against a fridge (accelerating)
• Pushing against a fridge (constant velocity)
29

30. Car going around a bend
SO: Identify the vertical and horizontal
forces on a vehicle moving with constant
velocity on a flat horizontal road.
2
mv
F friction 
r
30

31. Factors affecting cornering
2
mv
F friction 
r
r – radius of curvature
m – mass of vehicle
v - speed of vehicle
Ffriction – depends on
• tyres
• road surface
31

32. Banked Curves
Explain that when a vehicle travels round a banked curve at the
correct speed for the banking angle, the horizontal component of
the normal force on the vehicle (not the frictional force on the
tyres) causes the centripetal acceleration. 32

33. A car is going around a friction-free banked
curve. The radius of the curve is r.
FN sin  that points toward the center C
2
mv
FC  FN sin  
r
FN cos  and, since the car does not accelerate in the
vertical direction, this component must balance the
weight mg of the car.
FN cos  mg
33

34. FN sin  mv / r 2

FN cos mg
2
v
tan  
rg
At a speed that is too small for a given  , a car
would slide down a frictionless banked curve: at
a speed that is too large, a car would slide off the
top.
34

35. Want to watch the derivation?
http://www.youtube.com/watch?v=OlUlLglTEn4
Derive the equation relating the banking angle
to the speed of the vehicle and the radius of curvature .
35

36. Banking Angle Example
A curve has a radius of 50 m and a banking
angle of 15o. What is the ideal speed (no
friction required between cars tyres and the
surface) for a car on this curve?
Solve problems involving the use of
the equation
36

37. Banking Angle Example
A curve has a radius of 50 m and a banking angle of 15o.
What is the ideal speed (no friction required between cars
tyres and the surface) for a car on this curve?
37

38. Banking Angle Example
A curve has a radius of 50 m and a banking angle of 15o.
What is the ideal speed (no friction required between cars
tyres and the surface) for a car on this curve?
If the car negotiates
the bend at 11.5ms-1 it
can do so without
38
friction.

39. Conceptual Question
In a circus, a man hangs upside down from a trapeze, legs bent
over the bar and arms downward, holding his partner. Is it
harder for the man to hold his partner when the partner hangs
straight down and is stationary or when the partner is swinging
through the straight-down position?
39

40. When they are moving in a circular arc they have
centripetal acceleration. The acrobat exerts an
additional pull compared to when they are stationary.
40

41. Proportionality examples
A car is traveling in uniform circular motion on a section
of road whose radius is r. The road is slippery, and
the car is just on the verge of sliding.
(a) If the car’s speed was doubled, what would have to
be the smallest radius in order that the car does not
slide? Express your answer in terms of r.
(b) What would be your answer to part (a) if the car
were replaced by one that weighted twice as much?
41