1. Uniform circular motion

2. Learning outcomes

3. Misconceptions Teaching challenges
Common sense suggests there is an outward (centrifugal) force.
Students often get the impression that ‘centripetal force’ is a new force, when the term simply describes the direction of existing forces.
Teaching challenges
Convincing students that something travelling at a constant speed is accelerating.
Introducing the radian as a measurement unit for angles.
Analysis of the motion of moons and planets needs the relationship but this may not have been taught.

4. Newton’s conceptual leap
“The supreme act of imagination in the construction of modern dynamics”
- Richard Westfall (1971) Force in Newton’s Physics

5. Getting a feel for circular motion
In threes:
Do PP class experiment: Whirling a rubber bung on a string, answering associated questions (on small sheet).
To be followed by:
PP demo experiment Introducing circular motion

6. Examples Discuss, in pairs conker on a string clothes in a spin drier
blood sample in a centrifuge
child on a playground roundabout
car, bus or train going round a corner
Olympic sport ‘throwing the hammer’
cycle racing on an indoor track
Discuss, in pairs
In each case, what force keeps the object moving in a circle?

7. A video clip Bowling ball and mallet Discuss, in pairs
What does this video demonstrate about force and motion?
How does it relate to this diagram of a planetary orbit, from Newton’s Principia? 7

8. Vector analysis 1: acceleration of a falling object8

9. Straight line motion is easy9

10. Vector analysis 2: acceleration & velocity in different directions
Projectile motion: horizontal and vertical motions are independent, so analyse these separately.
Circular motion: the direction of motion is constantly changing. Use a more complex diagram that shows changes over very short time intervals.
Use Advancing Physics diagrams. 10

11. Centripetal acceleration
Magnitude
Direction constantly changing but always acts towards the centre of the circle.
and so force

12. Experimental test of F = mv²/r
Measure
tension, F
bung mass m
radius r
periodic time, T
Calculate mv2/r and compare with F.
VPLab simulation Circular motion

13. The Earth and Moon
If the force acting on an object is always at right angles to its velocity (momentum), then the object moves with constant speed in a circle.
Describe the force that keeps the Moon in orbit round the Earth as a centripetal force if you want, but remember it’s GRAVITY.
Be careful you don’t confuse students (or yourself).
Real forces can act centripetally. 13

14. What to measure?
When things moving in circles, or parts of circles (arcs),
you can often directly measure
angle of rotation
rate of rotation;
you must calculate
distance travelled
orbital speed.

15. In radian world
s = r θ defines the radian.
When θ = 1 radian, s = r.

16. The radian Another way of measuring angles
2π (~ 6) radians in a circle
How many radians in a right angle?
1 rad = degrees
57°17′45″
′ = minutes of arc (1/60 of a degree)
″ = seconds of arc (1/60 of a minute) 16

17. Angular velocity,   = /t linear motion rotational s  v  t
Another key quantity.
The number of radians swept out per second (rad s-1)
 = /t
linear
motion
rotational s  v  t
Yes, it’s not really a vector - pesudovector 17

18. Orbital speed from   and periodic time acceleration & force
periodic time, T = time for one revolution
acceleration & force

19. D R G ?
2π rad = 360 degrees

20. Rotation in a vertical circle
PP experiment Looping the loop
Rotating a bucket of water
Draw the free body force diagram.
Vector analysis shows how the forces acting (weight, central force such as tension) combine differently as the object circles round.

21. More free body force diagrams
conical pendulum
car/bike/train on a ramped curve
roller coaster

22. Artificial satellites
PP experiment:
Sketching a satellite orbit and predicting its period
This represents part of a circular orbit for a satellite at an altitude of 200 km.
‘Apparent weightlessness’: An orbiting spacecraft and its contents are in free fall.

23. Kepler’s third law
If for each planet you take an average radius, R, and time interval the planet takes to go once round its orbit (its year), T, then the ratio R3/T2 is the same for all planets.

24. Law of gravitation G = universal gravitational constant
= 6.67 × m3 kg-1 s-2
m1 and m2 = masses of interacting bodies
r = distance between their centres

25. Endpoints NOTE: In circular motion, F is always perpendicular to s.
This means that no energy is required e.g. to keep planets in orbit.