Uniform-circular-motion and two dimensional motion for high school

Uniform circular motion and two dimensional motion

1. Uniform circular
motion

2. Learning outcomes

3. Misconceptions
•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.
2
21
r
mm
GF

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
•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?

8. Vector analysis 1:
acceleration of a falling object

9. Straight line motion is easy

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.

11. Centripetal acceleration
Magnitude
Direction constantly changing but always acts towards
the centre of the circle.
and so force
r
v
a
2

r
mv
maF
2


12. Experimental test of F = mv²/r
Measure
• tension, F
• bung mass m
• radius r
• periodic time, T
Calculate mv
2
/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.

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 = 57.2958 degrees
57°17′45″
′ = minutes of arc (1/60 of a degree)
″ = seconds of arc (1/60 of a minute)

17. Angular velocity, 
Another key quantity.
The number of radians swept out per second (rad s
-1
)
= /t
linear
motion
rotational
motion
s 
v 
t t

18. Orbital speed from 




rv
tt
r


speed, orbital so
but ,
time
length arc
speed orbital
 and periodic time
periodic time, T = time for one revolution
f
T


 2
2
so ,2 revolution onefor 
acceleration & force

2
2
22
2
2



mrmaF
r
r
r
r
r
r
v
a



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. 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.
Artificial satellites

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 R
3
/T
2
is the same for all planets.

24. Law of gravitation
G = universal gravitational constant
= 6.67 × 10
-11
m
3
kg
-1
s
-2
m
1 and m
2 = masses of interacting bodies
r = distance between their centres
2
21
r
mm
GF
g


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.
sFW

