1. Circular Motion

2. Problem 1
A loop-the-loop machine has radius r of 18m.
Calculate the minimum speed with which a cart must enter the loop so that it does not fall off at the highest point.
Predict the speed at the top in this case.

3. Problem 2
In an amusement park ride a cart of mass 300kg and carrying four passengers each mass 60kg is dropped from a vertical height of 120 m along a frictionless path that leads into a loop-the-loop machine of radius 30m. The cart then enters a straight stretch from A to C where friction brings it to rest after a distance of 40 m
Determine the velocity of the cart at A.
Calculate the reaction force from the seat of the cart onto a passenger at B.
Determine the acceleration experienced by the cart from A to C (assumed constant)

4. The Law of Gravitation
Topic 6

5. Topic 6: Fields and forces
6.1 Gravitational force and field
State Newton’s universal law of gravitation.
Students should be aware that the
masses in the force law are point
masses. The force between two
spherical masses whose separation
is large compared to their radii
is the same as if the two spheres
were point masses with their
masses concentrated at the centers
of the spheres.

6. Earth exerts a downward force on you, & you exert an upward force on Earth. When there is such a large difference in the 2 masses, the reaction force (force you exert on the Earth) is undetectable, but for 2 objects with masses closer in size to each other, it can be significant.
The gravitational force one body exerts on a 2nd body , is directed toward the first body, and is equal and opposite to the force exerted by the second body on the first
Figure 6-2. Caption: The gravitational force one object exerts on a second object is directed toward the first object, and is equal and opposite to the force exerted by the second object on the first. 
Must be true from
Newton’s 3rd Law

7. Newton’s Universal Law of Gravitation
Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them. F12 = -F21  [(m1m2)/r2] Direction of this force:  Along the line joining the 2 masses

8. Comments
F12  Force exerted by particle 1
on particle 2
F21  Force exerted by particle 2
on particle 1
F21 = - F12
This tells us that the forces form a Newton’s 3rd Law action-reaction pair, as expected.
The negative sign in the above vector equation tells us that
particle 2 is attracted toward particle 1

9. More Comments
Gravity is a field force that always exists between 2 masses, regardless of the medium between them.
The gravitational force decreases rapidly as the distance between the 2 masses increases
This is an obvious consequence of the inverse square law

10. Example : Spacecraft at 2rE
A spacecraft at an altitude of twice the Earth radius
Earth Radius: rE = km
Earth Mass: ME =  1024 kg
FG = G(mME/r2)
Mass of the Space craft m
At surface r = rE
FG = weight
or mg = G[mME/(rE)2]
At r = rE
FG = G[mME/(2rE)2]
or (¼)mg = N

11. Example : Force on the Moon
Find the net force on the
Moon due to the gravitational
attraction of both the Earth &
the Sun, assuming they are at
right angles to each other.
ME =  1024kg
MM =  1022kg
MS =  1030 kg
rME =  108 m
rMS =  1011 m
F = FME + FMS

12. F = FME + FMS (vector sum) FME = G [(MMME)/ (rME)2] = 1
F = FME + FMS (vector sum) FME = G [(MMME)/ (rME)2] = 1.99  1020 N FMS = G [(MMMS)/(rMS)2] = 4.34  1020 N F = [ (FME)2 + (FMS)2] = 4.77  1020 N tan(θ) = 1.99/4.34  θ = 24.6º

13. Gravitational Field Strength
Force per
unit point
mass
Derive an expression for gravitational field strength at the surface of a planet, assuming that all its mass is concentrated at its centre.
Man’s weight = mg
BUT we know that this is equal to his gravitational attraction, so…
GMm = mg r2
Therefore:
GM = g r2
(this is a vector quantity)
Consider a man on the Earth:

14. Prob. 3
Estimate the force between the Sun and the Earth.

15. Prob.4
Determine the acceleration of free fall (the gravitational field strength) on a planet 10 times as massive as the Earth and with a radius 20 times as large.

16. Orbits and Gravity

17. Orbital Equation
Predicts the speed of the satellite at a particular radius

18. Angular speed

19. Orbital Period
Kepler’s Third Law