1. Handout III : Gravitation and Circular Motion
EE1 Particle Kinematics : Newton’s Legacy “I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.”
Chris Parkes
October

2. Gravitational Force
Myth of Newton & apple. He realised gravity is universal same for planets and apples
Any two masses m1,m2 attract each other with a gravitational force: F F m1 m2 r
Newton’s law of Gravity
Inverse square law 1/r2, r distance between masses
The gravitational constant G = 6.67 x Nm2/kg2
Explains motion of planets, moons and tides
mE=5.97x1024kg,
RE=6378km
Mass, radius of earth
Gravity on
earth’s surface Or
Hence,

3. Circular Motion Rotate in circle with constant angular speed 
360o = 2 radians
180o =  radians
90o = /2 radians
Circular Motion
Rotate in circle with constant angular speed 
R – radius of circle
s – distance moved along circumference
=t, angle  (radians) = s/R
Co-ordinates
x= R cos  = R cos t
y= R sin  = R sin t
Velocity
=t R s y
t=0 x
Acceleration

4. Magnitude and direction of motion
Velocity
v=R
And direction of velocity vector v
Is tangential to the circle v 
Acceleration a 
a= 2R=(R)2/R=v2/R
And direction of acceleration vector a
a= -2r
Acceleration is towards centre of circle

5. I is called moment of inertia
Angular Momentum
For a body moving in a circle of radius r at speed v, the angular momentum is
L=(mv)r = mr2= I 
The rate of change of angular momentum is
The product rF is called the torque of the Force
Work done by force is Fs =(Fr)(s/r)
= Torque  angle in radians
Power = rate of doing work
= Torque  Angular velocity
(using v=R)
I is called moment of inertia s  r

6. Force towards centre of circle
Particle is accelerating
So must be a Force
Accelerating towards centre of circle
So force is towards centre of circle
F=ma= mv2/R in direction –r
or using unit vector
Examples of central Force
Tension in a rope
Banked Corner
Gravity acting on a satellite

7. Satellites Centripetal Force provided by Gravity m R M
N.B. general solution is an ellipse not a circle - planets travel in ellipses around sun
Satellites
Centripetal Force provided by Gravity m R M
Distance in one revolution s = 2R, in time period T, v=s/T
T2R3 , Kepler’s 3rd Law
Special case of satellites – Geostationary orbit
Stay above same point on earth T=24 hours

8. Gravitational Potential Energy
How much work must we do to move m1 from r to infinity ?
When distance R
Work done in moving dR dW=FdR
Total work done r m1 m2
Choose Potential energy (PE) to be zero when at infinity
Then stored energy when at r is –W
-ve as attractive force, so PE must be maximal at 

9. Compare Gravitational P.E.
Relate to other expression that you know
Potential Energy falling distance h to earth’s surface = mgh
Uses:
Expression for g from earlier
g=GME/RE2
Binomial expansion given h<<RE
(1+)-1 = 1- +…..smaller terms…
Compare with Electrostatics:
Same form, but watch signs: attractive or repulsive force
attract
repel
Maximal at 
Minimal at

10. A final complication: what do we mean by mass ?
Newton’s 2nd law F = mI a
Law of Gravity
mI is inertial mass
mG, MG is gravitational mass
- like electric charge for gravity
Are these the same ?
Yes, but that took another 250 years till
Einstein’s theory of relativity to explain!