Newton's Legacy: Gravitation and Circular Motion

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 http://ppewww.ph.gla.ac.uk/~parkes/teaching/PK/PK.html October 2004

2. Myth of Newton & apple. He realised gravity is universal same for planets and apples m2 m1 Gravitational Force • Any two masses m1,m2 attract each other with a gravitational force: F F r Newton’s law of Gravity Inverse square law 1/r2, r distance between masses The gravitational constant G = 6.67 x 10-11 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. 360o = 2 radians 180o =  radians 90o = /2 radians =t 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 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. Angular Momentum (using v=R) • 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 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. 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. m2 m1 Gravitational Potential Energy • How much work must we do to move m1 from rto infinity ? • When distance R • Work done in moving dR dW=FdR • Total work done r 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 = mIa • Law of Gravity mI isinertial mass mG,MG isgravitational 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!