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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 2004 Handout III : Gravitation and Circular Motion

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Gravitational Force Myth of Newton & apple. He realised gravity is universal same for planets and apples Newton’s law of Gravity Inverse square law 1/r 2, r distance between masses The gravitational constant G = 6.67 x Nm 2 /kg 2 F F m1m1 m2m2 r Gravity on earth’s surface OrHence, m E =5.97x10 24 kg, R E =6378km Mass, radius of earth Explains motion of planets, moons and tides Any two masses m 1,m 2 attract each other with a gravitational force:

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Circular Motion x y =t=t R t=0 s 360 o = 2 radians 180 o = radians 90 o = /2 radians Acceleration 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

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

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For a body moving in a circle of radius r at speed v, the angular momentum is L=(mv) r = mr 2 = 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 Angular Momentum (using v= R) I is called moment of inertia s r

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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= mv 2 /R in direction –r or using unit vector Examples of central Force 1.Tension in a rope 2.Banked Corner 3.Gravity acting on a satellite

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

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Gravitational Potential Energy m1m1 m2m2 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 How much work must we do to move m 1 from r to infinity ? –When distance R –Work done in moving dR dW=FdR –Total work done

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Compare Gravitational P.E. Same form, but watch signs: attractive or repulsive force attractrepel Maximal at Minimal at Uses: 1)Expression for g from earlier g=GM E /R E 2 2)Binomial expansion given h<

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A final complication: what do we mean by mass ? Newton’s 2 nd law F = m I a Law of Gravity m I is inertial mass m G, M G 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!

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