* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Transparancies for Gravity & Circular Motion Section
Coriolis force wikipedia , lookup
Center of mass wikipedia , lookup
Laplace–Runge–Lenz vector wikipedia , lookup
Hunting oscillation wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Jerk (physics) wikipedia , lookup
Fictitious force wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Centrifugal force wikipedia , lookup
Classical mechanics wikipedia , lookup
Fundamental interaction wikipedia , lookup
Equivalence principle wikipedia , lookup
Mass versus weight wikipedia , lookup
Relativistic angular momentum wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Equations of motion wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Newton's laws of motion wikipedia , lookup
Classical central-force problem wikipedia , lookup
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.” October 2004 http://ppewww.ph.gla.ac.uk/~parkes/teaching/PK/PK.html Chris Parkes 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 m1m2 F G 2 r r m2 m1 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 24kg, m =5.97x10 mE m GmE E Gravity on m F G RE=6378km 2 2 earth’s surface RE RE Mass, radius of earth GmE 1 9.81ms Or F mg Hence, g 2 RE Circular Motion 360o = 2 radians 180o = radians 90o = /2 radians • 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 •Acceleration d v x ( R cos t ) R sin t dt d v y ( R sin t ) R cos t dt d d a x (v x ) ( R sin t ) R 2 cos t dt dt d d a y (v y ) ( R cos wt ) R 2 sin t dt dt y R s =t x t=0 Magnitude and direction of motion •Velocity v 2 vx v y R 2 w2 sin 2 t R 2 2 cos 2 t 2 R 2 2 2 v=R tan And direction of velocity vector v Is tangential to the circle vy vx cos t 1 sin t tan 90o •Acceleration 2 a a ax a y 2 v 2 R 2 w4 cos 2 t R 2 4 sin 2 t 4 R 2 a= 2R=(R)2/R=v2/R And direction of acceleration vector a a= -2r a x 2 x a y 2 y Acceleration is towards centre of circle Angular Momentum • For a body moving in a circle of radius r at speed v, the angular momentum is L=(mv)r = mr2= I (using v=R) I is called moment of inertia The rate of change of angular momentum is dL d dv dt (mvr) mr mra dt dt r ma r F r s – The product rF is called the torque of the Force • Work done by force is Fs =(Fr)(s/r) = Torque angle in radians Power d Torque Torque dt = rate of doing work = Torque Angular velocity Force towards centre of circle • Particle is accelerating – • • 1. 2. 3. So must be a Force Accelerating towards centre of circle – So force is towards centre of circle F=ma= mv2/R in direction –r 2 v or using unit vector F m rˆ r Examples of central Force Tension in a rope Banked Corner Gravity acting on a satellite N.B. general solution is an ellipse not a circle - planets travel in ellipses around sun Satellites •Centripetal Force provided by Gravity Mm mv2 F G 2 R R M 2 M v G v G R R m R M Distance in one revolution s = 2R, in time period T, v=s/T R T 2R / v 2R GM T2R3 , Kepler’s 3rd Law •Special case of satellites – Geostationary orbit •Stay above same point on earth T=24 hours 3 24 60 60 2 R 42,000km R2 GM E Gravitational Potential Energy • How much work must we do to move m1 from r to infinity ? m1m2 F G 2 R – When distance R r m2 m1 – Work done in moving dR dW=FdR – Total work done m1m2 W G r R 2 dR W Gm1m2 ( R1 )r Gm1m2 ( 1 ) Gm1m2 ( 1r ) W Gm1m2 r Choose Potential energy (PE) to be zero when at infinity m1m2 Then stored energy when at r is –W PE G r -ve as attractive force, so PE must be maximal at Compare Gravitational P.E. • Relate to other expression that you know Potential Energy falling distance h to earth’s surface = mgh Uses: 1) Expression for g from earlier • g=GME/RE2 M m M m G E G E RE RE h GM E 1 m 1 1 h / RE 2) Binomial expansion given h<<RE RE • (1+)-1 = 1- +…..smaller terms… mgRE (1 1 h / RE smaller ..terms...) mgh •Compare with Electrostatics: Same form, but watch signs: attractive or repulsive force attract m1m2 F G 2 r repel Q1Q2 F k 2 r Maximal at m1m2 PE G r Minimal at Q1Q2 PE k r A final complication: what do we mean by mass ? • Newton’s 2nd law F = mI a mI is inertial mass • Law of Gravity mG M G F G rˆ 2 r 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!