3Gravity The moon is performing circular motion round the earth. The centripetal force comes from the gravity.vFcmoonearth
4GravityNewton found that the gravity on the moon is the same force making an apple fall.WGround
5Newton’s Law of Gravitation Objects attract each other with gravitational force.In the diagram,m1 and m2 are the masses of the objects and r is the distance between them.m1m2Fr
6Newton’s Law of Gravitation Every particle of matter attracts every other particle with a force whose magnitude isG is a universal constantG = 6.67 m3kg-1s-2m1m2FrNote that the law applies to particles only.
15Gravitational FieldA gravitational field is a region in which any mass will experience a gravitational force.A uniform gravitational field is a field in which the gravitational force in independent of the position.
16Field strength, gThe gravitational field strength, g, is the gravitational force per unit mass on a test mass.F is the gravitational forcem is the mass of the test massg is a vector, in the same direction of F.SI unit of g is Nkg-1.test massFm
17Field strength, g SI unit of g is Nkg-1. The gravitational field strength, g, is the gravitational force per unit mass on a test mass.F is the gravitational forcem is the mass of the test masstest massFmSI unit of g isNkg-1.
18Field strength, g, outside an isolated sphere of mass M The gravitational field strength, g, outside an isolated sphere of mass M isMrfield strengthat XXOProve it by placing a test mass m at a point X with distance rfrom the centre of the isolated sphere M.
19Example 4The field strength of the earth at the position of the moon.
20Field strength, g Unit of g is Nkg-1. g is also a measure of the acceleration of the test mass.g is also the acceleration due to gravity, unit is ms-2.
21Field strength, g Field strength, g. Unit Nkg-1. A measure of the strength of the gravitational field.Acceleration due to gravity, g.Unit ms-2.A description of the motion of a test mass in free fall.
22Field linesWe can represent the field strength by drawing field lines.The field lines for a planet are radially inward.planetRadial field
23Field linesWe can represent the field strength by drawing field lines.The field lines for a uniform field are parallel.Uniform fieldearth’s surface
24Field linesThe density of the field lines indicates the relative field strength.g1= 10 Nkg-1g2= 5 Nkg-1
25Field linesThe arrow and the tangent to the field lines indicates the direction of the force acting on the test mass.direction of the forcetest mass
26The earth’s gravitational field Mass of the earth Me 5.98 1024 kgRadius of the earth Re 6.37 106 mORe
27Gravity on the earth’s surface, go The gravitational field go near the earth’s surface is uniform andThe value of go 9.8 Nkg-1
28Example 5The gravity on the earth’s surface, go.
29Apparent WeightUse a spring-balance to measure the weight of a body.Depending on the case, the measured weight R (the apparent weight) is not equal to the gravitational force mgo.Rmgo
30Apparent WeightThe reading on the spring-balance is affected by the following factors:The density of the earth crust is not uniform.The earth is not a perfect sphere.The earth is rotating.
31Apparent Weight The density of the earth crust is not uniform. Places have different density underneath. Thus the gravitational force is not uniform.
32Apparent Weight 2. The earth is not a perfect sphere. Points at the poles are closer to the centre than points on the equators.rpole < requatorgpole > gequatorN-poleEquatorS-pole
33Apparent Weight X Y 3. The earth is rotating. Except at the pole, all points on earth are performing circular motion with the same angular velocity . However the radii of the circles may be different.XY
34Apparent Weight 3. The earth is rotating. Consider a mass m is at point X with latitude .The radius of the circle is r = Re.cos .mYXrReO
35Apparent Weight 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force.FcmXYrReONote that Fnet points to Y.
36Apparent Weight R R Fc X Y r m mgo O 3. The earth is rotating. The net force on the mass m must be equal to the centripetal force.So the apparent weight (normal reaction) R does not cancel the gravitational force mgo.FcXYrmmgoO
37Apparent Weight R R 3. The earth is rotating. The apparent weight R is not equal to the gravitational force mgo in magnitude.FcXYrmmgoO
38Apparent weight R on the equator mgoRThe apparent field strengthon the equator is
39Apparent weight R at the poles mgoThe apparent field strengthat the poles is
41Apparent weight at latitude FcXYrmmgoONote that the apparent weight Ris not exactly along the line throughthe centre of the earth.
42Variation of g with height and depth Outside the earth at height h.rmhReMegOh = height of the mass m from the earth’s surface
43Variation of g with height and depth Outside the earth at height h.rmhReMegOwhere go is the field strength on the earth’s surface.
44Variation of g with height and depth Outside the earth at height h.rmhReMegOwhere go is the field strength on the earth’ssurface.
45Variation of g with height and depth Outside the earth at height h close to the earth’s surface. h<<Re.rmhReMegOwhere go is the field strength on the earth’ssurface.
46Variation of g with height and depth Below the earth’s surface.RerOdMeOnly the core withcolour gives thegravitational force.gr = Re-d
47Variation of g with height and depth Below the earth’s surface.RerOdMeFind the mass Mr ofgr = Re-d
48Variation of g with height and depth Below the earth’s surface.RerOdMegr = Re-d
49Variation of g with height and depth Below the earth’s surface.RerOdMegg rr = Re-d
50Variation of g with height and depth r < Re , g r.r > Re ,earthggor distance from the centreof the earthRe
51Gravitational potential energy Up Object inside a gravitational field has gravitational potential energy.When object falls towards the earth, it gains kinetic energy and loses gravitational potential energy.This objectpossesses Upearth
52Zero potential energyBy convention, the gravitational potential energy of the object is zero when its separation x from the centre of the earth is .Up = 0earthOx
53Negative potential energy For separation less than r, the gravitational potential energy of the object is less than zero. So it is negative.Up < 0earthOr
54Gravitational potential energy Up Definition 1It is the negative of the work done by the gravitational force FG as the object moves from infinity to that point.earthFGOrdx
55Gravitational potential energy Up Definition 1earthFGOrdx
56Gravitational potential energy Up Definition 2It is the negative of the work done by the external force F to bring the object from that point to infinity.MeearthFOrdxm
57Gravitational potential energy Up Definition 2MeearthFOrdxm
74Gravitational potential V Definition:The gravitational potential at a point is the gravitational potential energy per unit test mass.whereU is the gravitational potentialenergy of a mass m at the point
75Gravitational potential V Definition:The gravitational potential at a point is the gravitational potential energy per unit test mass.unit of V is J kg-1
76Gravitational potential V Example 12 – to find the change in gravitational potential energy.ΔU = U – UoIf ΔU >0, there is a gain in U.If ΔU <0, there is a loss in U.
77EquipotentialsEquipotentials are lines or surfaces on which all points have the same potential.The equipotentials are always perpendicular to the field lines.
78EquipotentialsThe equipotentials around the earth are imaginary spherical shells centered at the earth’s centre.
107Satellites Natural satellites – e.g. moon. Artificial satellites – e.g. communication satellites,weather satellites.
108Geosynchronous satellites A geosynchronous satellite is above the earth’s equator.It rotates about the earth with the same angular speed as the earth and in the same direction.It seems stationary by observers on earth.
110Geosynchronous satellites Find the radius of the orbit of a geosynchornoussatellite.ωequatoraxissatellitehRersh + Re = rs
111Geosynchronous satellites rs = 4.23×107 mωequatoraxissatellitehRerh + Re = rs
112Geosynchronous satellites h = 3.59×107 mωequatoraxissatellitehRerh + Re = rs
113Parking Orbit Note that there is only one such orbit. It is called a parking orbit.ωequatoraxissatellitehRerh + Re = rs
114Satellites Near the Earth’s surface Assume that the orbit is circular with radius r Re , the radius of the earth.The gravitational field strength go is almost a constant (9.8 N kg-1).The gravitational force provides the required centripetal force.
115Satellites Near the Earth’s surface Find vrvrsatelliter Reearth
116Energy and Satellite Motion Find v and the kinetic energy Uk of the satellite.rsatelliteearth Mevm
117Energy and Satellite Motion The satellite in the orbit possesses both kinetic energy and gravitational energy.rsatelliteearth Mevm
118Energy and Satellite Motion earth MevmNote that Uk > 0
119Energy and Satellite Motion Find Up the gravitational potential of the satellite.rsatelliteearth Mevm
120Energy and Satellite Motion earth MevmNote that Up < 0
121Energy and Satellite Motion Find U, the total energy of the satellite.rsatelliteearth Mevm
122Energy and Satellite Motion earth MevmNote that U < 0
123Energy and Satellite Motion U : Up : Uk = -1 : -2 : 1rsatelliteearth Mevm
124Falling to the earth The satellite may lose energy due to air resistance. The total energy becomes morenegative and r becomes less.rsatelliteearth Mevm
125Falling to the earth The satellite follows a spiral path towards earth Mevm
126Falling to the earthAs r decreases, the kinetic energy of the satelliteincreases and the satellite moves faster.rsatelliteearth Mevm
127Falling to the earthExample 17 – Loss of energy
128Weightlessness in spacecraft vvmgThe astronaut isweightless.
129Weightlessness in spacecraft We fell our weight because there is normal reaction on us.Normal reactiongroundmg
130Weightlessness in spacecraft If there is not any normal reaction on us, we feel weightless. e.g. free fallingmg
131Weightlessness in spacecraft vThe gravitational forcemg on the astronaut isthe required centripetalforce. He does not requireany normal reaction toact on him.mg
132Weightlessness in spacecraft The astronaut isweightless.vmg