3 Gravity The moon is performing circular motion round the earth. The centripetal force comes from the gravity.vFcmoonearth
4 GravityNewton found that the gravity on the moon is the same force making an apple fall.WGround
5 Newton’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
6 Newton’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.
15 Gravitational 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.
16 Field 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
17 Field 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.
18 Field 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.
19 Example 4The field strength of the earth at the position of the moon.
20 Field 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.
21 Field 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.
22 Field linesWe can represent the field strength by drawing field lines.The field lines for a planet are radially inward.planetRadial field
23 Field linesWe can represent the field strength by drawing field lines.The field lines for a uniform field are parallel.Uniform fieldearth’s surface
24 Field linesThe density of the field lines indicates the relative field strength.g1= 10 Nkg-1g2= 5 Nkg-1
25 Field 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
26 The earth’s gravitational field Mass of the earth Me 5.98 1024 kgRadius of the earth Re 6.37 106 mORe
27 Gravity 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
28 Example 5The gravity on the earth’s surface, go.
29 Apparent 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
30 Apparent 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.
31 Apparent Weight The density of the earth crust is not uniform. Places have different density underneath. Thus the gravitational force is not uniform.
32 Apparent 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
33 Apparent 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
34 Apparent 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
35 Apparent 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.
36 Apparent 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
37 Apparent Weight R R 3. The earth is rotating. The apparent weight R is not equal to the gravitational force mgo in magnitude.FcXYrmmgoO
38 Apparent weight R on the equator mgoRThe apparent field strengthon the equator is
39 Apparent weight R at the poles mgoThe apparent field strengthat the poles is
41 Apparent weight at latitude FcXYrmmgoONote that the apparent weight Ris not exactly along the line throughthe centre of the earth.
42 Variation of g with height and depth Outside the earth at height h.rmhReMegOh = height of the mass m from the earth’s surface
43 Variation of g with height and depth Outside the earth at height h.rmhReMegOwhere go is the field strength on the earth’s surface.
44 Variation of g with height and depth Outside the earth at height h.rmhReMegOwhere go is the field strength on the earth’ssurface.
45 Variation 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.
46 Variation of g with height and depth Below the earth’s surface.RerOdMeOnly the core withcolour gives thegravitational force.gr = Re-d
47 Variation of g with height and depth Below the earth’s surface.RerOdMeFind the mass Mr ofgr = Re-d
48 Variation of g with height and depth Below the earth’s surface.RerOdMegr = Re-d
49 Variation of g with height and depth Below the earth’s surface.RerOdMegg rr = Re-d
50 Variation of g with height and depth r < Re , g r.r > Re ,earthggor distance from the centreof the earthRe
51 Gravitational 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
52 Zero 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
53 Negative potential energy For separation less than r, the gravitational potential energy of the object is less than zero. So it is negative.Up < 0earthOr
54 Gravitational 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
55 Gravitational potential energy Up Definition 1earthFGOrdx
56 Gravitational 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
57 Gravitational potential energy Up Definition 2MeearthFOrdxm
74 Gravitational 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
75 Gravitational 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
76 Gravitational 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.
77 EquipotentialsEquipotentials are lines or surfaces on which all points have the same potential.The equipotentials are always perpendicular to the field lines.
78 EquipotentialsThe equipotentials around the earth are imaginary spherical shells centered at the earth’s centre.
107 Satellites Natural satellites – e.g. moon. Artificial satellites – e.g. communication satellites,weather satellites.
108 Geosynchronous 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.
112 Geosynchronous satellites h = 3.59×107 mωequatoraxissatellitehRerh + Re = rs
113 Parking Orbit Note that there is only one such orbit. It is called a parking orbit.ωequatoraxissatellitehRerh + Re = rs
114 Satellites 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.
115 Satellites Near the Earth’s surface Find vrvrsatelliter Reearth
116 Energy and Satellite Motion Find v and the kinetic energy Uk of the satellite.rsatelliteearth Mevm
117 Energy and Satellite Motion The satellite in the orbit possesses both kinetic energy and gravitational energy.rsatelliteearth Mevm
118 Energy and Satellite Motion earth MevmNote that Uk > 0
119 Energy and Satellite Motion Find Up the gravitational potential of the satellite.rsatelliteearth Mevm
120 Energy and Satellite Motion earth MevmNote that Up < 0
121 Energy and Satellite Motion Find U, the total energy of the satellite.rsatelliteearth Mevm
122 Energy and Satellite Motion earth MevmNote that U < 0
123 Energy and Satellite Motion U : Up : Uk = -1 : -2 : 1rsatelliteearth Mevm
124 Falling to the earth The satellite may lose energy due to air resistance. The total energy becomes morenegative and r becomes less.rsatelliteearth Mevm
125 Falling to the earth The satellite follows a spiral path towards earth Mevm
126 Falling to the earthAs r decreases, the kinetic energy of the satelliteincreases and the satellite moves faster.rsatelliteearth Mevm
127 Falling to the earthExample 17 – Loss of energy
128 Weightlessness in spacecraft vvmgThe astronaut isweightless.
129 Weightlessness in spacecraft We fell our weight because there is normal reaction on us.Normal reactiongroundmg
130 Weightlessness in spacecraft If there is not any normal reaction on us, we feel weightless. e.g. free fallingmg
131 Weightlessness in spacecraft vThe gravitational forcemg on the astronaut isthe required centripetalforce. He does not requireany normal reaction toact on him.mg
132 Weightlessness in spacecraft The astronaut isweightless.vmg