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# F21cos-F21sin(13341010)cos(60)-(13341010)sin(6...

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F21cos-F21sin(13341010)cos(60)-(13341010)sin(6...

Notes for Ch&Mat.E CHAPTER 13

Gravitation

NEWTON’S LAW OF GRAVITATION

F ? m F ? m 121212

F ? mm 12

1 Also, F ? 2r

Newton’s law of universal gravitation:

;Gmm12ˆ Fr12212r

1122where the Gravitational constant G6.67×10 N)m/kg.

F means the force acted on m by m 1212

The negative sign indicates attraction force between the two objects. Thus in

a general form no subscript is used.

Gmm12ˆ r2r

Spherical Mass Distribution

Although in general, Newtons law of gravitation applies only to point particles, there is an exception. According to the “point mass theorem” a spherically symmetric mass distribution (such as a uniform sphere or shell) attracts a point particle as if all its mass were concentrated as the center.

Thus, we may take r to be the distance from the point particle to the center.

1This result is a consequence of the nature of the force law. Another 2r

consequence of the inverse square nature of the force law is that a point particle inside a uniform spherical shell experiences no net force.

-1-

CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E Principle of superposition

If there are several particles present, the net force on a given particle, say m, is 1the vector sum of the individual forces due to each of the other masses.

;;;;

？……？ FFFF12131N1

;;;;

FFFF1213141

The gravitational force between m and 1

m is not affected by the presence of a 2

body placed between them.

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CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E

Example 13-1

m2 kg 2

;

? F2

L2m

m4 kg m3 kg 13

11Gmm6.6710(4)(2)?1012F1.334×10 (N) 2122r2

;10ˆˆˆFcosFsin(1.334×10)cos(60?)iiFj212121

10ˆ(1.334×10)sin(60?) j

;1110ˆˆ6.67×101.155×10 Fij21

11Gmm6.6710(2)(3)?1023F1.0005×10 (N) 2322r2

;1010ˆˆˆˆFcosFsin(1.0005×10)cos(60?)(1.0005×10)sin(60?) Fijij232323

;1111 ˆˆ5.0025×108.665×10Fij23

;;;

FFF22123

11101111ˆˆˆˆ(6.67×101.155×10)(5.0025×108.665×10) ijij

;1110ˆˆ(1.668×10)(2.021×10) Fij2

-3-

CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E GRAVITATIONAL AND INERTIAL MASS

Newtons Second law:

Fma ? mm inertial mass I

Newtons Law of gravitation:

GmM ? mm Gravitational mass FG2r

Gravitational force can be expressed as

mGFmg ? Fma ? ag GImI

Basically, mm GI

Attempt to distinguish m and m made by Newton GI

For a simple pendulum of length L,

mLIT2

mgG

When different bobs of materials were used,

LT2

g

i.e, m m GI

Principle of equivalence

No experiment can distinguish the effect of a gravitational force from that of an

inertial force in an accelerated frame. When a ball is released, it accelerates toward the

floor of the cabin. The astronaut can not tell whether the acceleration relative to the rocket was

leftcaused by the force of gravity or,

righta consequence of the acceleration of the

rocket in gravity-free space.

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CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E THE GRAVITATIONAL FIELD STRENGTH

Consider a stationary particle of mass M exerts a force on a particle of mass m,

which moves from a position to another, as shown in the figure.

The extent of mass m affected by the mass M is called force field.

Note the force between the two particles is

GMmˆ r2r

; and m g

GM;ˆ Magnitude of force field /m = rg2r

ˆ1. Origin of is at M. r

;2. The quantity , measured in N/kg, is called gravitational field g

strength, the gravitational force per unit mass.

3. The gravitational force on any particle of mass m, i.e., weight, is

given by

;m gW

-5-

CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E

Acceleration due to Gravity

Consider a particle of m at some latitude on the earth,

;Gravitational strength: g

;The gravitational force m is directed to the g

center and serves two function:

1. It causes the particle to fall with acceleration

;. g

;2. It produces the centripetal acceleration . ac

;;From m Fa

;;;mm() gagc

;;Vectors and are parallel only at the poles and at the equator. gg

;At the pole, 0, ggapp c

2624?(6.4?10)4R22At the equator, a0.034 (m/s)3.4 (cm/s) c22T(86400)

gga gg3.4 eecee

gggg3.4 pepe

Measurement shows gg5.2 pe

gg1.8 ? earth is not a perfect sphere. pe

GMTheoretically, g, R6357 km and R6378 km pe2R

gg6.41.8 pe

The difference is due to:

(1) Equation used here is only applicable to uniform spherical distribution.

(2) The density of the earth in radial direction is a non-uniform distribution.

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CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E KEPLER’S LAWS OF PLANETARY MOTION

Law 1.

The planets move around the sun in elliptical orbits with the sun at one

focus.

major axis: AP2a

P: Perihelion A: aphelion

Law 2.

The line joining the sun to a planet sweeps out equal areas in equal times.

Area SABArea SCD

Law 3

The square of the period of a planet is proportional to the cube of its mean

distance from the sun.

mean distancesemimajor axisa

23Ta

: constant for a fixed star system.

-7-

CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E

Energy in an Elliptical Orbit

At perihelion and aphelion, The conservation of angular momentum

mrvmrv ? rvrvAAppAAPP

The conservation of mechanical energy

GMmGMm1122mm vvpA22rrAP

1122 ? 2GM() vvpArrpA

2rr112AA (1)2GM() vvvPAA2rrrrpPAP

22rrrrpA2PA()2GM() vA2rrrAPP

(r;r)(rr)(rr)2PAPAPA2GM vArrPA

r2P()2GM() vrrAPArA

Again, 2a rrAP

GMr2P vAarA

GMr2Av Similarly, parP

-8-

CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E

Bound and Unbound Trajectories

A cannonball is fired from the peak of a very tall tower with speed v

Escape velocity: v esc

GMm12 Total energy Emv 22r

Conservative of mechanical energy;EE (EE) ifat the surfacer?~

and E0 f

GMm2GM12m0 ? v vescesc2r2r

Velocity at orbital of radius r: v c

2GMGMvAt orbital of radius r mmg and g ? v crrr

Velocity Phenomena Energy

0 Very small The projectile strikes the earth.

The orbit is elliptical, with the peak as the apogee vv0 c (遠地點).

vv0 The orbit is circular. c

vvv0 The orbit is elliptical, with the peak as the perigee. cesc

vv0 The object is unbounded, and the path is a parabola. esc

vv0 The object is unbound, and the path is a hyperbola. esc

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CHAPTER 13 GRAVITATION

Notes for Ch&Mat.E

CONTINUOUS DISTRIBUTIONS OF MASS

The gravitational field strength g due to an extended object

Gdm dg 2r

GTotal gravitational field g dm?2r

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CHAPTER 13 GRAVITATION

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