Chapter 13

Gravitation

Newton’s law of gravitation

• Any two (or more) massive bodies attract each other

• Gravitational force (Newton's law of gravitation)

• Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant

rr

mmGF ˆ

221

Gravitation and the superposition principle

• For a group of interacting particles, the net gravitational force on one of the particles is

• For a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral

n

iinet FF

21,1

body

body FdF

,1

Chapter 13Problem 5

Three uniform spheres of mass 2.00 kg, 4.00 kg and 6.00 kg are placed at the corners of a right triangle. Calculate the resultant gravitational force on the 4.00-kg object, assuming the spheres are isolated from the rest of the Universe.

Shell theorem

• For a particle interacting with a uniform spherical shell of matter

• Result of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center

shell

shell FdF

,1

Gravity force near the surface of Earth

• Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface

• Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth

g = 9.8 m/s2

• This formula is derived for stationary Earth of ideal spherical shape and uniform density

jR

mmGF

Earth

EarthEarth

ˆ2

1,1

jmgjm

R

Gm

Earth

Earth ˆˆ112

Gravity force near the surface of Earth

In reality g is not a constant because:

Earth is rotating, Earth is approximately an ellipsoid with a non-uniform density

Gravitational field

• A gravitational field exists at every point in space

• When a particle is placed at a point where there is gravitational field, the particle experiences a force

• The field exerts a force on the particle

• The gravitational field is defined as:

• The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle

m

Fg g

Gravitational field

• The presence of the test particle is not necessary for the field to exist

• The source particle creates the field

• The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field

• The magnitude is that of the freefall acceleration at that location

gR

Gm

Earth

Earth 2

Gravitational potential energy

• Gravitation is a conservative force (work done by it is path-independent)

• For conservative forces (Ch. 8):

f

i

r

r

rdFU

f

i

r

r

Earth drr

mGm2

1

fiEarth rr

mGm11

1

Gravitational potential energy

• To remove a particle from initial position to infinity

• Assuming U∞ = 0

fiEarthif rr

mGmUUU11

1

i

Earth

iEarthi r

mGm

rmGmUU 11

11

i

Earthii r

mGmrU 1)(

r

mGmrU 21)(

Gravitational potential energy

r

mGmrU 21)(

Escape speed

• Accounting for the shape of Earth, projectile motion (Ch. 4) has to be modified:

gRvgR

vac

2

Escape speed

• Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)

002

12

1 planet

planet

R

mGmvm

ffii UKUK

planet

planetescape R

Gmv

2

Escape speed

• If for some astronomical object

• Nothing (even light) can escape from the surface of this object – a black hole

csmR

Gmv

object

objectescape /103

2 8

Chapter 13Problem 30

(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system, if it starts at the Earth’s orbit? (b) Voyager 1 achieved a maximum speed of 125 000 km/h on its way to photographJupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

Kepler’s laws

Three Kepler’s laws• 1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus• 2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals• 3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit

Johannes Kepler(1571-1630)

Tycho Brahe/Tyge Ottesen

Brahe de Knudstrup(1546-1601)

First Kepler’s law

• Elliptical orbits of planets are described by a

semimajor axis a and an eccentricity e

• For most planets, the eccentricities are very small

(Earth's e is 0.00167)

Second Kepler’s law

• For a star-planet system, the total angular momentum is constant (no external torques)

• For the elementary area swept by vector

rpL

))((2

1 rdrdA dt

dr

dt

dA 2

2

m

L

dt

dA

2

))(( mvr ))(( rmr 2mr const

2

2r

r

Third Kepler’s law

• For a circular orbit and the Newton’s Second law

• From the definition of a period

• For elliptic orbits

))(( 22

rmr

GMmmaF

2

22 42

TT

32

r

GM

32

2 4r

GMT

32

2 4a

GMT

Satellites

• For a circular orbit and the Newton’s Second law

• Kinetic energy of a satellite

• Total mechanical energy of a satellite

r

vm

r

GMm 2

2)(maF

2

U

2

2mvK

UKE r

GMm

r

GMm

2 r

GMm

2 K

r

GMm

2

Satellites

• For an elliptic orbit it can be shown

• Orbits with different e but the same a have the same total mechanical energy

a

GMmE

2

Chapter 13Problem 26

At the Earth’s surface a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.

Questions?

Answers to the even-numbered problems

Chapter 13

Problem 22.67 × 10−7 m/s2

Answers to the even-numbered problems

Chapter 13

Problem 43.00 kg and 2.00 kg

Answers to the even-numbered problems

Chapter 13

Problem 10(a) 7.61 cm/s2

(b) 363 s(c) 3.08 km(d) 28.9 m/s at 72.9° below the horizontal

Answers to the even-numbered problems

Chapter 13

Problem 24(a) −4.77 × 109 J(b) 569 N down(c) 569 N up