Chapter 13

Universal Gravitation

Intro

• Prior to 1687-– Vast amounts of data collected on planetary

motion.– Little understanding of the forces involved

• 1687 and after– Newton publishes the Philosophiæ Naturalis

Principia Mathematica– Law of Gravity is applied “Universally”

13.1 Newton’s Law of Universal Gravitation

• Newton’s Law of Universal gravitation was the first time “earthly” and “heavenly” motions were unified.

• The Law states-– Every particle in the Universe attracts every other

particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

13.1

or• G is the universal gravitational constant

G = 6.673 x 10-11 N.m2 / kg2

• The second equation indicates the force is attractive (opposite of the radial vector)

• One example of an “Inverse Square Law”

221

rmmGFg 122

2112 r̂F

rmmG

13.1

• Law of gravity describes an action reaction pair, both objects acted on by equal and opposite forces.

• Also for spherical mass distributions, the force of gravity is the same as if allmass was concentrated at the center.

13.1

• Quick Quizzes pg 392• Example 13.1

13.2 Measuring the Gravitation Constant

• Henry Cavendish (1798)• Two small masses attached by a rod, hung by a thin

wire.• Two large masses, in fixed positions.• Light deflection from mirror is measured• Repeated with different masses

13.3 Free Fall Accleration

• On the surface of the earth the force of gravity is defined as

• Applying universal gravitation we can determine the free fall acceleration anywhere.

mgFg

221

rmmGmg

13.3

• Using earth for example, and the falling object as mass 2.

• Mass 2 cancels (as it should)

22

2E

E

RmMGgm

2E

E

RGMg

13.3

• And for Altitudes above the surface

• The force of gravity decreases with altitude.• As RE + h approaches ∞, mg approaches zero.

Quick Quiz pg 395Examples 13.2, 13.3

2hRGMgE

E

13.4 Kepler’s Laws and the Motion of Planets

• Observation of the moon, planets, and stars has taken place for thousands of years.• Early astronomers considered the universe to be Geocentric, formalized in the Ptolemaic Model (Claudius Ptolemy, 100-170 AD).

13.4

13.4

• In 1543, Nicolaus Copernicus (1473-1543) establishes the 1st comprehensive Heliocentric Model where the planets revolve about the sun. (The Copernican Model)

13.4

13.4

• Tycho Brahe (1546-1601) set out to determine how the heavens were constructed, so developed a system to determine accurate locations for the visible planets and over 700 stars using only a sextant and a compass.

13.4

• Brahe died before he couldfinish his observations and fully develop the Tychonic System, but he passed on thevolumes of data collected to his assistant, Johannes Kepler.

13.4

• Johannes Kepler (1571-1630)using Brahe’s data (specifically for Mars about the Sun)Developed a model for Planetary motion, stated in three simple laws.

13.4

• Kepler’s Laws1. All planets move in elliptical orbits with the sun

at one focus. (Elliptical Orbits)2. The radius vector drawn from the sun to a planet

sweeps out equal areas in equal time intervals.(Equal Area in Equal Time)

3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit. (_______)32 rT

13.4

• Kepler’s 1st Law- Circular orbits are only a special case, elliptical are the general case.

• On ellipses- curved wherethe total distance to two Focal points in is a constant.

(r1 + r2) = constant

13.4

• The long axis through the center and foci is the major axis (2a), and distance to the center is the semimajor axis (a).

• The short axis through thecenter is the minor axis (2b)and the distance to the center is the semiminoraxis (b)

13.4

• The eccentricity of the ellipse is defined as

• Where c is the distance the focal points to the center. As b decreases, c increases.

• For circles, both foci are at the center, therefore, a = b, c = 0, and e = 0.

ace

13.4

• Most of the planets have very low e orbits (nearly circular) with Pluto (not a planet anymore) having the highest eccentricity.epluto = 0.25ehalley = 0.97

13.4

• For planetary orbits, the points closest and furthest from the sun are call the Perihelion and Aphelion respectively, (perigee and apogee for orbits around the earth).

• Perihelion distance = a - c• Aphelion distance = a + c• Bound objects must follow elliptical orbits• Unbound objects either follow parabolic or

hyperbolic paths. 1e

13.4

13.4

• Kepler’s 2nd Law- speaks about the changing speeds due the changing distances from the sun.

• Comes from the conservation of Angular Momentum (L) – There is no net torque on an orbiting object,

because force and lever arm are parallel.

13.4

13.4

• Kepler’s 3rd Law (T2 ~ r3) can be verified with Universal Gravitation for Circular Orbits

• M2 cancels, and we have g = ac

camrmmG 2221

rv

rmG

2

21

13.4

• In for v, we can plug circumference/period

• Simplifying and rearranging

rTr

rmG

2

21 2

2

2

21 4

Tr

rmG

3

1

22 4 r

GmT

13.4

• For a circular orbit, the radius is the semimajor axis, we can apply to ellipses…

• Where K is a constant for the object being orbited.

• Conveniently, when discussing the sun, K = 1 when T is in Earth Years and a is in AU (astronomical unit = avg earth sun distance).

33

1

22 4 Kaa

GmT

13.4

• Quick Quizzes p. 399• Examples 13.4-13.5

13.5 Gravitational Field

• Gravitational Field- a method for explaining how objects can apply forces over a distance.

• Objects with mass create a “gravitational field” in the surrounding space. Object is called a source particle

• The field is equal to the force on a test particle in the field, divided by the mass of the test particle.

13.5

• The Gravitational Field is another way discussing theacceleration due to gravity felt by all objects in the field.

rF

g ˆ2r

GMmg

13.6 Gravitational Potential Energy

• Remember Gravity is a conservative force.• So work done by gravity = opposite change in

potential energy. (ΔU = - W)

f

i

r

rdrrFU )(

13.6

• We can integrate Fg over changes in r

• Becomes

f

i

r

rdrrFU )(

2)(rmGMrF E

f

i

r

rE drr

mGMU 2

1

f

i

r

rE rmGMU

1

13.6

• So we have

ifEif rrmGMUU 11

rmGMU E

13.6

• Why is it negative?– Gravity is an attractive force– An external force must do positive work to

increase the separation between objects. – Ug = 0 at an infinite distance away.– U becomes less negative as r increases.

It is a potential well, or energy deficit.

13.7 Energy Considerations in Planetary and Satellite Motion

• The total energy for a planet in orbit

UKE

rGMmmvE 2

21

13.7

• Using Universal Gravitation we can put Kinetic Energy in gravitational terms.

• Multiplying by r and dividing by 2 givesrmv

rGMm 2

2

221

2mv

rGMm

13.7

• Sub in the previous energy equation.

• Becomes…

• Therefore…

rGMmmvE 2

21

rGMm

rGMmE 2

rGMmE2

13.7

• The energy of a circular orbit is a constant, just like angular momentum.

• It is negative, because the object in orbit is bound,

• The absolute value is the energy need to escape orbit (to have zero energy at r = ∞)

13.7

• Escape speed is the velocity required at a given distance from the gravitational center needed to reach infinity with zero velocity.

• The velocity that gives the system a total energy of zero (unbound).

RGMvesc2

17.3

Quick Quiz p 406Example 13.7, 13.8

13.7

• On black holes-– Massive Stars, collapse under gravitational forces

post Supernova. – The highly dense material (singularity) has an

extremely strong gravitational force. – At (and inside) the critical (Schwarzschild) radius,

the escape speed is equal to (and greater than) the speed of light (c = 3.0 x 108

m/s). – This Spherical boundary is called the Event Horizon.

13.7

13.7

• Binary System of a star and black hole.

13.7

• While nothing can escape from inside the event horizon, matter that is accreting experiences friction, increasing temperature, causing a release of high enery radiation (up to x-ray).