Astronomy 114 - University of Massachusetts...

Astronomy 114

Lecture 7: Newton’s Law of Gravity, Tidal Force

Martin D. Weinberg

[email protected]

UMass/Astronomy Department

A114: Lecture 7—12 Feb 2007 Read: Ch. 4,5 Astronomy 114—1/17

Announcements

Problem Set #1 solutions posted

Problem Set #2 posted last Friday, due this Friday

Today is Add/Drop day!


Announcements

Problem Set #1 solutions posted

Problem Set #2 posted last Friday, due this Friday

Today is Add/Drop day!

Today:

Newton’s Law of Gravity

Tidal Force

Wednesday: LIGHT, Chap. 5


Newton’s Law of Gravity (1/6)

Newton described the force of gravity mathematically

Explains Kepler’s laws

Every body in the Universe attracts every other bodywith a force proportional to the product of theirmasses and inversely proportional to the square ofthe distance between them:

Fgravity =Gm1m2

r2

G is the same here as it is in a distant galaxy. It is a physical

constant of the Universe.



Acceleration

Velocity

Planet

Resulting

trajectory

Combined with Laws ofMotion: explains orbits

Kepler’s Three Laws



Newton discovered that orbiting bodies may followany one of a family of curves called conic sections

The ellipse is only one possibility



Bound, finite orbits: circle, ellipse

Unbound, infinite orbits: parabola, hyperbola



Planets obey the same laws as objects on Earth




Kepler’s laws: explained by force of gravity

Planets orbit around the center of mass of theSolar System

Since most of the mass is the Sun, Sun is veryclose to center of mass

Third law depends on the sum of the two masses:

P2 =

[

4π2

G(m1 + m2)

]

a3




Kepler’s laws: explained by force of gravity

Planets orbit around the center of mass of theSolar System

Since most of the mass is the Sun, Sun is veryclose to center of mass

Third law depends on the sum of the two masses:

P2 =

[

4π2

G(m1 + m2)

]

a3

New types of unbound orbits—hyperbolas andparabolas—in addition to ellipses


Food for thought . . .

Inertial and gravitational mass . . . Are they the same?




Action at a distance?




Action at a distance?

Why does an astronaut in orbit feel weightlessness?


More consequences: tidal forces (1/5)

Does every point on the Earth feel the samegravitation force from the Sun (or Moon)?




Differential force in near side and far side





Stretches body along line joining body to Sun





Stretches body along line joining body to Sun

Compresses body in 2 perpendicular directions

Results in “football” shape (prolate spheroid)



Both Sun and Moon influence tides on Earth

Which is bigger?



Both Sun and Moon influence tides on Earth

Which is bigger?

About the same (but not quite):

Sun is more massive but farther away

Moon is less massive but closer

Moon causes 70%, Sun causes 30%



Orientation of Moon and Sun relative to Sun–Earthdirection determines the strength of the tide




Tidal force is reinforced when Sun-Moon-Earth arealong same line (Spring Tide)




Tidal force is reinforced when Sun-Moon-Earth arealong same line (Spring Tide)

Tidal force is diminished if Sun-Earth force andMoon-Earth force are perpendicular (Neap Tide)



At what time of day to neap tides occur?

a. Near sunrise

b. Near sunset

c. Near noon

d. Near midnight

e. More than one of theabove



Moon is receding fromEarth

Rotating bulge onEarth acceleratesMoon in orbit Earth

Moon



Moon is receding fromEarth

Rotating bulge onEarth acceleratesMoon in orbit

Why does moon keepsame side towardEarth?

Attraction ofMoon’s tidal bulgeby Earth locks withMoon’s revolution

Earth

Moon


Calculations with Kepler’s 3rd Law [1]

Newton’s generalization:

P2 =

[

4π2

G(m1 + m2)

]

a3




P2 =

[

4π2

G(m1 + m2)

]

a3

Set m1 = Msun, m2 = Mearth then P = 1year and a = 1AU.Since Msun ≫ Mearth, m1 + m2 ≈ Msun.

P2

(year)2=

(

Msun

m1 + m2

)

a3

(year)3

or




P2 =

[

4π2

G(m1 + m2)

]

a3

Set m1 = Msun, m2 = Mearth then P = 1year and a = 1AU.Since Msun ≫ Mearth, m1 + m2 ≈ Msun.

P2

(year)2=

(

Msun

m1 + m2

)

a3

(year)3

or

P (year)2 =(

Msun

m1 + m2

)

a(AU)3



May solve for the period P of the planet, given a:

P (year) =

√

Msun

m1 + m2

a(AU)3/2




P (year) =

√

Msun

m1 + m2

a(AU)3/2

Example: radius of Mars’ orbit given the period

a(AU) = P2/3(year) = (1.88)2/3 = 1.52AU




P (year) =

√

Msun

m1 + m2

a(AU)3/2


a(AU) = P2/3(year) = (1.88)2/3 = 1.52AU

Example: Quadruple mass of Sun, keep radius thesame. How does period of Earth orbit change?




P (year) =

√

Msun

m1 + m2

a(AU)3/2


a(AU) = P2/3(year) = (1.88)2/3 = 1.52AU

Example: Quadruple mass of Sun, keep radius thesame. How does period of Earth orbit change?

P (year) =

√

1

4a(AU)3/2 =

1

2year


Newton’s 3rd law: how do rockets work?

1. People see: huge flame and hot gas pouring out theback

2. Assume: rocket pushing against the ground or the air


Newton’s 3rd law: how do rockets work?

1. People see: huge flame and hot gas pouring out theback

2. Assume: rocket pushing against the ground or the air

Wrong!

Controlled explosion

Material is ejected from nozzle

By 3rd law, rocket is accelerated in opposite direction

Rocket would work regardless of what is short outthe back!


Definitions (1/2)

scalar: a simple numerical value

vector: quantity described by both numerical valueand a direction

velocity: the speed and direction of an object [vector]

acceleration: a rate of change of velocity [vector]

inertia: property of mass by which it resists change inits motion

momentum: a measure of an object’s inertia, equal toproduct of object’s mass and velocity [vector]


Definitions (2/2)

force: something which changes the momentum of anobject, equal to rate of change of momentum [vector]

mass: a measure of the total amount of material (e.g.atoms) in an object [scalar]

weight: “downward” force on an object due to gravity[scalar]


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