Lecture 8

• More on gravity and its consequences– Orbits– Tides and tidal forces– The Three Kepler laws revisited

• Assigned reading: Chapter 5.2

Announcements

• I will be away on Friday 22. Dr. Calzetti will replace me.

• I am leaving town tomorrow after class:– Cannot see students on Wednesday

afternoon, Thursday and Friday

• Homework #3 is due in class on Friday 22

• Quiz #2 is today• Students who still have to take Quiz #1,

pleasee come and talk to me• Still lots of student with no password

Assigned Reading

• Chapter 5 up, but to not including 5.3

Gravity

• What keeps us on the rotating Earth?• Why don’t planets move in straight lines,

but orbit around the Sun instead?

… so why don’t planets just fall into the sun?

M1M2

… because they miss (that is, they have enough tangential

velocity to always miss)

M1M2

v

This is the concept of an orbit.

Fg Fg

Why doesn't the earth fall to the sun?

• It has a velocity and it has inertia!

• Force of gravity causes change in the direction of velocity --- acceleration.

• The earth is falling towards the sun all the time!

V=8km/s

The best way to get comfortable with orbits is to do the tutorial at the textbook website (it’s also a good study aid for the exam).

Orbital Velocity

• In orbit, force of gravity and centrifugal force balance each other:

–mv2/r = GMm/r2

• Solving for v gives:

•v = [GM/r]1/2

• For example, in the case of the Moon:

•v = 1.02 km/s ~ 3,600 km/h

Why don't they fall?

They are circuling Earth at a speed of 8 km/s!

Mass and Weight• Mass is a measure of how much

material is in an object.• Weight is a measure of the gravitational

force exerted on that material.• Thus, mass is constant for an object,

but weight depends on the location of the object.

• Your mass is the same on the moon, but your weight on the surface of the moon is smaller

Escape Velocity

• Kinetic Energy (energy due to motion):• Ek = ½ m v2

• Potential Energy (energy due to position):

• Eg = GMm/r• To escape, Kinetic Energy has to be

larger (or at least equal) than Potential Energy:

• ½ m v2 >= GMm/r• Solving for v:

• vesc = [2GM/r]1/2 • For example, to escape Earth:

• vesc = 11.2 km/s = 40,320 km/h

1 The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

2 Planets move proportionally faster in their orbits when they are nearer the Sun.

3 More distant planets take proportionally longer to orbit the Sun

Kepler’s Laws of Planetary Motion

Kepler’s Three Laws of Orbits

1. The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

Kepler’s Three Laws of Orbits

2. As a planet moves around it’s orbit, it sweeps out equal areas in equal times.

1 month

1 m

onth

Kepler’s Three Laws of Orbits

3. A planet’s Period (the time it takes to complete one orbit) is related to its average distance to the sun.

(orbital period in years)2 = (average distance in AU)3

P2 = a3

Notice that there is nothing stated about theplanet’s or Sun’s mass here!

Tides

• Tides occur because of the gravitational pull of the Moon on the Earth.

• The Moon pulls more strongly the closer side of Earth than the one further away.

• It literally stretches Earth• Water (and air) get stretched much

more easily than rock.• This, in essence, is what makes tides• Note that the Sun does the same, too

Let’s build this one step at a time

Moon

Exaggerated viewof tides

high tidehigh tide

low tide

low tide

Looking downon the Earth

We have two high tides because of the stretching action

Moon

The Moon exerts a stronger gravitational pullon the near side of the Earth than on the far side of the Earth. This causes the Earth tostretch!

Tides

Rotation of EarthExaggerated viewof tides

high tide

high tide

low tide

low tide

The tides aren’t quite aligned with the Earth-Moon line because it takes time for the waterto slosh over.

Earth's rotation slows down by 0.0023 s/100 years.Only 900 million years ago, Earth' day was 18 hrs long.The moon's orbit is growing larger by about 4 cm/yr.

Friction drags the tidal bulges eastward out of the direct earth-moon line

Discussion Question

• Why does the Moon always show the same face to the Earth? (hint: think of the tidal pull of the Earth on the Moon)

Earth

Moon

The near faceis pulled harderthan the far face.

Earth

Moon

The n

ear

face

is p

ulle

d h

ard

er

than t

he f

ar

face

.

Spring Tides

Occur at every new and full moon

Neap tides

Occur at every first- and third-quarter moon

221

d

MMGFg Survey Question

If our Sun mysteriously turned into a black hole of the same mass but 10 times smaller diameter, what would change about the Earth’s orbit?

1) it would be 10 times smaller in radius2) it would spiral into the black hole3) nothing would change4) it would spiral away from the black

hole5) it would be 10 times larger in radius