Lecture 7

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

• Assigned reading: Down to end of Chapter 5.2 (no Relativity)

Announcements

• Late-enrolled students who still have to pass HW1 should start to wrap up their work. – I will post solutions to HW1 as soon as all

Homework papers have been handed over.

Gravity

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

but orbit around the Sun instead?

The Universal Law of Gravity

F = - G Mm

r2

(G is the Universal constant of gravity.

Mother Nature has set its value, and we

cannot change it)

m

M

r

The strength of the force only depends on M, m and r

mh

D

The Universal Law of Gravity

F = - G Mm

r2

(G is the Universal constant of gravity.

Mother Nature has set its value, and we

cannot change it)

m

M

r

Now suppose that Earth shrinks in size, but keeps its mass (gravitational collapse).

Would the force of gravity on the Moon change?

Would the force of gravity on the human change?

mh

D’

The Universal Law of Gravity

F = - G Mm

r2

(G is the Universal constant of gravity.

Mother Nature has set its value, and we

cannot change it)

m

M

r

Now suppose that Earth shrinks in size, but keeps its mass (gravitational collapse).

Would the force of gravity on the Moon change?No, same M and m, and same r

Would the force of gravity on the humans change?Yes, same M, same mh, but different D

mh

D’

… so what is an orbit?

Suppose you dig a hole through the center of Earth and pump all the air off.

And then toss an object into the hole.

Can you visualize the motion of that stone?

… so what is an orbit?

Suppose you dig a hole through the center of Earth and pump all the air off.

And then toss an object into the hole.

Can you visualize the motion of that stone?

The object will move up and down, periodically, for ever.

Its speed will be highest when it transits at the center of Earth…

… and the slowest in proximity of the surface, when it stops and reverts its motion

THAT IS AN ORBIT!

(with velocity always having the same direction and alternating sense)

V=8km/s

Now, let’s make an orbit whose velocity does change direction

If I do not give the object enough speed, its trajectory will eventually intersect the ground

At the right speed, namely the ORBITAL VELOCITY, the vertical downward motion and the horizontal outward motion combine to produce the circular orbit

V=8km/s

Suppose you shrink Earth keeping the same mass.

Now the trajectory of even the slowest moving object does bot intersect ground.

That is an elliptic orbit!

The circular orbit is only a special case of an orbit where the speed is always the same

In no circular orbits the speed changes along the orbit

At the right speed, namely the ORBITAL VELOCITY, the vertical downward motion and the horizontal outward motion combine to produce the circular orbit

V=8km/s

Important: an object in orbit is free falling. Thus, it feels no gravity, since it is falling onto the source of gravity

At the right speed, namely the ORBITAL VELOCITY, the vertical downward motion and the horizontal outward motion combine to produce the orbit

Geosynchronous Orbits

… 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!

Orbital Velocity

• Another way to understand orbits: 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

Quiz Astronauts inside the space shuttle

float around because ____they are falling in the same way as the space shuttle.

If you are in a free-falling elevator, you are massless. (true or false)false

You are a shuttle astronaut returning after attempting to fix the ISS with a hammer. As you are jetting back to your shuttle, your lifeline breaks, your jets run out of fuel, your radio goes dead, and you miss the shuttle. To get back safely, you should:

• use a swimming motion with your arms and legs

• throw the hammer at the shuttle to get someone’s attention

• throw the hammer away from the shuttle• make a hammering motion in the direction

of the shuttle• make a hammering motion away from the

shuttle

V=8km/s

Escape velocity: how can I free myself from the pull of gravity of a given body?

Can I accelerate myself at such a speed that I will start spiraling out and eventually abandon the planet instead of keep orbiting it? How fast is such speed?

Gravity: depends on mass and on distanceSo, what is escape velocity?

F = - G Mm

r2

You can change the force of gravity either by varying the mass(es), or by varying the distance, or both

m

M

r

Now suppose that Earth shrinks in size, but keeps its mass (gravitational collapse).

Would the force of gravity on the Moon change?No, same M and m, and same r

Would the force of gravity on the humans change?Yes, same M, same mh, but different D

mh

D’

Such an escape velocity will have to depend how far away I am from the center of gravity of the body that generate the gravity field.

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

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 difference in pull causes the Earth to stretch!

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 as a result.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

Friction wastes energy, and this energy comes at the expense of Earth’s rotational

energy

Acceleration of the Moon’s Orbital Motion

Earth’s tidal bulges are slightly tilted in the direction

of Earth’s rotation.

Gravitational force pulls the moon slightly

forward along its orbit.

Spring and Neap Tides

The Sun is also producing tidal effects, about

half as strong as the Moon.

• Near Full and New Moon,

those two effects add up to cause

spring tides.

• Near first and third quarter, the two effects work at a right angle, causing neap

tides.

Spring tides

Neap tides

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)

• The friction of changing Moon’s tidal bulge dissipated rotational energy, and put Moon’s rotation to such a value that the tidal bulge does not move any longer: the synchronous rotation with orbit

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

.

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

Angular Momentum

• Depends on the geometry, the mass, and the rotational velocity of an object.

• Angular momentum is conserved.– A spinning wheel wants to keep spinning.– A stationary wheel wants to keep still.

• Angular momentum is also a vector quantity – this means that the direction of the axis of rotation is significant and resistant to change. Its INTENSITY is:

•P = m·v·r• m is the mass, v the speed, and r the distance

to the center of rotation

Everyday Examples of the Conservation of Angular

Momentum

• Riding a bike

• Spinning a basketball on your finger

• A spinning ice skater

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 Revisited

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

Figuring out orbital velocities with angular momentum

• The angular momentum of an object (like a planet) moving in a circle (like an orbit!) is:

L = m·v·rAngular Momentum is always conserved

r

v

mm = mass of planetv = velocity of planetr = orbital radius of planet

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!

Newton's laws of motion imply Kepler's Laws.In orbit, centrifugal force balances gravitational force

Fc = mv2/r v = 2r/P v2 = 42r2/P2

+Fg = GmM/r2 mv2/r = GmM/r2

----> 42r2/P2 m/r = GmM/r2

----> r3= G/42 M P2

If you express P in years and r in AU, then the term G/42 cancels out and you have Kepler Third Law.