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? M1M1 M2M2

… because they miss (that is, they have enough tangential velocity to always miss) M1M1 M2M2 v This is the concept of an orbit. FgFg FgFg

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: –mv 2 /r = GMm/r 2 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

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

Escape Velocity Kinetic Energy (energy due to motion): Ek = ½ m v 2 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 v 2 >= GMm/r Solving for v: v esc = [2GM/r] 1/2 For example, to escape Earth: v esc = 11.2 km/s = 40,320 km/h

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.

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

Figuring out orbital velocities with angular momentum The angular momentum of an object (like a planet) moving in a circle (like an orbit!) is: P = m·v·r r v m m = mass of planet v = velocity of planet r = orbital radius of planet

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

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 P 2 = a 3 Notice that there is nothing stated about the planet’s or Sun’s mass here!

Newton's laws of motion imply Kepler's Laws. In orbit, centrifugal force balances gravitational force F c = mv 2 /r v = 2  r/P v 2 = 4  2 r 2 /P 2 + F g = GmM/r 2 mv 2 /r = GmM/r > 4  2 r 2 /P 2 m/r = GmM/r > r 3 = G/4  2 M P 2 If you express P in years and r in AU, then the term G/4  2 cancels out and you have Kepler Third Law.

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 view of tides high tide low tide Looking down on the Earth

We have two high tides because of the stretching action Moon The Moon exerts a stronger gravitational pull on the near side of the Earth than on the far side of the Earth. This causes the Earth to stretch!

Tides Rotation of Earth Exaggerated view of tides high tide low tide The tides aren’t quite aligned with the Earth- Moon line because it takes time for the water to slosh over.

Earth's rotation slows down by 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 face is pulled harder than the far face.

Earth Moon The near face is pulled harder than the far face.

Spring Tides Occur at every new and full moon

Neap tides Occur at every first- and third-quarter moon

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 radius 2) it would spiral into the black hole 3) nothing would change 4) it would spiral away from the black hole 5) it would be 10 times larger in radius