A block -

Apologies for late practice problems.

1.  Calculate the acceleration due to gravity on Mars.  It's a is roughly 1.5 AU. It's mass is about 1/10 earth's and its radius is roughly half of earth.  Also find how long it takes to orbit the sun.

2.  Revisit the rocket problem.  Mass is .100 kg and the engine provides 12-N of thrust for 0.25 seconds. Find acceleration, max v and max height

Answers to recent HW

1. a.  11.86 years

b.  24.77 m/s/s

c.  approx 13,000 m/s

d.  4.2 x 10^23 N

e and f.  5.151 AU and 5.249 AU

2.  Assume that the net force is 75 N.  I'll let you know if this is the case otherwise on the test.

a.  a = 1875 m/s/s, so Vf = 188 m/s

b.  1794 m

c.  19 sec

Fun!

https://www.fourmilab.ch/solar/

https://www.fourmilab.ch/earthview/

http://www.theplanetstoday.com/

http://www.solarsystemscope.com/

https://in-the-sky.org/solarsystem.php


And for some size and scale perspective:


http://htwins.net/scale2/
you can choose your language

http://scaleofuniverse.com/
same as above

http://xkcd.com/482/

http://xkcd.com/1331/
This is just cool.

http://workshop.chromeexperiments.com/stars/

http://www.youtube.com/watch?v=8yCzzTkDSMo

Jack Horkheimer (R.I.P.), for your interest.  

Test practice

Some pre-test problems.  Test will be next Thursday (E block) and Friday (A block).

These will NOT be collected.

1.  Consider the planet Jupiter.  The semi-major axis of orbit is 5.2 AU and its mass is 318 times that of Earth.  The radius of Jupiter is about 11.2 times that of Earth.  Use that information to find the following:

a.  time for Jupiter to orbit the Sun once

b.  surface gravity on Jupiter * (A block hasn't covered this yet - see notes)

c.  approximate speed of Jupiter in orbit around the Sun (in m/s).  You can approximate the orbit as circular, or use the fact that the semi-minor axis is 5.1973 AU.
d.  the force of gravity (on average) between Jupiter and the Sun (m = 2 x 10^30 kg)

The following will not be on the upcoming test/quest.  Now, if you know that the eccentricity of Jupiter’s orbit is 0.049, you can find other things.  By the way, the easiest way to relate eccentricity (unlike the definition I gave you in class) is:

ecc = f/a

In this definition, f is the distance between the center of the ellipse and either focal point.  The semi-major axis is still a.  Find the following:

e.  the closest that Jupiter gets to the Sun (in AU)

f.  the furthest that Jupiter is from the Sun (in AU)

2.  Newton review

A spring-loaded ball (mass = 0.04 kg) is shot up into the air.  The ball and spring are in contact for 0.1 seconds  and during this time, the spring exerts an average force of 75 N on the ball.  Find:

a.  the maximum speed of the ball (“muzzle velocity”)

b.  how high the ball can travel

c.  how long it takes the ball to reach max height

The upcoming test will likely have 3 questions:  two like the above, and a third where you choose 1-2 things to write about (out of 3 or 4).  Other topics worth reviewing are:  the rocket lab (and all the discussion on drag that we had), Kepler's 3 laws, and Newton's 3 laws (the theory).

FYI:

https://www.youtube.com/watch?v=C4pcg7bXgmU

Newton and the law of universal gravitation

Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:

All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:

F = G m1 m2 / d^2

or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.

Big G = 6.67 x 10^-11, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.

This is an INVERSE SQUARE law, meaning that:

- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.

Weight

Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):

g = G m(planet) / d^2

Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).

Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.

If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.

The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 2.5 m/s/s.

Kepler's laws of planetary motion - revisited.

http://astro.unl.edu/naap/pos/animations/kepler.swf



Note that these laws apply equally well to all orbiting bodies (moons, satellites, comets, etc.)

1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.

2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.

3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the amount of time (P, for period) to go around the Sun in a very peculiar fashion:

a^3 = P^2

That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:

- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles

- the unit of time is the (Earth) year

The image below calls period P:

Example problem:  Consider an asteroid with a semi-major axis of orbit of 4 AU. We can quickly calculate that its period (P) of orbit is 8 years (since 4 cubed equals 8 squared).

Likewise for Pluto: a = 40 AU. P works out to be around 250 years.

Note that for the equation to be an equality, the units MUST be AU and Earth years.


Cool, for fun:
http://galileo.phys.virginia.edu/classes/109N/more_stuff/flashlets/kepler6.htm

A neat video - I rather liked this

http://www.vox.com/science-and-health/2016/11/29/13769152/physics-subdiscipline-video

Kepler's laws "lab" - due next week (Tuesday A, Wednesday E)

Laboratory – Kepler’s Laws and Celestial Motion


In this lab, you will learn about the three laws determined by Johannes
Kepler in the early 17th Century.  Kepler was employed by Tycho Brahe,
a Danish nobleman often considered the last of the great pre-telescope
observers.  Brahe amassed thousands of pieces of data related to star
and planet positions.  After Brahe’s death in 1601, Kepler began a
long arduous process of piecing this data into universal laws – or at
least what he thought were universal laws (Einstein shed greater light
on these in the 20th century).  

Answer the questions below based on your observations and reading.

Questions – use diagrams where helpful.

1.  What is the shape of a planetary orbit?
2.  Where is the sun in this orbit?
3.  Where or what is aphelion?
4.  Where or what is perihelion?
5.  What is the relationship between a circle and an ellipse?  That is,
one is a special case of the other; explain.
6.  What is eccentricity?
7.  What is Kepler’s 1st Law?
8.  Kepler’s 2nd Law refers to an “area” swept out by a planet.  What does
this mean?
9.  What does Kepler’s 2nd Law predict for the speeds of planets as they
are nearer to and farther from the sun?
10.  When exactly is the Earth closest to the Sun?
11.  How is this related to the seasons?  That is, is this why we have
seasons?  If not, what causes seasons?
12.  What is Kepler’s 2nd Law?
13.  What is an Astronomical Unit (AU)?
14.  What is the period of Earth’s orbit?
15.  What is a semi-major axis of an elliptical orbit?  What does “semi”
mean in mathematics?
16.  What is Kepler’s 3rd Law?

17.  Give some general concluding remarks regarding Kepler’s laws.