Program assessment - we will fill this out during our final class

https://www.surveymonkey.com/s/GMQDTR3

Newton and Gravitation

Newton

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 25 m/s/s.

Kepler's Laws

Kepler's Laws

Kepler's laws of planetary motion

http://astro.unl.edu/naap/ssm/animations/ptolemaic.swf

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


Johannes Kepler, 1571-1630

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 to go around the Sun in a very peculiar fashion:

a^3 = T^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 (rather than T), but the meaning is the same:

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

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


The applets I referenced::

http://www.physics.sjsu.edu/tomley/kepler.html

http://www.physics.sjsu.edu/tomley/Kepler12.html
for Kepler's laws, primarily the 2nd law

http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html
for our discussion on geocentrism and how retrograde motion appears within this conceptual framework

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

Homework - due NEXT class!

Also, make sure that your lab group has submitted the complete informal Mechanics lab.

Phinal Physics Phun-signment!  (Get it?)

1.  You have a model rocket with a mass of 0.050 kg.  It contains an engine that delivers a thrust force of 10-N for 0.8 seconds.  Answer the following questions based on this data.

a.  What is the weight of this rocket?

b.  What is the NET force acting on this rocket during the thrust period?  (Keep in mind that the engine pushes up, while the weight acts down.)

c.  What will be the acceleration of this rocket?

d.  If we assume that the rocket starts at rest, what will be the final velocity of the rocket before the engine stops burning?

e.  How high will the rocket go during this time period?

f.  Will the rocket continue to rise after the engine burns out?  If not, how far additionally will it rise?


2.  Consider standing on a scale in an elevator.  The scale has a spring inside of it, causing an "effective weight" measurement to be given.  A 130-lb (while at rest) woman stands on the scale in the elevator.  Answer the questions below thusly:  equal to 130 lb, less than 130 lb, greater than 130 lb, 0 lb.

a.  If the elevator is moving upward at a constant velocity, what will the scale read?

b.  If the elevator is moving downward at a constant velocity, what will the scale read?

c.  If the elevator is moving upward at a constant acceleration, what will the scale read?

d.  If the elevator is moving downward at a constant acceleration, what will the scale read?

e.  If the elevator were freely falling (uh oh....), what will the scale read?

3.  Pick one idea or topic that has captured your imagination during the motion/mechanics unit.  Describe or discuss it briefly.

Newton!

Newton and his laws of motion.

Newton, Philosophiae naturalis principia mathematica (1687) Translated by Andrew Motte (1729)

Lex. I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.


Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.


Lex. II. Mutationem motus proportionalem esse vi motrici impressae, & fieri secundum lineam rectam qua vis illa imprimitur.


The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.


If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.


Lex. III. Actioni contrariam semper & aequalem esse reactionent: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi.


To every action there is always opposed an equal reaction; or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.


Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I  may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.

>

And now, in more contemporary language:

1.  Newton's First Law (inertia)

An object will keep doing what it is doing, unless there is reason for it to do otherwise.

The means, it will stay at rest OR it will keep moving at a constant velocity, unless acted on by an unbalanced force.

2.  Newton's Second Law

An unbalanced force (F) causes an object to accelerate (a).

That means, if you apply a force to an object, and that force is unbalanced (greater than any resisting force), the object will accelerate.

Symbolically:

F = m a

That's a linear relationship.

Greater F means greater a.  However, if the force is constant, but the mass in increased, the resulting acceleration will be less:

a = F / m

That's an inverse relationship.

We have a NEW unit for force.  Since force = mass x acceleration, the units are:

kg m / s^2

which we define as a newton (N).  It's about 0.22 lb.

There is a special type of force that is important to mention now - the force due purely to gravity.  It is called Weight.  Since F = m a, and a is the acceleration due to gravity (or g):

W = m g

Note that this implies that:  weight can change, depending on the value of the gravitational acceleration.  That is, being near the surface of the Earth (where g is approximately 9.8 m/s/s) will give you a particular weight value, the one you are most used to.  However, at higher altitudes, your weight will be slightly less.  And on the Moon, where g is 1/6 that of the Earth's surface, your weight will be 1/6 that of Earth.  For example, if you weight 180 pounds on Earth, you'll weight 30 pounds on the Moon!


3.  Newton's Third Law

To every action, there is opposed an equal reaction.  Forces always exist in pairs.  Examples:

You move forward by pushing backward on the Earth - the Earth pushes YOU forward.  Strange, isn't it?

A rocket engine pushes hot gases out of one end - the gases push the rocket forward.

If you fire a rifle or pistol, the firearm "kicks" back on you.

Since the two objects (m and M, let's say) experience the same force:

m A = M a

That's a little trick to convey in letters but, the larger object (M) will experience the smaller acceleration (a), while the smaller object (M) experiences the larger acceleration (A).

Conversion factor HW

Create two interesting units of speed - show how to create their conversion factors from m/s.

Test practice

Find the time for a ball to fall 7-m from rest.  If the ball had been kicked horizontally at 4 m/s, how far would it have traveled?

Heading west in a motorboat at 15 m/s, you encounter a 5 m/s current north.  At the end of 400-m, how far off course are you?  And how much time did you spend to get there?

Gravity practice problems (with answers)

A reminder that you call the initial direction of motion positive, whether it is up or down.  Then gravitational acceleration (which is always down) could be either + or -, depending on your system.

1.     If you drop a ball from a 50-m tower, how long does it take to hit the ground and how fast is it traveling right before it hits the ground?  (3.2 seconds, 31.3 m/s)

2.     Throw a ball straight up at 25 m/s. 

a.     How long will it take to reach the top of the arc?  (2.6 s)

b.     How long will it take to return to your hand?  (5.1 s)

c.      How high does it go?  (31.9 m)

d.     Where exactly is the ball at the 4-second point and how fast is it traveling?  (21.6 m above ground, -14.2 m/s)

Quiz Friday - practice problems here

1.  A ball is dropped from rest from atop a 60-m tall tower.  If it undergoes a standard acceleration due to gravity of  9.8 m/s/s, find the following:

a.  how long it takes to hit the ground

b.  the speed it will have right before it hits the ground

2.  A car is traveling at 30 m/s when the driver applies the brakes, bringing the car to a halt in 3 seconds.  Find:

a.  the acceleration during this braking period

b.  how far the car travels during this period

HW for Monday (equations of motion practice)

Use the equations of motion (where helpful) to solve the following problems.

1.  Consider a car, capable of accelerating from rest at 5 m/s/s.  If it accelerates uniformly for 8 seconds, find:

a.  the velocity after 8 seconds
b.  how far the car has gone in this time

Now assume that the driver applies the brakes and the car uniformly comes to a halt in 3 seconds.

c.  What is the acceleration during this braking time?
d.  How far does the car go during this time?

Draw two graphs that represent:

a.  the displacement vs. time for the entire trip
b.  the velocity vs. time for the entire trip


2.  Imagine that a falling body accelerates at 10 m/s/s.  It is released from a high tower:

a.  If it takes 3.5 seconds to hit the ground, how fast is it traveling immediately before hitting the ground?
b.  If it took 3.5 seconds to hit the ground, how tall was the tower from which it was released?

Keep working on your lab report.  Your draft is due Monday, and the final lab is due Wednesday.

Lab hw

Work on the lab report for Tuesday.  Make sure you have the graph (with line and slope) for the ticker tape part, and calculated velocity for the photogate part.  

Follow the lab sheet for the rest.  The draft will be due 2 classes after our next class.

Weekend - yay!

HW for Monday

1.  Review the distinction between vector and scalar quantities.  Make sure you know the difference between them.

2.  Review the difference between distance and displacement, as well as speed and velocity.

3.  So, we calculated average speeds and average velocities in class.  Here is a question to think about.  My average speed on the way to school was 24 miles per hour.  Imagine that instead of driving safely and obeying the law, I drove at a CONSTANT speed of 24 miles per hour for the entire trip from home to school.  Would I arrive at the same time or not?  Discuss.

4.  What do you suppose "instantaneous speed" or instantaneous velocity is?  How could we determine it experimentally for a car or person?

5.  Imagine a car driving straight at a constant speed of 25 m/s for 5 minutes.  Draw an approximate graph that represents the displacement vs. time for the entire trip.

6.  Two ropes are attached to a boulder that is stuck in the mud.  One person pulls on the boulder with a force of 100-N south.  Another person pulls on the boulder with a force of 250-N east.
a.  Draw this as a problem.
b.  Solve for the total force acting on the boulder - magnitude (number) and angle.  You may need SOH-CAH-TOA to find the angle.

Arduino info

Here is a download link for code for several Arduino projects in the Sik guide:

sparkfun.com/sikcode

And the guide is here:

http://dlnmh9ip6v2uc.cloudfront.net/datasheets/Kits/SFE03-0012-SIK.Guide-300dpi-01.pdf

HW for Friday

THIS WILL BE COLLECTED.

Static electricity review:

1.  Consider a chunk of charge equal to 5 coulombs.

a.  How many protons is this?

b.  If another chunk of charge equal to -5 coulombs is brought to a point 0.1-m away from the first charge, what is the force between the two chunks of charge?

c.  Draw the electric field between these two chunks of charge.


2.  In the circuit below, find the total resistance, battery current and fill in the chart.


3.  In this collection of equal-resistance light bulbs below, list the bulbs in order of brightness (brightest first).

Practice problem

Solve for total resistance and total (battery) current.

Then fill in table.

Today's combination circuit

HW

Lab calculations

You've (ideally) finished the calculations for the series part of the experiment:

total resistance
percent difference from the theoretical (65 ohms)

If you haven't yet done the parallel calculations, do them:

- total experimental resistance (battery voltage divided by battery current)
- total theoretical resistance - use this formula:
   1/Rp = 1/R1 + 1/R2

The easiest way to proceed is to use the x^-1 key on your calculator.  Don't forget to take the inverse of the sum of the inverses.

- percent difference between theoretical resistance and experimental resistance

Keep working on lab.  You'll still need the usual things:

title
purpose
data table(s)
calculations
conclusion (use the lab questions as a 'jumping off' point - make sure you answer the questions in bold)
sources of error, ways to improve, etc.

Lab writeup questions

Folks,

You have some data for a series circuit and a parallel circuit.  I will give you some "leading questions" to help you frame your formulation of Kirchoff's Rules.  The questions in bold are to be answered in your lab report.

I.  Series Circuit

Think about these things: 

- What do you find to be true about the voltages?  

- What do you find to be true about the currents?  

- How do the individual voltages compare to the battery voltage?

- How do the individual currents compare to the battery current?

Write a general rule about voltages in a series circuit.

Write a general rule about currents in a series circuit.

Calculations (you may have to wait until next class for these):

What is the total resistance of this circuit (according to the values of the resistors given in class from the stripes)?

What is the total resistance of this circuit according to calculation?  (R = battery voltage divided by battery current)

II.  Parallel Circuit

- What do you find to be true about the voltages?  

- What do you find to be true about the currents?  

- How do the individual voltages compare to the battery voltage?

- How do the individual currents compare to the battery current?

Write a general rule about voltages in a parallel circuit.

Write a general rule about currents in a parallel circuit.

Calculations (you may have to wait until next class for these):

What is the total resistance of this circuit (according to the values of the resistors given in class from the stripes)?

What is the total resistance of this circuit according to calculation?  (R = battery voltage divided by battery current)

Other general questions to answer:

Did you notice if resistors got hot?  Why might this be?

What exactly is Ohm's Law?  Does it apply to all electrical components?

What do you think "tolerance" is, as applied to resistors.  For example, if a resistor is said to have a 10% tolerance, what can that mean?

Give sources of error in this experiment.

Graph HW

Graph your lab data:

I vs. R

That is, I (on the y-axis) vs. R (on the x-axis).

What do you make of this relationship?  Does it make sense in light of the definition of resistance?

Ohm-work (get it?)

Please review definitions of voltage, current, and resistance (along with their units).

1.  How long would it take a circuit carrying 4 amps to pass 50 coulombs of charge?

2.  A 9-V battery is attached to a resistor.  As a result, 0.1-A of current passes through it.  What is the resistance of the resistor?

3.  Look up the schematic symbols for battery, wire and resistor.  Draw a basic circuit which depicts the bulb and battery arrangement we made in class.

hw

Come to class with a definition of energy that is better than my "blocks" analogy.

He

How do batteries work?  Find out and write about it in your notes.

HW - electric field

Look up the concept of electric field and determine how one draws them and what significance they have.

HW

Late post, so do these if you have time:

1.  What are each of the quantities in the Coulomb's law equation, and what are their units?

2.  a.  Calculate the force that exists between two clusters of charge (each 10 micro coulombs), 0.1-m apart.  The prefix "micro" means x 10^-6.  Be sure to use scientific notation (EE key --> 2nd comma) on your calculator).
b.  Is this force attractive or repulsive?  How do you know?

3.  If you have a cluster of charge that is 5 C in magnitude, how many protons is this?  Recall that 1 C of charge is 6.25 x 10^18 protons.

4.  Thinking about the inverse square law - if the distance between two charges is changed to five times the original distance, what exactly happens to the force between them (compared to the original force)?

5.  If the distance between two charges is changed to half the original distance, what exactly happens to the force between them (compared to the original force)?

*6.  The radius of a typical Hydrogen atom is 53 pm (5.3 x 10^-11 m) -- the distance between a proton and electron in "orbit" around it.  What is the force between these two particles?  Recall that the charge of a proton and electron is the same, though one is negative and the other is positive.

HW

Find a definition for charge, including the unit for it.

And if time, find out something about Coulomb's law.

HW to TURN IN on Thursday

1.  Find out about a few examples of how holography is used today.  Or if you're feeling ambitious, suggest some uses for it.

2.  Diffraction question

You are sending light through a diffraction grating that has 300,000 slits/openings per meter.  You aim a blue laser at it (wavelength = 475 x 10^-9 m).  The laser and grating is 1-m from a wall.  Find:

a.  the value for d (the distance between the slits)
b.  the diffraction angle for a first order (n = 1) image
c.  the distance between the central image (n = 0) and the first (n = 1) image.  Use trig here.
d.  Draw this situation.
e.  If you used a red laser, would the dot spacing change?  If so, how?

HW

Find out something about holography - what it is, how it works, applications, etc.

Diffraction

Please do a little reading about light diffraction.  Find out what it is, and if possible, find a mathematical expression that represents it.