PLEASE keep checking your instructions. I will be making some addditions. Yesterday's changes will be in GREEN and TODAY'S in ORANGE.
Don't Forget to bring homework and "First Results" of your complete experiments to class (same format as your project proposal)
Here is an attachment that shows you how to use an EXCEL SPREADSHEET to do repetitive calculations easily and accurately (if you get the equations right :) ). This will be a huge help when you do your calculations for your science fair project. You can click on the various cells to see how the numbers were calculated.
https://sites.google.com/site/yhbscience/3%20BODY%20program%20for%20Sun%2C%20Earth%2C%20%26%20Moon%20%28CIRCULAR%20ORBITS%29.xlsx?attredirects=0&d=1
Here is HOMEWORK for next week (due May 1, 2014)
https://sites.google.com/site/yhbscience/Class15.Homework10--Exploration%20of%20Physics%20Simulators.docx?attredirects=0&d=1
New Orbit Equation: I just figured out how to calculate the tangential velocity for a circular orbit in My Solar System 2.04:
Vtang = 100 sqrt[(M+m)/r] ß Savannah and Ashur !
BOTH bodies will make circular orbits around the center of mass of the two bodies (this is the balance point if you glued both bodies to a massless rod). If m << M (m is much smaller than M) then you can leave it out (depends on the accuracy you want )
M & m = masses of the bodies
r = distance between centers of the bodies
100 = an arbitrary constant that the program uses
EXAMPLE: M,m = 800, 200; r = 250; à Vtang = 100 sqrt[(800+200)/250] = 200
Try it - it works.
PLEASE NOTE: Go back and read your instructions from Class 11. I am not going to repeat everything there in these new instructions -- Class 11 has additional information to what is given below !!! THX!!!
(Mon Apr 21) Finished: Ashur
(Sun Apr 20) Finished: Samuel, Kai, Layel, and Savannah.
(Fri Apr 18) Finished: Elie and Jonas
(Thu Apr 17) Finished: Tristen
-- IN THIS SECTION I WILL DISCUSS YOUR SCIENCE FAIR WORK. The green label will be general comments for everyone, and the violet label will have individual comments for each student (although it might be helpful if you all read all of the sections). I WILL KEEP ADDING TO THE COMMENTS WITHOUT PUTTING IN DATES, SO PLEASE KEEP UP WITH THE CHANGES OVER THE NEXT WEEK. THX.
GENERAL COMMENTS:
1. The format of your poster [6 sheets of letter-size (8 1/2 x 11) paper] will be the same as for your Proposal and your First Results submissions (except that you don't need 6 sheets of paper for the preliminary work, only for when you are finished all your experiments)
2. Every poster must show your results in a graphical format. This is like the graph we did in class using the Microsoft EXCEL Spreadsheet. Remember, you have a column of x-axis data and a corresponding column of y-axis data. Every x value has a y value. If x is age (years) and y is height (inches) we might measure five children and type in the data as:
Age (years) Height (inches)
---------------- ---------------------
10 48
11 53
13 59
14 62
16 68
Then we make a plot (the details may change for different computers and software), by doing the following (i) drag the cursor to make a rectangle around just the numbers. (ii) click "insert", (iii) click on "scatter", (iv) click on whichever style of plot we want (I usually use just the points with no lines, but you might like something else).
After that, you have to dress up the plot: put in a plot title at the top, and axis labels for the x and y axes, just like the ones in yellow above. You may also need to add a trend line. This is usually a straight line (linear) or a curve (quadratic curve = parabola, etc.). Not sure of some of these words? Start by typing them in the Google Search box. For example:
"Excel plot title", or "How do I make a plot title in Excel?"
"What is a quadratic curve?" or "What is a parabola"
You can also search in the Excel program itself. Mine has a blue question mark (some might have the word "HELP" or something else. If I click on the ? and enter "Excel plot title," I get a bunch of links. The first one says:
Article: Create and change a column, bar, pie, line, or scatter chart (or graph) in Excel 2007 or Excel 2010.
Now, that looks real helpful, doesn't it?
3. TO BE CONTINUED
INDIVIDUAL COMMENTS: I'll make my comments here in two parts: I. BIG PICTURE, and II. SPECIFICS. First I will give everyone the big picture, and then go back for the details:
TRISTEN:
I bought some adhesive-backed aluminum-foil tape. Maybe you can use it to roll yourself a tube-diameter-reducing insert. You can roll it onto a small cylinder of any diameter to set the ID of the insert, and stop rolling when you have reached an insert OD that matched the ID of your siphon tube. To keep the insert from sticking to the rolling core, you could bend back the first few millimeters of the tape and adhere it to itself and flatten that part out. That way the insert will not adhere to the core, but you keep rolling, the tape will stick to itself and not unroll !
I chose aluminum because it wouldn't absorb the liquid and swell up.
I'll bring some tape to class.
I. BIG PICTURE: We went over your project fairly carefully at our place.
a. Your goal is to investigate the average liquid flow created in your siphon. You will measure this for:
b. various differences in the two liquid levels;
c. various tube diameters; and
d. various liquids of different viscosity (resistance to flow).
e. something of your own choice.
II. SPECIFICS:
a. Flow = volume/time at any instant of time. Average flow = total volume/total time. Since the liquid levels in your siphon will be changing, the flow will be changing too. Therefore you need to measure average flow. The units of flow could be cubic cm / s, or liquid ounces / s, or whatever you want.
b. The level difference, LD = H1 - H2, where H1 = height of the higher liquid surface level, and H2 = height of lower liquid surface level. the LD can never be negative because the siphon stops pumping when the LD = 0. So, when you have the upper container at the highest position, LD could start out at, say, 30" and stop when LD is, say, 24" and the siphon starts to suck in air. (You have to stop it instantly. You can squeeze the tube if it's flexible enough, or else quickly raise the lower container until it is higher the the upper one.)
You can specify the average LD for each of your experiments.
c. You can reduce the aperature of the flow tube two ways. The hard way is by replacing the entire tube with one with a smaller inner diameter. The easier way is to put a miniature "washer" in the tube. Since they don't make washers that small, you could put a short (1") section of a smaller tube inside, or maybe roll up a 1" x 2" piece of plastic sheet, or aluminum foil or something to make your aperture. The hobby store at White Station and Summers has small diameter tubes. Try to do a first class job on this or you get leaks and inconsistent results.
d. You can try water, sugar water, salt water, oil, detergent water, etc. The way kids usually do science projects is to try a bunch of different fluids and see what they look like, but a more scientific approach would be just choose one liquid (like sugar water) and measure it for different concentrations of sugar (pure water, 1 tsp sugar/cup, 2 tsp sugar/cup etc. that would make a nice plot with concentration on the x axis and average flow (all at the same height, of course) on the y axis.
I expect sugar and salt water will flow slower than pure water (not sure), but it would also be nice if you can find something that flows faster (lower viscosity): maybe oil or detergent water.
e. You do it.
OTHER THOUGHTS: Don't place the end of your tubes too close to the bottom of the containers or you will obstruct (and slow down) your flow. Also it would be good if your tube were secure in their mountings, but you had a way to slip in small sections of tube -- or -- I just thought of something else. Can you get a short piece of wire and wrap it around (4-5 times) a narrow cylinder to make a coil. The OD (outer diameter) of the coil would = the ID (inner diameter) of your tube, and the ID of the coil would be your new tube diameter. You could slip the coil inside your tube (assuming you made the siphon to allow for this!). You could also use different diameter wires (smaller wrapping cylinders) to make coils with small IDs.
You need to think about what works best.
ELIE:
I. BIG PICTURE:
a. Your first goal is to demonstrate what happens when a sets of bodies collide (in your case they are suspended as pendulums, but the same thing would happen if the bodies were hockey pucks sliding on ice, or other situations)
b. You should study all the basic types of collisions, including:
(i) one group moving - one group at rest
(ii) two outer groups moving inwards with maybe some stationary balls in between them.
(iii) can you think of any other ways to study the motion?
c. This is more difficult, but you should try (with parental help) because it is so important. WHY
do the balls behave this way? For the 2m--3r case, why do 2 balls come out together; why not just one ball going much faster?
II. SPECIFICS:
a. you have five bodies (steel ball bearings or barrel weights) mounted so they are in a perfect straight line and just touch each other when they are at rest. The bodies should be fixed with glue or something so they can't slip around (they have to stay in a horizontal straight line).
b. I know of two types of motion you can investigate (there could be more):
(i) one group moving (m) - one group at rest (r): EXAMPLE: pull out one ball, let it hit four stationary balls ---> 1m--4r. The other possibilities are obviously: 2m--3r, 3m--2r, and 4m--1r.
(ii) two outside groups moving (toward the center) with the possibility of stationary balls in between them. EXAMPLE: pull out ball 1 and ball 5 and release them (balls 2,3,4 are stationary) ---> 1m--3r--1m. Other possibilities (there are a lot): 2m--1r--2m, 1m--2r--2m, 1m------1r--3m, 1m--0r--4r, 2m--0r--3r. (NOTE that in every case, the total of the numbers must be 5 (because there are 5 balls).
(iii) You can try different ideas. Some crazy ideas are, can you let the outer balls go at different times (first the ones on the left, then the ones on the right? Can you have a helper get the middle group moving and then release the outer groups?
c. The reason the balls behave this way is that they are making (almost) elastic collisions (elastic means that no, or almost no energy is lost --things just bounce of with the same energy they had.) Shooting a bullet into a ball of clay where it gets trapped is an example of an inelastic coillision: energy is lost when the clay gets deformed and the lost energy turns to heat, sound, etc. With hard steel balls, a tiny bit of energy is lost making the clink sound and also due to air resistance and bending the strings. We are going to consider this loss as negligible.
Now...IMPORTANT... in a elastic collision, BOTH momentum and energy are conserved (that means they are the same just after the collision as they were just before the collision.
OK, so what is momentum andf energy, and what kind are we talking about?
MOMENTUM (p) = attribute of a moving body that determines how much force is needed to stop it in a given time. (The more momentum it has, the harder you have to push to stop it.)
ENERGY = ability to do work: move things, lift things, heat things up, bend things, etc.
KINETIC ENERGY (T) = energy of a moving body (chemical energy = energy in a battery...)
(Sorry about the symbols, p and T, for momentum and kinetic energy, but that's what they use.)
Mathematically, if a body has total mass m and velocity v, then
p = mv (momentum) and
T = ½ mv2 (kinetic energy)
Now we can begin to see why the balls behave they way they do:
EXAMPLE: For simplicity, we will forget about units, and assume m(1 ball) = 1, m(2 balls) =2, etc.
lets also assume that a certain velocity v =1, twice that velocity = 2, etc.
WE WILL ASSUME DIFFERENT THINGS AND CHECK IF MOMENTUM AND KINETIC ENERGY ARE CONSERVED (STAY THE SAME) BEFORE AND AFTER THE COLLISION!!
(A) ASSUME 2BALLS IN --> 2BALLS OUT ( WITH SAME VELOCITY... NOTE- THAT p AND T OF ALL THE STATIONARY BALLS = 0, BECAUSE v = 0 FOR THEM!)
Before collision: p = (2m)v = 2 x 1 = 2 After: p = (2m)v = 2 x 1 = 2 so p is CONSERVED!!!!
Before collision: T = 1/2 (2m)v2 = 1/2 x 2 x 12 = 1 After: T = 1/2 (2m)v2 = 1/2 x 2 x 12 = 1 so T is CONSERVED!!!! (This is what actually happens with 2 balls in, 2 out)
MOMENTUM AND KINETIC ENERGY ARE BOTH CONSERVED!!!
(B) NOW ASSUME 2BALLS IN WITH VELOCITY V --> 1BALL OUT WITH VELOCITY 2V
Before collision: p = (2m)v = 2 x 1 = 2 After: p = m(2v) = 1 x 2 = 2 so p is CONSERVED!!!!
Before collision: T = 1/2(2m)v2 = 1/2 x 2 x 12 = 1 After: T = 1/2 m(2v)2 = 1/2 x 1 x (2)2 =1/2 x 1 x (4) = 2 so T is NOT CONSERVED!!!! (That's why we don't see this happen!!!)
MOMENTUM AND KINETIC ENERGY ARE NOT BOTH CONSERVED!!!
JONAS:
I. BIG PICTURE: You have built an apparatus that will allow you to study a very interesting type of wave motion. I know you have 12 independent (what one pendulum is doing does not affect its neighbors) pendula now, but I'm hoping we can still have one with 16 pendula because that's what most people use. Anyway, here is the basic idea which is pretty simple (that math details get a little more complicated).
a. In a given time, say 30 seconds, the 1st pendulum makes exactly 60 cycles (full swings); the second pendulum, exactly 59 cycles, the 3rd pendulum, exactly 58 cycles, and so on.
b. You need a way to maintain the exact length of each of your pendulums, once you get the right values.
c. You need to explain what causes the patterns.
d.. Something else you think of.
II. SPECIFICS:
a. The hard part here is the word "exactly." If the second pendulum, for example, makes 58.9 or 59.1 cycles instead of 59, the edges of your patterns will become very ragged and won't look good. We are talking about an error of only 0.1 cycle out of 59. How much is that? 0.1/59 = 0.17%. That's less than two-tenths of one percent!
How do you adjust the period of a pendulum? You change the length. As we discovered (or should have discovered) in our class and homework experiments, the period of a pendulum is proportional to the square-root of the length (PERIOD ~ √L). That actually helps a little because if we make a 1% error in the length of the pendulum, the error in the period is only 0.5%.
The best way to get accurate pendulum periods is not by super-accurate measurements of the length of the pendulum. The best way is to get within, say, 2% of the right period, and then use a mechanical method to FINE-TUNE the length. The best way is probably to use an adjustment screw on EACH pendulum. So, if I turn the screw, say, 1/4 turn, it increases (or reduces) the string length about 1/100". Maybe you can get sufficient accuracy without adjustment screws. It's worth a try anyway.
NOTE - if your apparatus is too low to allow 60 cycles in 30 seconds, you could try 40 cycles in 30 seconds, or something else. The only thing that matters is that you get nice patterns.
b. Maintaining the exact length of each pendulum is not so easy. Basically, NOTHING CAN MOVE OR SLIP. It may work that you can set the wooden clamp on top of your apparatus just tight enough that it is possible to carefully pull any string a tiny bit tighter or looser WITHOUT AFFECTING THE OTHER STRINGS. Then, when you get everything right you can FULLY TIGHTEN UP THE CLAMP so nothing will slip.
When you are ready, you get a small, straight board (or a ruler, or something) and pull all the bobs towards you to line them up for the same swing distance, Then you quickly move the board away and down so all bobs are released at exactly the same time.
c. The starting patterns, which look like sine waves, are caused by the fact that since each bob is moving a little faster than its longer neighber, it gets advanced in its phase (or position within one cycle) by the same amount. But after a while the waves breaks up into strange patterns and sometimes multiple straight line. See if you can find out what causes this.
d. Let's see what you come up with.
SAMUEL:
I. BIG PICTURE: You are studying the normal (fundamental) modes (patterns, configurations) of a system (group) of FOUR coupled pendula (pendulums). "Coupled" means that when one pendulum moves, it can make nearby pendula move (due to connecting strips or rubber bands). So it can transfer some of its kinetic energy to its neighbors. Later, they will transfer their energy back again. HOWEVER, when the system is operating in any of its NORMAL modes, their is NO ENERGY TRANSFER: the system moves exactly the same way in every swing. The number of normal modes = the number of pendula. So, a single pendulum has only one normal mode: back and forth, back and forth, etc. In fact, (only) for the case of one pendulum: the normal mode is the ONLY way it can go (because there are no other pendula to transfer its kinetic energy to!
All systems with more than one pendulum can operate in either a normal mode OR in a MIXED mode. In a mixed mode, every swing is different from the previous one because kinetic energy is being transferred around, until after many swings, the system comes back to the original swing and so on after that. Every mixed mode is actually a combination of two or more normal modes (but it may not be easy to identify which modes and how much of each of them!)
So, your goal is to:
a. FIND the four normal modes of your system and be able to demonstrate it to others.
b. See if you EXPLAIN what characteristic all the normal modes have that distinguishes them from any of the mixed modes.
c. FIND a few (there are an infinite number of them) interesting mixed modes and be able to demonstrate them to others.
d. Show what happens when the STRENGTH of the coupling changes (by lowering, raising, or even removing the coupling bar)
e. Anything else interesting that you are able to find.
II. SPECIFICS:
You don't need a lot of specifics for each of your goals since they are straighforward:
a. Carefully measure the periods of the normal modes. (Can you see any relationship of the numerical values? This might be hard -- don't worry)
c. Carefully measure the periods of some mixed modes (This is the time from start to when the overall configuration first repeats itself.
Also, I'll give you some overall comments and suggestions.
(i) Make sure all your pendula have exactly the same length;
(ii) Make sure your coupling bar is exactly horizontal **;
(iii) Make sure everything is tight and nothing slips;
(iv) It's not easy for one person to get four pendula into each of the starting positions. Maybe you (or we) could make a gizmo with four adjustable vertical squares (or something) that you use to position the bobs, and then lower it to release them.
(v) In your poster, you could use a row of four position vectors, to show the starting position of the bobs. Here are two examples:
-1- -2-
----> ---->
---->
<----
----> ---->
----> <----
** Actually, seeing what happens when the coupling bar is tilted might be very interesting because the strength of the coupling will change from pendulum to pendulum! I never saw this type of motion before! How does the response of the system change compared with when the bar is horizontal?
KAI:
I. BIG PICTURE: You are studying the difference in the behavior and properties of the planar (swings back and forth) and the conical (swings in a "perfect" circle) pendula. Actually, there is only one physical pendulum, but you make it swing two different ways. Your goals are:
a: To compare the periods of a planar and a conical pendulum relative to changes in mass only (this means the other two parameters: length and angle stay the same;
b. To compare the periods of a planar and a conical pendulum relative to changes in length only;
c. To compare the periods of a planar and a conical pendulum relative to changes in angle only;
d. Make sure you use EXCEL to make your plots (or, if you have to, use a handwritten set of x and y axes and plot by hand)
II. SPECIFICS:
a. It is better to make the conical pendulum swing by grabbing the support board and moving the pivot around in a small circle (rather than by grabbing the string), because you may get an ellipse - not a circle - if you grab the string. TRY DIFFERENT WAYS TO GET THE BEST CIRCULAR MOTION.
The math of how a planar pendulum and a conical one each move is very advanced (college level). I can give it to you, but you may not understand it. Let's hold off for now. In the meantime, you should go back to Class 11 and do the detailed experiments I told you to do.
Do as much as you can for your "First Results" report due on Thursday.
b. For the length changes you can use the two nails I told you about to make sure the planar and conical pendula have the same length. Also make sure the angles are the same (this is the same as making sure the side displacement in the x-direction of the bob relative to the stationary (vertical) position is the same) for both types of pendula.
c. For the angle study, the mass and length of the two types of pendula have to be the same. It will be hard to get conical pendulum to rise up to near 90 degrees (horizontal string) because the force and the speed both get huge. Do the best you can without hurting anyone with a flying bob.
LAYEL:
I. BIG PICTURE: You're studying a rigid (180 degree ) pendulum with the capability of swinging in a tilted (non-vertical) plane. You will try to verify the Law of Conservation of Energy. You know how to do the experiments except maybe for the Conservation of Energy part, so I'll explain that part.
a. In a vertical plane (the ususal situation) accurately determine the period of your rigid pendulum versus the swing angle.
b. For several angles of the tilted plane accurately determine the period of your rigid pendulum versus the swing angle. (Just to be clear: when you have the tilt angle set so that the bob swings in a perfectly horizontal plane so it won't move on it own -- that is a ninety degree tilt angle
c. Check the Law of Conservation of Energy. You have to measure the speed of a moving bob, and you will probably get more accurate results using a tilted pendulum, since things will be moving slower.
d. Do something else that you find interesting.
II. SPECIFICS:
a. For small angles (under, say 45 deg) the loss in height (potential energy) during each cycle (full swing) is small, so you can take the AVERAGE of the starting angle and the final angle to represent the "angle" of that experiment. For large angle, this is not so good. For example, suppose the maximum angle of swing <1> is 135 deg, <2> is 100 deg, ..... <9> is 70 deg and <10> is 65 deg. Swing <1> lost 35 deg, but swing <9> only lost 5 deg. The average period of the ten swings associated with the average angle of the ten swings (means when you are done, you just assume your average period would be the same as if there were no height losses AND all ten swings happened at the average swing angle of the ten swings) may not give the same number as average period of ten experiments each experiment representing ONE swing of the pendulum, where all ten are taken starting at the average angle. (I know this is confusing - please ask a parent to explain it). Anyway when you are doing a series of, say, ten consecutive swings, and the change in angle (not the angle, but the amount it changes from swing to swing) is getting smaller and smaller for each swing, you have to be careful. Either make sure what we said above is actually TRUE or don't do a series - just measure one swing (or even a half swing: from full left to full right). Then repeat that a bunch of times and average the periods. If you just do a half swing, you will have to double your measured number to determine the full period (left-to-right-to-left again) - got it? Sorry, this would be a lot easier to show you than to explain in words
b. For a tilted plane, the action will slow down (making all measurements easier). You can make plots of your results for a few tilt angles and compare with the vertical plane. You can also check if the period of a vertical swing when the maximum height (when you start) minus the minimum height (the lowest point in the swing) is, say, 10" (25.4 cm) is the same as the period of a tilted swing that has the same difference in heights.
c. OK, here if the LCE we spoke about in Class 11:
Assuming no friction in the bearing (not quite right) and no air resistance (definitely not right), the starting potential energy (PE) of your swing would never diminish and the pendulum would go forever (like the one on a grandfather's clock). As the pendulum descended, the PE would gradually be converted to kinetic energy (KE, sometimes labelled T) increasing the speed. When you reached the bottom of the swing, all the energy would be converted to kinetic energy (max speed) and the potential energy would be at its minimum. If we measure the vertical height from this point, the height would be zero and the PE would also be zero. Then for the rising part of the swing, the KE would decrease to zero and the PE would rise back to the exact value it started with. This would go on forever. However, you DO have friction and air resistance, so things gradually peter out. The height of each swing gets less and less until the maximum potential energy also goes to zero and the show stops.
We will start out assuming NO ENERGY IS LOST:
E = TOTAL energy
<--
KE = ½ m v2
PE = mgh
Where m = bob mass (assuming rod and bearing mass
is negligible – not really true)
v = velocity or speed of of the bob
g = the acceleration of gravity (9.8 m/s/s or 32.17 ft/s/s)
h = vertical height of center of pendulum bob
Now LCE says that if there are no energy
losses, E = PE + KE = constant value
So, if KE = 0 at top (start) of swing, and PE
= 0 at bottom of swing, then
KE (bottom) = PE (top) and
½ m v2 = mgh
v2 = 2gh
So, if we quadruple (multiple by 4) h, the
height of the swing, we should quadruple v2 and thereby double v, the speed
at the bottom. THIS IS WHAT YOU ARE
GOING TO CHECK. If you can prove this
(or demonstrate that it is APPROXIMATELY TRUE), you will have verified the Law
of Conservation of Energy!
Let see how you do it. We have to raise the bob a certain distance (say, 5") release it and measure its speed at the bottom of the swing. There is no easy way to do that (that I know of) exactly at the bottom of the swing, but we can measure the average speed over a horizontal distance starting a few inches (say, 8") before reaching the bottom and ending 8" past the bottom. Then we can raise the bob 4 x 5" = 20" and measure the speed again and see if it doubles. Maybe you will have to change the two basic numbers (5" and 8") to something else to get better results.
So, maybe you can put a board or piece of paper under the pendulum and draw two dark, thick (easily visible) parallel lines, one at the start point (where you start your timer) and one where you stop the timer. Measure 10 times and get a mean and SDM. If the SDM is too large (say, greater than 5% of the mean) you will have to try changing the base numbers (you should do that anyway to get the smallest value of SDM / mean). Another way to reduce SDM / mean is to average 20 or more runs instead of 10. Even better is to do both of these things!
NOTE - if you are using a tilted pendulum, you will have to reduce the vertical height of your release points, but NOT necessarily the horizontal marker-line separation distance.
Check your results and see if LCE is true or almost true! See if you can explain in a little detail what caused any discrepancies.
SAVANNAH:
I. BIG PICTURE:
As for all the other students, you should go back and read your instructions from Class 11.
Basically, your goal is to find out how large a "Moon's" orbit can get before:
a. the influence of the "Sun" begins to disturb its orbit, and
b. the gravity of Sun becomes so strong it actually pulls the Moon away from the "Earth" and either it goes into orbits around the Sun but not the Earth, OR it gets kicked (catapulted) out into space (perhaps never to return) AND POSSIBLY
c. if the orbit is too large, the Moon may crash into the Sun, or crash into the Earth, or even eventually come back and re-enter an orbit around the Earth again!
d. or something happens or something else you want to try.
II. SPECIFICS: Everything in green comes after I found the new orbit equation!
We will proceed by selecting arbitrary values for the masses of the Sun, Earth and Moon. If you want to make the masses in the program PROPORTIONAL to the true masses of these bodies, that is great. You can find the masses easily using Google. You can start by setting the mass of the sun to, say, 1,000 and choose the mass of the Earth and Moon so they are in the same ratio to this mass as the real Earth and Moon are to the real Sun's mass.
You may already be past this point, but if you're not, you could just make the Earth 1000 times lighter than the Sun and make the Moon 1000 times lighter than the Earth. Also, you make the Earth's orbit fill the entire screen, while at the same time, the Moon's orbit will be small, but larger enough to see it. For example:
M x y Vx Vy
1000 -100 -50 0 0
1 150 -50 0 200
0.001 155 -50 0 250 --> 245 (244.74) from new eqn
Vtang = 44.74 and have to add Vyearth = 200 !! 250 wasn't a bad guess :)
I tried the 245 and the 250 in the Simulator (245 gives more circular moon orbit!)
Now you can check if the Moon's orbit is circular. One way is to turn on the trace and see if it comes the same distance inside and outside the Earth's orbit. Another (slower but more accurate) way is to turn off the trace and keep stopping and starting the motion. When you stop the motion, use the cursor to give the x and y components of both the position and velocity of both bodies (aren't you glad we did vectors?), so you can subtract the Earth's components from the Moon's to see if the orbit around the Earth is a circle and also find the Moon's velocity around the Earth. When you put the cursor on the Earth you get the Earth's information, same for Moon!!
For relative position (Moon's position relative to the Earth):
Radius = Rmoon = sqrt(xrel2
+ yrel2) where xrel = xmoon -xearth (same for y), so
Rmoon = sqrt(xrel2 + yrel2) = sqrt[(xmoon - xearth)2 + (ymoon – yearth)2]
Same thing for velocity (Moon's velocity relative to the Earth)
Vmoon = sqrt(Vxrel2
+ Vyrel2) = sqrt[(Vxmoon - Vxearth)2 + (Vymoon –
Vyearth)2]
The ENTIRE discussion of circular orbits is simplified if you use the new orbit equation I just gave you at the top of the blog that lets you calculate Vtang!! Just remember 2 things when you are talking about the Moon's orbit around the Earth:
[1] r in the equation is the distance between the Moon and the Earth, and
[2] the Vtang that you get is the relative velocity between the Moon and the Earth, so if we start with the Vx-components equal, then Vtang = Vymoon - Vyearth , not Vymoon. In other words, what you put into the Simulator box for Vymoon is (=) Vtang + Vyearth (see, we are just re-writing the equation in the previous line). We still need to discuss this, but it just got a lot easier with the new orbit equation!
As the Moon moves to different positions around the Earth, you can keep checking to see if R remains the same. If it does, you have a circular orbit. (You should also check that the Earth's orbit around the Sun is a circle. That's easy because the Sun is at (-100,-50) so you just turn on the grid and make sure the Earth is 2 1/2 grid blocks from the Sun at it's N, S, E and W positions.
Now, you can keep increasing x(Moon) and Vy(Moon) to get larger and larger orbits. Now, when you increase the size of the Moon's orbit, how much should you decrease the Moon's velocity? REMEMBER, we are NOT talking about the Moon's actual velocity. We are talking about it's relative velocity around the Earth. That is the quantity Vmoon. One way to make this easier is to check when BOTH the Moon and Earth have the same x OR y velocity components.
if Vxmoon = Vxearth, then Vmoon = Vymoon - Vyearth !!! OR
if Vymoon = Vyearth, then Vmoon = Vxmoon - Vxearth !!!
(It's the same for position as it is for velocity
if xmoon = xearth, then Rmoon = ymoon - yearth !!! OR
if ymoon = yearth, then Rmoon = xmoon - xearth !!!
IF the Sun's effect on the Moon is not too strong (not sure about this), once we know the relative velocity for a circular orbit for one orbital radius, we can use Kepler's Third Law to find the relative velocity needed at another orbital radius. I won't go through the math now, but it turns out that
V2 α 1/r (this means velocity squared is proportional to the reciprocal of the radius. More simply:
V α 1/√r (
V prop to recip of square-root of orbit radius)
EXAMPLE: I have an Earth mass of 200 (with a tiny Moon mass), the radius is 200 and the velocity for a circular orbit is 100. Suppose I increase the radius to 225, what happens to the velocity to keep the orbit circular?
Vnew / Vold = √(Rold/Rnew) BE CAREFUL!!
Vnew = Vold x √(Rold/Rnew)
So Vnew = 100 x √(200/225) = 94.28 When the orbit gets bigger, then velocity gets smaller!
So now, we can try to do the subsections:
a. We check that our setup is right (both orbits are circular). Then we increase the Moon's orbit relative to the Earth by, say, 10% or so, and use our eqn above to calculate the new relative (relative, relative) velocity of the Moon.
THIS IS VERY TRICKY. I WILL HAVE TO SIT DOWN WITH YOU AND SHOW YOU, OTHERWISE YOU WILL INCREASE THE RADIUS OF THE MOON'S ORBIT AROUND THE EARTH BUT THEN REDUCE THE RELATIVE VELOCITY BY THE WRONG AMOUNT. IF THAT HAPPENS, THE MOON WILL GO OUT OF ORBIT - NOT BECAUSE OF THE SUN - BUT BECAUSE THE NEW ORBITAL VELOCITY IS WRONG. MAYBE I CAN WRITE YOU AN EXCEL PROGRAM TO DO THE CALCULATIONS! IT'S TOO HARD TO DO THIS WITH A CALCULATOR.
Then we check that we still have a circular orbit. If so, the sun is not affecting us yet. Go another 10% or so, until we see the Moon starting to be pulled in (the inside part of the orbit moves farther from the Earth). RECORD this orbital radius as the first orbital size where the Sun's influence begins to be felt.
b. Keep increasing the radius, following the velocity formula, until Moon is pulled away from the Earth. RECORD this distance as the first size where the Earth is pulled out of orbit.
c. Keep increasing the orbit until something crazy happens. RECORD this orbit.
d. Do what it says in the BIG PICTURE.
ASHUR:I. BIG PICTURE:
Your goal is to use the Orbit Simulator (My Solar System 2.04) study what happens when a lighter body is in the gravitational field (or influence) of a heavier body. In particular you will study a planet that starts with no radial component but only has a tangential component to its velocity around the Sun. If we think of all possible values for the planet's starting velocity (which we will call Vy), we can identify nine special velocities or velocity domains.
a. Use the Orbit Simulator to study the various special velocities and different domains that occur when we increase the tangential velocity from zero to the maximum possible velocity.
b. Use the Simulator to check the Law of Conservation of Angular Momentum (LCAM)
c. Anything else I want to add about this subject.
II. SPECIFICS:
TURN ON TRACES AND SYSTEM CENTERED.
a. The special values of the tangential velocity are:
1) V1: this means that Vy = 0. And since Vx also = 0, then the total velocity V (or Vplanet) also = 0. ( And V1 = 0) Thus the planet will move in a straight line towards to the Sun and crash into it.
2) V2: this means that Vy = all speeds higher than V1 (0) but less than V3 (orbit speed). Thus V2 represents a lot of different speeds and we call this a DOMAIN. We can call it the V2 Domain or the Crash Domain, since all speeds in this domain will result in the planet crashing into the Sun.
3) V3: this is the minimum value of Vy whereby the planet will just miss the Sun and go into an elliptical orbit.
4) V4: this means that Vy = all speeds between V3 (orbit speed) and V5 (circular orbit).
We call this the Inferior Elliptical Orbit Domain, because all orbits are ellipses where the far end of the orbit is closer to the Sun than the starting point.
5) V5: this value of Vy gives a circular orbit. NOW, you can use the Vtang equation at the top of the BLOG (see EXAMPLE in Savannah's section above). So you can now calculate the exact speed for a perfect circle (not just approximate it by trial and error). This will really help you!
6) V6: this means that Vy = all speeds between V5 (circular orbit) and V7 (parabolic orbit).
We call this the Superior Elliptical Orbit Domain, because all orbits are ellipses where the far end of the orbit is farther from the Sun than the starting point.
7) V7: this value of Vy gives a parabolic orbit. The best way to approximate this is to turn on TRACES and GRID and SYSTEM CENTERED. If the Sun is making an ellipse or any curved line (may take a long time to see -- be patient), then the Moon is still in an elliptical orbit. When the Sun gradually goes off the screen in what looks like a straight line, you know that the Moon (and the Sun) are in parabolic or hyperbolic (Google, Google) orbits. The parabolic one is the first one you reach, all higher speeds are hyperbolic). *********** I need to think more about this. I want to make sure that the parabola is not reached WHILE THE LINES ARE STILL CURVED. Let's hold back a little here...............................
8) V8: this means that Vy = all speeds between V7 (parabolic orbit) and V9 (maximum possible speed = speed of light) .We call this the Hyperbolic Orbit Domain, because all orbits are hyperbolas (I think!!!) CHECK that they are not parabolas.
9) V9: this value of Vy is the maximum possible speed = the speed of light (186,000 miles/s or 300,000 km/s !!) According to Einstein's Special Theory of Relativity, nothing can travel faster than light!!
So, what do you actually have to do?
(i) Set up the Simulator:
Don't worry about trying to use the actual mass of the Sun. Start with Msun = 1000. Then set the mass of the planet (Earth) so that 1000/Mearth = (true Msun) / (true Mearth). In other words Mearth (in simulator) = 1000 x (true Mearth)/(true Msun). You can can find the true masses using Google (make sure both are in the same units). The Mearth you get will be tiny. If it's too small, you can just use Mearth = 1 (or 0.1) to make it more visible on the screen. As long as it is very small compared with the Sun, all your Vy values will still be correct!
Choose the distance of the Earth to the Sun = 250. This will make the circular orbit the size of the computer screen in the vertical direction. In order to center everything and fill the screen, you should start out with the numbers below and you can change later if necessary.
M x y Vx Vy
1000 -100 -50 0 0 Sun
?? 150 -50 0 ?? Earth
- Turn the grid, traces, and system centering ON. Use the maximum accuracy setting on the Simulator.
(ii) Find the critical values of Vy Earth
- Find all of the critical speeds (V3, V5 and V7) using the Simulator. This goes into the Simulator by changing the Vy of the Earth above. Start at zero, and systematically increase Vy Earth by, say, adding 10 every time. Suppose you got a crash at Vy = 230, but no crash at Vy = 240. Then you go back and try 231, 232, 233, etc. until you find V3.
To get V5, you keep increasing by 10 until you go from an inferior ellipse to a superior ellipse. Read about how to determine exact x, y, Vx, Vy, values by stopping the program and using the cursor (also read Savannah's instructions about this)
Put actual values of masses and experiments, etc!!!!!!!!!!!!!!!!!!!
It's going to be hard to find V7 using the Simulator, because you can't tell when the shape of the orbit changes from an ellipse (closed) to a parabola (open) on the far end, because the far end of the orbit is off the screen. Probably, the best you can do is to approximate it as the speed where the Earth never comes back even after you wait OVERNIGHT (MAKE SURE THE TRACE IS TURNED ON). This will be an excellent approximation!
b. Check the LCAM. Whenever the planet's velocity is perpendicular (at 90 degrees) to the line between the planet and the Sun, it is easy to check the LCAM:
J = angular momentum = mass x velocity x distance.
NOTE:- for an elliptical orbit this only happens at two points (the starting and the far point) [for a circular orbit it happens everywhere!]
Calculate J at the start and far points of one of your inferior ellipses. See if they are the same. Repeat for a few other inerior ellipses. Repeat for several superior ellipses.
c. Whatever you want to do.
(Fri Apr 11) -- HERE'S THE HOMEWORK 9 (DUE APRIL 24)
W A R N I N G !!!! Do NOT use a GLASS container for the homework!! USE A CARDBOARD OR PLASTIC CUP. -- I tried it with a glass jar, and the glass broke apart in my hands while I was shaking it !! Thank G-d, I was not cut.
https://sites.google.com/site/yhbscience/Class15.Homework9--FourCoinToss.docx?attredirects=0&d=1
(Sun Apr 6) -- Misty rain all day but we finished our running events . Tried to send you some stuff but Blogspot doesn't like the hotel computer's browser and we don't have our laptops, so we will have to wait till we get home on Tues. Sorry. :(
(Sat Apr 5) -- on iPhone. Had wireless issues in our room yesterday. Moving to a new room!!
NO CLASSES ON APR 3, 10 OR 17 !!
PLEASE KEEP CHECKING, BECAUSE:
- I will provide each person individual comments about the Science Fair Project Proposal you submitted
- I will provide more suggestions and details about how to present your research results using EXCEL GRAPHS
- The "First Results on My Science Fair Project" will be due on April 24
- I may assign easy homework that will be due on April 24
- I may have other information or notices for you
Please spend as much time as you can completing and adjusting your experimental apparatus or (if you're doing an orbit project) becoming more familiar with My Solar System 2.04 -- and then start taking as much of the data as you can for your "First Results on My Science Fair Project" (I will be telling you how to write the report: the structure will be the same as the one in your Project Proposal).
REMEMBER, THERE WILL ONLY BE THREE WEEKS FROM WHEN I SEE YOUR FIRST RESULTS TO WHEN THE JUDGES, PARENTS, OTHER MEMBERS AND VISITORS WILL BE SEEING YOUR FINAL RESULTS. I know you will make us all proud of you. I think you are all going to do great jobs!!!
I will miss you all for the next three weeks!!! Chag Sameach!!
Mr. Frank
