class 051116

18-34-375

11-09-453

Quiz:

What were your maximum and minimum KE for a 1 kg mass in your 'best' attempt at a circular orbit at the distance you were specifically assigned?

How much PE change would be observed for that orbit?

The following assignments should be completed by next class:

• Chapter 3:
• 121:  1-4, 9, 13, 17, 18, 22, 25
• 201:  1-5, 9, 12, 13, 16, 19, 20, 22, 25, 30, 33
• Chapter 4
• 121: 1-4, 19-22, 25, 31, 36-39
• 201: 21, 23, 25, 26, 29, 31, 34, 39,44,47,49,55,58
• Intro Prob Sets
• Set 6, #7-8
• Set 8, #'s 1-6
• Chapter 5
• 121: 1-4, 9, 28-29, 43, 44
• 201: 3, 6, 9, 14, 18, 26, 30, 31, 36, 40, 43, 46, 50
• 201:  Chapter 7: 47, 48, 50

Quiz: 

By setting F = G m m_Earth / r^2 equal to the centripetal force m v^2 / r, determine the velocity required for an orbit at your given radius.

(G = 6.667 * 10^-11 N m^2 / kg^2, Earth radius is 6380 km, Earth mass is 5.98 * 10^24 kg)

How does this compare with the velocity you observed in the simulation for the circular orbit at your given radius?

How much variation was there in your orbital velocity, and how much variation in your distance from the center of the Earth, when you constructed this orbit?  *****

Go to the simulation and set up your orbit with your calculated values, using the default time factor 1000, and see what the max and min orbital velocities are:

• Observe vMax and vMin

Also observe your max and min orbital radii:

• Observe rMax and rMin

Slow the simulation down to Time Factor 100 and refine your observations.

How much difference is there in the KE of a 1 kg object between its max and min velocities?

How much difference is there in the gravitational PE of a 1 kg object between its max and min orbital distances, based on max and min orbital distances?

If you were to accidentally give a satellite a velocity which is 1% too great, then what maximum and minimum orbital velocity would you expect?  How much PE change would you expect in your orbit?

Take the quiz and submit it in the usual manner.  If necessary see notes from 051109.

• What impulse did you use, and what maximum and minimum speeds do you observe over an orbit with your best attempt to achieve a circular orbit at 1.5 Earth radii?
• How much KE change does that indicate for a 100 kg satellite?
• How much would the distance from the center of the planet have to change to achieve a PE change equal to your KE change?
• By how much do you therefore conclude your orbital 'altitude' changed?

• Answer the same questions for 2 Earth radii.
• Answer once more for the orbit you were assigned to investigate.

Now set up an orbit that starts at 2 Earth radii and 'skims' the surface of the Earth.

• note the max and min distances (distance is in the box just to the right of the 'circle radius' box)
• Take data sufficient to answer the question:  What are the max and min KE's of a 1 kg mass in this orbit?

Repeat for the orbital distance you were assigned.

For each orbit, what is the PE change from max to min distance?

Then proceed to the gravitational simulation:

Instructions for gravitational simulation:

The program will be found at the Sup Study ... site under Course Documents > Downloads > Physics I.   Download and/or run it in order to see the buttons and boxes described here:

The array of boxes and buttons at the right side of the screen contains information about the planet, the satellite and the time scale of the simulation.

Planet mass is the mass of the planet in multiples of the mass of the Earth. The default values assume that the planet is Earth, so the default planet mass is 1. You can enter any planet mass you wish. For example the Moon has a mass about 0.0123 times that of the Earth, the Sun has a mass which is about 340,000 times that of the Earth. If you wanted to simulate and orbit around the Sun or the Moon you would enter 0.0123 or 340,000 in this box.

Planet radius is given in multiples of the radius of the Earth. Since the default planet is Earth the planet radius has default value 1. If you wanted to simulate the Moon you might enter 0.26, which represents the fact that the Moon has a radius about 0.26 times that of the Earth. If you what and to simulate the Sun you might enter 1100, since the Sun has a radius about 1100 times that of the Earth.

Time factor is the factor but which the simulation is speed up. The default value of the time factor is 1,000, which means that everything runs about 1000 times faster than actual. This means, for example, that a low-Earth orbit will take place in about six seconds rather than the actual approximate time of 6,000 seconds.

Screen scale is the distance from the center of the picture to the edges, in Earth radii. The default value is 3, which works well for low and moderate Earth orbits. However if you are trying to investigate orbits which move further than 3 Earth radii from the center of the planet you need to adjust the screen scale accordingly or the satellite or projectile might not show up on the screen.

Initial distance is the distance of your satellite or projectile from the center of the Earth. This distance is set to 1.02, which is around the minimum distance at which it is possible to orbit at least a few times without encountering significant atmosphere. You can set it for any distance you wish. [ Note that this simulation ignores atmospheric drag and will work just fine for orbits inside the atmosphere. In fact it ignores any sort of interference at all so orbits low enough to encounter mountains will work just find here. Not only that, but this program implicitly assumes that all the mass of the planet is concentrated at its center and even allows orbits inside the surface of the planet. The only problem arises if you get very very close to the center of the planet, in which case the simulation breaks down and spits the satellite or projectile out at very high velocity in a straight line (which is just an anomaly of the simulation and would not really happen in any circumstance). ]

Initial angular position is the angle in radians made with the positive x axis (which is directed toward the right, as is standard for many applications) by a line segment from the center of the planet to the initial position of your satellite or projectile. Note that there are approximately six radians (actually 2 pi, closer to 6.28 radians) around a circle.

The impulse of the 'burn' is actually impulse per kg. Recall that the impulse of a force acting on an object, which is the product F `dt of the average force and time interval during which it acts, gives the change in the momentum of the object. It follows that the impulse kg is in fact the change in the velocity of the object. Note that we are here assuming that the 'burn' does not significantly change the mass of the object; this is not always the case with actual satellites and certainly is not the case with a rocket boosting a satellite into orbit. The default impulse is 8000, which will give the satellite or projectile a velocity of 8000 m/s, a bit in excess of the velocity required to achieve a circular low-Earth orbit.

To deliver an impulse you first choose the magnitude of the impulse, then click on the Forward, Backward, To Right or To Left button.

The direction of the initial impulse depends on the goal of the simulation. If we wish to achieve a circular orbit then because of the geometry of a circle (at every point the circle is perpendicular to the radial line from the center to that point) the impulse must be at a right angle to the initial angular position; otherwise circularity is in the first instant violated. Since a right angle is 1/4 of the angle around a circle, the right angle is 2 pi / 4 radians = pi / 2 radians, or approximately 1.57 radians. On the other hand if we wish to shoot a projectile 'straight up' from the surface of the Earth we must 'fire' it in the direction directly away from the center of the planet, which means that we must 'fire' along the radial line from the center to our starting point. This means that the initial direction must be the same as the initial angular position.

Clock time is displayed as the simulation runs. Clock time is the actual simulation time since the 'run' started.

Circle radius is the radius of a circular orbit you might be trying to achieve, in Earth radii.  If the number in this box is not zero then when you click Run Simulation the program will place a red circle of this radius, centered at the center of the planet, on the screen.

Realtime interval is the 'real world' time in minutes since the simulation began.

Speed is the speed of the satellite or projectile in meters/second.

The Run Simulation button is used to begin the simulation. When the simulation is begun the planet will show in blue the center of the screen and the satellite will show in white.

The first eight buttons in the rightmost column are used to deliver an impulse to the satellite or projectile.

The top four buttons deliver the impulse forward, i.e., in the direction of velocity of the object, or backward in the direction directly opposite that of the object's velocity, or to the right (defined to be at a right angle to the right as perceived by an individual facing the direction of motion) or to the left.

The default impulse is 0. The magnitude of the impulse is chosen by clicking one of the next four buttons. Once clicked this impulse is 'set' until another impulse button is clicked, so that it is possible with successive clicks to deliver any reasonable chosen impulse.

The Pause Simulation button, as you might expect, allows you to pause the simulation.  There are two reasons you might want to do this.   One is to simply have a look at the numbers in the boxes, another might be to change the numbers and restart the simulation without erasing the existing screen.

The Continue button will continue the program after a pause; if you haven't changed anything in the boxes the program simply picks up where it left off.

The Run (don't clear) button restarts the simulation after a pause, without erasing the existing screen.

 

The 2d planet button will create a second planet (e.g., the Moon), but first you have to go down to text boxes at the bottom of this column (just above the Apply button) and enter the necessary information. The default message in each box will tell you what you need to know, but those messages are repeated here.  Note that only the first word or two of each message actually shows in the box.

• The first of the two text boxes contains the message 'second planet mass as multiple of Earth mass (Moon = .0123)', telling you to enter the mass of the second planet, for example .0123 if you mean the Moon. Enter just the number with no punctuation (except a decimal place if required) and no letters or words.
• The second box contains the message 'second planet dist as multiple of Earth radius (Moon = 60.2)', meaning that you should enter the number 60.2 if you want to simulate the Moon, or any appropriate number if you want to set up some other situation.

Be sure you have the correct numerical information in these boxes.  If you don't the program is likely to crash. 

After entering this information you can click on the 2d planet button, which will give you a message telling you that the 2d planet has been created. If the simulation is already running the second planet should appear, provided the screen scale can accommodate it. If not, or if you wish to make other changes, you may change the screen scale and any other information you wish then click on Run Simulation.

Suggestions for first-time use:

Without changing anything click on Run Simulation. You will see a blue circle representing the Earth and a moving white dot representing the successive positions of a satellite. The satellite completes its orbit 1000 times as fast as an actual satellite, due to the default time factor.

While the simulation is running click on 'Impulse 100' then on Forward and see what happens to the orbit.

Click on Forward a few more times and see what happens to the orbit.

Click on 'Impulse 1000' then on 'forward' and see what happens. Then click on 'right' and on 'left' and see what happens.

Any time the simulation gets out of control you can started over by clicking on Run Simulation. See if you can figure out the most efficient way to move from the default orbit into a 'higher' circular orbit. 

Do this: 

• See how efficiently (i.e., in how few clicks) you can get the satellite to the edge of the screen using 100 (kg m/s) / kg impulses, starting from the default orbit.

Start the simulation over:

• If you start from the basic orbit and supply a 1000 (kg m/s) / kg impulse, in such a way as to maximize the KE change, what happens to the maximum and minimum KE of the projectile?

• If you supply a second 1000 (kg  / s) / kg impulse, in such a way as to maximize the KE change, what happens to the max and min KE of the projectile?

• If you supply a third 1000 (kg  / s) / kg impulse, in such a way as to maximize the KE change, what happens to the max and min KE of the projectile?

• What do you think would happen to the max and min KE if you supplied a fourth 1000 (kg m/s) / kg impulse?

Change the initial impulse:

• Note that the default impulse of 8000, which gives the satellite an initial velocity of 8000 m/s, gives an orbit which is not quite circular. Change the 8000 to 9000 and click the Run Simulation button to see how the orbit changes as a result of the greater initial velocity. Then change the impulse to 7000 and click the Run Simulation button to see what happens. Then see if you can find the initial velocity that gives you a good circular orbit.

Achieve circular orbits at 1.5 and 2 Earth radii:

• Change the init distance number to 1.5 and click on Run Simulation.  What happens to the shape of the orbit? 

  Change the initial impulse and click again on Run Simulation. 

  Use trial and error to answer the following:  What initial impulse is required at 1.5 Earth radii to achieve the circular orbit?

   

• Repeat for init distance equal to 2 Earth radii.

Put the satellite into a circular orbit and investigate:

• What immediate effect does a single impulse to the right or left have on the speed of the satellite moving in a circular orbit?

'Shoot' a projectile 'straight up' from the surface:  To shoot 'straight up' you shoot straight out along a radial line (see direction of the initial impulse in the description of the program above). 

• First set the number in the Circle Radius box to 2. 
• To position the projectile on the surface set Initial Distance at 1, to place the projectile at 1 Earth radius from the center.
• Set Initial Angular Position to 1, which will place the projectile at the 1-radian position.
• Set Direction of Initial Impulse also to 1, which will 'shoot' the projectile in the 1 radian direction.  This will 'shoot' the projectile straight out from the Earth, which from the perspective of an observer at that position will appear to be 'straight up'.
• Reduce the Initial Impulse from its default value of 8000 to 6000.
• Click on Run Simulation.  See how far out from the planet the projectile goes before 'falling back'.

 

• Repeat for Initial Angular Positions of 2, 3, 4, 5 and 6 radians, with initial impulses of 4000, 5000, 7000, 8000 and 9000.  Be sure in every case to set the Direction of Initial Impulse so that the projectile 'shoots' straight away from the Earth.

 

• See which initial impulse gets the projectile closest to the radius-2 red circle.  Estimate the impulse you would need to exactly reach that circle without 'overshooting' it and test your estimate.

The Experiment:

To start with you will determine the velocity required for a circular orbit at a distance of 1.2 Earth radii. 

Set Initial Distance at 1.2.  Leave the remaining settings as they are and click Run Simulation.

Determine from the shape of the resulting orbit whether the initial velocity is too high or too low and change the number in the Initial Velocity box to a value you believe will bring the orbit closer to a circular shape.  The click Run Simulation.

Repeat until you have achieved a good circular orbit and record the velocity.

Now find the velocity with which a projectile would have to be 'shot' from the surface of the Earth, ignoring air resistance, to get 'up' to the altitude of the orbit you have just created. 

• Pick a reasonable initial velocity. 
• Position the initial angular position at 0 radians and make an appropriate selection for the direction of the initial velocity (see the initial introduction to the program under Suggestions for first-time use above). 
• Set the Time Factor to about 200 in order to give yourself time to judge when the object has reached is maximum altitude.
• Pause the simulation before the object has much time to fall back to Earth because if the projectile falls back to Earth then through to the center you might like the result but it won't help you.

Repeat but change your initial angular position to 1 radian (and of course change the initial direction of the 'shot' accordingly), and adjust your velocity to get closer to your goal.  Adjust the time of the simulation to allow the projectile to stop moving outward and begin falling back. Continue changing the intial angular position and your angular velocity until you manage to just reach the circular orbit you created in the first part of the experiment.

Repeat this procedure for the orbital radius you have been specifically assigned.

That is, determine the velocity necessary to maintain that orbit and the velocity required to achieve the altitude of that orbit.

If you have not been assigned an orbital radius use 1.2 + .1 * (number of letters in your first name).

Now achieve an elliptical orbit that just skims the surface of the Earth.

Starting from a position at 2 Earth radii from the center, adjust your initial velocity until you have an orbit that at its 'lowest' point just touches the Earth's surface.  Observe everything you can about the motion of the satellite in this orbit, including the maximum velocity.

Analysis

For each radius you investigated determine, by setting centripetal acceleration equal to gravitational acceleration, the 'actual' velocity for that circular orbit.

Compare with the velocities you obtained.

According to your observations:

• How much KE per kg is required in your initial 'shot' to get to the 'altitude' of each orbit?  How much does the PE of the projectile change as it 'climbs' to each orbit?
• What is the KE per kg in each orbit?
• By how much does the PE therefore change between the two circular orbits you investigated?
• By how much does the KE change between the two circular orbits you investigated?
• How are the PE and KE changes related?

According to theory:

• How much KE per kg is required in your initial 'shot' to get to the 'altitude' of each orbit?  How much does the PE of the projectile change as it 'climbs' to each orbit?
• What is the KE per kg in each orbit?
• By how much does the PE therefore change between the two circular orbits you investigated?
• By how much does the KE change between the two circular orbits you investigated?
• How are the PE and KE changes related?

For the elliptical orbit, based on your observations:

• How does the velocity of the satellite at the 'initial' position 2 Earth radii from the center compare to the velocity as it 'skims' the surface?
• What happens to the KE between the extree furthest distance from center and the closest approach to center?
• What happens to the PE between the extreme furthest distance from center and the closest approach to center?
• Are your observations consistent with the conservation of energy?

For the elliptical orbit, based on theory:

• By how much should the KE of the satellite change from 'initial' position 2 Earth radii from the center and the point where it 'skims' the surface?
• What therefore should be the velocity of the object as it 'skims' the surface, assuming that the initial velocity you gave it was accurate?  How does this velocity compare to the velocity you observed?

Speculate on what sorts of strategies are required to get a Space Shuttle into a circular orbit at an 'altitude' of 400 km.  Keep in mind that rocket fuel doesn't carry enough energy to even get its own mass into orbit.

Speculate on what strategies are required to get a spacecraft to the Moon, which is about 60 Earth radii away, and back.  Note that you can if you wish experiment with this situation by following the instructions for 2d Planet; just be careful to set your original parameters so that the Moon is visible.