Class 091130

The gravitational simulation program is at

http://vhcc2.vhcc.edu/ph1fall9/dos_simulations/grav_field_simulation_v1.exe

When you run the program a form appears with boxes in which you can specify

There are buttons to deliver impulses, run simulation, pause simulation, continue, run (don't clear).

Notice the initial settings:

You can skip the next several paragraphs for now.  They describe the different settings in more detail.

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.

`q001.  For now, do the following:

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You've probably noticed that the simulation isn't realistic in the sense that an object can orbit inside the Earth.  Due to the nonconservative forces that occur inside the atmosphere, then inside the Earth itself, this is clearly not realistic.  Think of the object as orbiting a point mass located at the center of the Earth, with mass equal to that of the Earth.

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This occurs when the initial impulse directed at an angle which lies somewhere between 1.1 and 1.3 radians.

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Pendulum motion

The synopsis below summarizes the pictures and equations presented during class:

If a pendulum of length L and mass m is displaced a horizontal distance x from its equilibrium position, then the pendulum mass is subject to the downward force m g of gravity, and the tension in the pendulum string. 

T_x = T cos(theta) = T * (- x / L) = - T x / L

and, since T = m g, that

T_x = - m g x / L.

Again, this is fairly easy to follow with the diagrams and pictures you should have in your notes.  Your text also shows this.

T_x is, as mentioned, the net force acting on the pendulum mass. 

The equation T_x = - m g x / L is therefore of the form F_net = - k x, with k = m g / L.

`q002.  omega = sqrt(k / m), where k is the constant and F_net = - k x. 

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`q003.  What is the period of a pendulum of mass .8 kg, with force constant k = 2 N / m?

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`q004.  Here is a list of definitions you should know.  Give your best definition of each, and explain as best you can what each means, and give an example of how each might be used.  Give examples of how each has been applied to experimental situations:

average rate

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average velocity

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average acceleration

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force

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KE

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PE

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impulse

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momentum

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gravitational force

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moment of inertia

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torque

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angular position

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angular velocity

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angular acceleration

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simple harmonic motion

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theorems/laws thread

`q005.  Formulas you just need to know.  Identify each formula and tell what every symbol means, as well as what the formula means, and give examples of how it could be used.  Give examples of how each has been applied to experimental situations

a_cent = v^2 / r

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omega = sqrt(k/m)

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F = G M m / r^2

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PE = -G M m / r

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`q006.  Formulas you could be excused for memorizing.  Identify each formula and tell what every symbol means, as well as what the formula means, and give examples of how it could be used.  Give examples of how each has been applied to experimental situations.  Give your excuse for memorizing each.

v = sqrt( G M / r)

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PE_elastic = 1/2 k x^2

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`ds = v0 `dt + .5 a `dt^2

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KE = 1/2 m v^2

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vf^2 = v0^2 + 2 a `ds

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F_ave * `dt = `d ( m v)

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`q007.  Formulas there is no excuse whatsoever for memorizing, even though you might end up doing so.  Identify each formula and tell what every symbol means, as well as what the formula means, and give examples of how it could be used.  Give examples of how each has been applied to experimental situations.  Explain why there is no excuse for memorizing the formula, even though you're probably going to do so anyway:

`ds = (v0 + vf) / 2 * `dt

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This is just a rearranegment of the definition of average velocity.

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vf = v0 + a `dt

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This is just a rearrangement of the definition of average acceleration.

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omega_f^2 = omega_0^2 + 2 alpha `dTheta

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This is a simple rearrangment of the fourth equation of uniformly accelerated motion into rotational terms.

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PE = m g h

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This formula is unnecessary and misleading.  It is unnecessary because `dPE is equal and opposite to the work done by gravitational force, which is easy enough to calculate in situations when gravitational acceleration is uniform.  It is misleading because students universally misuse it when analyzing situations in which gravitational acceleration is not uniform (e.g., for satellites and astronomical bodies); it is even commonly used for situations involving elastic PE.

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`dW_NC_ON = `dPE + `dKE

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Anyone who has applied conservation of energy to dozens of situations will know this very well.  Anyone who hasn't is not likely to apply it correctly on a test or in analyzing a lab situation.

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`dW_net_ON = `dKE

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See the note on the preceding formula.

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Homework:

Your label for this assignment: 

ic_class_091130

Copy and paste this label into the form.

Answer the questions posed above.

You have already seen most of the ideas in the qa's and Introductory Problem Set mentioned below.  If you work through these documents as assigned, you will get plenty of practice and should develop good expertise with these concepts.

Do qa's #32 on moments of inertia and 33 on rotational KE and angular momentum.

qa's 34 and 35, on SHM, are also listed below:

Introductory Problem Set 9 consists of 17 problems on simple harmonic motion.  You will be expected to work through these problems by the first of next week.  http://vhmthphy.vhcc.edu/ph1introsets/default.htm .