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Calculating and Displaying Gravitational Force
OBJECTIVES
In a future program you will model the motion of a spacecraft traveling around a planet and a moon, which
will require repeated calculation of gravitational forces.
The objectives of the current exercise are:
- to learn how to instruct VPython to calculate the vector gravitational force on an object due to
another object;
- to write the instructions in a symbolic form that can later be used in an iterative calculation to
predict the motion of the spacecraft even though the gravitational force is changing in magnitude
and direction; and
- to create and scale arrows to represent the gravitational force on an object.
TIME
You should plan to finish this activity in 40 minutes or less.
GROUP ROLES
Before you begin, you should agree on the responsibilities of the Manager and the Recorder for this activity.
(Your TA may assign those roles.)
PLANNING
On a whiteboard or a piece of paper, draw a diagram like the one below. Each numbered location represents
the position of a different spacecraft. (A single spacecraft near a planet would not move in a straight line.)
- At each numbered location, draw an arrow representing the gravitational force on a spacecraft at
that location, due to the planet.
- Make sure the direction of your arrows is correct, and that the length of the arrow is proportional to
the magnitude of the quantity it represents.
1 2 3 4 5
planet
Look at your diagram. Does it make sense? Compare your work to that of another group.
REVIEW: COMPUTER PROGRAM ORGANIZATION
- A computer program consists of a sequence of instructions.
- The computer carries out the instructions one by one, in the order in which they appear, and stops
when it reaches the end.
- Each instruction must be entered exactly correctly (as if it were an instruction to your calculator).
- If the computer encounters an error in an instruction (such as a typing error), it will stop running
and print a red error message.
A typical program has four sections:
1. Initial setup statements
2. Definitions of constants (if needed)
3. Creation of objects and specification of initial conditions
4. Calculations
1 Setup statements
Using VIDLE for VPython, create a new file and save it to your own space. Make sure to
add “.py” to the file name.
Enter the following two statements in the VIDLE editor window:
from future import division, print function
from visual import *
Both Python and VPython are undergoing continuous improvement. For example, before Python version 3.0, 1/2 was truncated to zero, but beginning with Python 3.0, 1/2 means 0.5. Also, starting with Python 3.0, print statements must be in the form print(’hello’) rather than the older form print ’hello’. The first statement in your program tells older versions of Python to treat 1/2 as 0.5, and makes the new form of print statements work on older versions of Python. You don’t need this statement if you are using Python 3.0 or later, but it doesn’t hurt, because it is simply ignored by later versions of Python.
2 Constants
Since you will be calculating a gravitational force, you will need the constant G (without units):
G = 6.67e-
You should also put the masses of the planet and the spacecraft in this section. Define constants to represent:
- the mass of the spacecraft (15e3 kg) (you could call this mcraft, for example)
- the mass of the planet (6e24 kg) (you might call it mplanet)
- The relative position vector You know the vector positions of the two objects, which are craft.pos and planet.pos. - Add a statement to your program to calculate a vector r that points from the planet to the spacecraft, representing the vector ~r. Think about what you know about calculating relative position vectors between two objects. Don’t use any numbers, just symbols. The point is for VPython to do the numerical calculations for a variety of positions of the planet and the spacecraft.
r =?
- The magnitude of the relative position vector In order to instruct the computer to calculate the magnitude |~r| of the relative position vector, you need to know the following: - The components of a vector in VPython are given by its x, y, and z attributes. For example, craft.pos.x is the x component of the position of the spacecraft, and r.y is the y component of the vector r that you created. - To calculate the square of a quantity, use two asterisks. For example, 3**2 gives 9. - To calculate the square root of a quantity, use the function sqrt. For example, sqrt(9) is 3.
Knowing these features of VPython,
- Add a statement to your program to calculate the magnitude |~r| of the relative position vector:
rmag =?
- The magnitude of the gravitational force Using the quantity |~r| that you just calculated (rmag), and the masses mcraft and mplanet (or whatever you chose to call them), - Add a statement to your program to calculate the magnitude |~Fgrav| of the gravitational force. You should define G near the start of your program. Use this symbol G rather than a number in calculating |~Fgrav|:
Fmag =?
- The unit vector
- You know both the vector ~r (r) and its magnitude (rmag). Use these quantities to add a statement to your program to calculate the unit vector ˆr (pronounced “r-hat”):
rhat =?
- The gravitational force as a vector Now you have everything you need to be able to calculate the gravitational force that the planet exerts on the spacecraft: ~Fgrav = |~Fgrav|Fˆgrav = −|~Fgrav| ˆr. - Add a statement to your program to calculate the net vector force acting on the spacecraft. We’ll assume that the planet is the only object near enough to have a significant effect on the spacecraft.
Fnet =?
- Add a print statement to your program to show the components of the net force:
print(’Fnet =’, Fnet)
- Run the program and ask yourself whether the signs of the force components makes sense (then make changes to your program if necessary).
Visualizing the force vector with an arrow
Having calculated the gravitational force vector, we want to visualize it by displaying an arrow representing the vector.
- Add a statement to your program to create a yellow arrow on the screen that represents the gravitational force acting on the spacecraft (object 2). Choose pos and axis attributes so that the tail of the arrow is on the spacecraft and the arrow points toward the planet. Do NOT use numbers! Write the values for the pos and axis attributes symbolically, in terms of the quantities you have already calculated.
If you have calculated the gravitational force correctly, you probably don’t see an arrow! The force is so small you have to scale it up to be able to see it, by multiplying by a scalar factor. How do you pick a scale factor?
6 Scalefactors
If you haven’t already done so, watch VPython Instructional Videos: 5. Scalefactors at http://www.youtube.com/VPythonVideos.
Use what you have learned from this video to determine an appropriate scalefactor for this force arrow. Remember to define the scalefactor in the # Constants section of your code, and use it later in the program where the force arrow is defined. The same scalefactor should be used for all force arrows.
The force at other positions
In order to see how the force on the spacecraft would vary at different positions, add code to create 4 other spacecraft (spheres) at the locations listed below, and to calculate and display the force on each spacecraft. 〈− 6 .5e7, 6 .5e7, 0 〉 m 〈 0 , 6 .5e7, 0 〉 m 〈 6 .5e7, 6 .5e7, 0 〉 m 〈13e7, 6 .5e7, 0 〉 m
- Run your program so you see your display showing the spacecraft and the planet.
- If the arrows get too big, reduce your scalar Fscale so that the arrows don’t run into each other.
- You must use the same scalefactor with all of your arrows, so that all the force vectors are consistent.
Look at your display. Does it make sense? Does it look like the diagram you drew earlier? Compare your result to that of another group.
The Recorder should turn in the program to eLC, and make sure everyone has a copy of the program.
