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Calculating gravitational force with VPython
Every program that models the motion of physical objects has two main parts:
- Before the loop : The first part of the program tells the computer to: a. Create numerical values for constants we might need b. Create 3D objects c. Give them initial positions and momenta
- The “while” loop : The second part of the program, the loop, contains the statements that the computer reads to tell it how to increment the position of the objects over and over again, making them move on screen. These statements tell the computer: a. How to calculate the net force on the objects (the subject of this lab activity) b. How to calculate the new momentum of each object, using the net force and the momentum principle c. How to find the new positions of the objects, using the momenta
In your fan cart program you studied motion with a constant force. But in the orbit of a planet around a star, or two stars orbiting each other (a “binary star”), the gravitational forces on the objects may change continually in magnitude and direction.
In this lab activity you will learn how to make VPython calculate gravitational forces, as vectors. Knowing how to do this will enable you to write programs that predict the motion of objects in space, for if you can calculate the net force on an object, you can update its momentum just as you did in the fan cart program. Knowing the momentum lets you update the position. Repeating these momentum and position updates will trace out a complete trajectory.
1 Creating the objects
The NEAR spacecraft flew past the asteroid Mathilde (for details, see Problem 2.11, p. 70 of the textbook). Create spheres representing Mathilde and the spacecraft: Mathilde
1200 km
NEAR spacecraft 10 km/s
from visual import * craft = sphere(pos=vector(-1.2e6,0.6e6,0), radius=1e4, color=color.cyan) Mathilde = sphere(pos=vector(0,-0.6e6,0), radius=5e4, color=color.white)
- Run the program to see Mathilde, with the spacecraft at the upper left. The size of Mathilde is about right, but the size of the spacecraft has been greatly exaggerated so that you can see it.
2 Giving the objects physical attributes
In addition to geometrical attributes such as pos and radius , you can give objects physical attributes such as mass. The mass of the NEAR spacecraft was 805 kg; add the following statement to your program:
craft.m = 805
The mass of Mathilde was estimated from its size as seen in photos taken by the spacecraft, and the density of typical rocks. The density of rocks is about 3000 kg/m^3 , and Mathilde appeared to be about 70 km by 50 km by 50 km in the photos (7e4 m by 5e4 m by 5e4 m). Add a statement that calculates the estimated mass of Mathilde in kilograms. Write out the calculation in the program and let Python do the calculation for you; don’t use your calculator. This lets you see in the program how the mass was calculated.
Mathilde.m =?
3 Calculating the gravitational force on the spacecraft
Two spheres with masses m 1 and m 2 attract each other with a force given by r
r
mm
F G 2 ˆ
12
1 2
grav v
v
= , where
G = 6.7e-11 N·m^2 /kg 2
r 12
v
is a relative position vector pointing from one object to the other
r ˆ^ is a unit vector pointing from one object toward the other
In the following sections you will calculate the gravitational force that the Mathilde asteroid exerts on the NEAR spacecraft, by
• calculating the relative position vector r 12
v
that points from one object to the other
• calculating its magnitude r 12
v
• using r 12
v
to calculate the magnitude of the gravitational force (^2) 12
1 2 grav
r
mm
F G v
v
• calculating the unit vector r ˆ , which gives the direction of the gravitational force
• calculating the vector gravitational force, the product of its magnitude and direction: F grav F grav r ˆ
v v
3.1 Calculate the relative position vector You know the vector positions of the two objects, which are craft.pos and Mathilde.pos.
- Add a statement to your program to calculate a vector r that points from Mathilde to the spacecraft. Think about what you know about calculating relative position vectors between two objects. Don’t use any numbers, just symbols.
r =?
3.2 Calculate the magnitude of the relative position vector
In order to calculate the magnitude r 12
v
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 a square of a quantity, use two asterisks. For example, 32** 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 12
v
of the relative position vector:
rmag =?
3.3 Calculate the magnitude of the gravitational force
Using the quantity r 12
v
that you just calculated ( rmag ), and the masses craft.m and Mathilde.m ,
• Add a statement to your program to calculate the magnitude F grav
v
of the gravitational force. You should define G near the start of your program and use it here.
Fmag =?
3.4 Calculate the direction of the gravitational force You have calculated a vector that points from Mathilde to the spacecraft. The direction of this vector is along the
line of the gravitational forces between the two objects, so if we knew the direction of r 12
v
in the form of its unit
vector r ˆ^ , we could use it to get the direction of the gravitational force.
- Add a statement to your program to display an arrow representing the force that the spacecraft exerts on Mathilde. Put the tail of the arrow on Mathilde. Don’t add any new calculations, and don’t use any numbers, just symbols.
arr2 = arrow(pos=?, axis=?, color=color.yellow)
A question to ponder: In a ten second time interval, how would the change in the momentum of the spacecraft compare to the change in the momentum of Mathilde? How would the change in the velocity of the spacecraft compare to the change in the velocity of Mathilde? How would the total momentum (the momentum of the spacecraft plus the momentum of Mathilde) change during this time interval? Your instructor will ask you about these issues.
CHECKPOINT G1: Ask an instructor to check your work for credit. You can read ahead while you’re waiting to be checked off.
6 The force at other positions
In order to see how the force on the spacecraft would vary at different positions of the spacecraft relative to Mathilde, calculate and display the force on the spacecraft (and on Mathilde) for four other locations.
- Add the following statement to your program, to place the craft at a new position:
craft.pos = vector(-0.6e6,0.6e6,0)
- Copy and paste your previous calculations at the end of the program, and run the program. You should see arrows representing force vectors for two different locations of the spacecraft.
- Once you have seen the two pairs of force vector arrows displayed correctly, repeat these additions to the program for the positions < 0, 0.6e6, 0 >, < 0.6e6, 0.6e6, 0 >, and < 1.2e6, 0.6e6, 0 >.
- 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 Fscale with all of your arrows, so that all the force vectors are consistent with each other.
CHECKPOINT G2: Ask an instructor to check your work for credit. You can read ahead while you’re waiting to be checked off.
7 Playing around
You might like to display the force arrows for some additional positions of the spacecraft.
In order to emphasize the physics rather than programming techniques, we had you make multiple copies of the calculational statements, which is clunky, and not feasible if there are hundreds of positions involved. It is not difficult to have just one copy of the calculations in a while loop where you increment the x component of the position of the spacecraft each time through the loop. Try it!
An advanced topic is to modify your program in such a way that you can drag the spacecraft around with the mouse, displaying the forces as you move, in which case you don’t want to create new arrows all the time, just update the attributes of the two existing arrows. To see how to handle mouse interactions, study the VPython demos that use the mouse, and see the “Visual” documentation on the IDLE help menu. Choose “Reference manual”, then “Mouse Interactions”. You may want to make the spacecraft bigger to make it easier to select and drag it.
Another advanced topic is to define a function (“def” in Python) that calculates the forces and displays them, then “call” that function repeatedly. We won’t need this technique in this course, but if you get interested in programming you might like to study the Python documentation available on the IDLE help menu. Python is an important and powerful programming language; we will barely scratch the surface of its capabilities.
