Theory

Law of Gravitation

Gravity and gravitational pull are terms, which we come across often. Newton formulated the law of gravitation, which can be stated as:

“Every particle of matter in the universe attracts every other particle with a gravitational force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them”.

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Mathematically the law can be represented as:

F = G m1 m2 / r2

Where,
F = gravitational force
m1 and m2 = masses of the two particles
r = distance between the two particles
G (gravitational constant) = 6.67 x 10-11 N m2/kg2

This means that any two bodies in the universe attract each other, but the extent of attraction depends upon their masses and how close (or far) they are from each other. In other words, if the mass of one body is doubled, the gravitational attraction between the two bodies is also doubled. However, if the distance between the two bodies is doubled, the force of attraction then reduces by four times.

In our solar system, the planets revolve around the sun because of the gravitational pull exerted by the sun on the planets. The planets thus move around the sun in a fixed path known as the ‘orbit’. It was discovered by an astronomer called Johannes Kepler in the 16th century that the planets do not move around the sun in circles but follow an elliptical path. Because of the elliptical path, the distance between the sun and the planets keeps on changing and so does its speed. A planet moves faster when it is closer to the sun and slower when it is away from the sun.

Similar to the planetary motion, a natural satellite like moon also revolves round earth because of the gravitational pull exerted by the earth on it. The same concept is used to keep the man-made or artificial satellites in orbit around the earth. An artificial satellite however needs to be launched with a certain velocity for the given altitude, so that it moves in a circular orbit around the earth.

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To understand how a satellite continually moves in its orbit around the earth, let us take an example of a stone thrown horizontally from top of a tower. Since the stone is thrown horizontally, it travels a certain distance in a curved path before hitting the ground. The distance the stone travels before hitting the ground depends on the speed at which it is thrown (launch velocity), greater the speed, greater the distance. The image above shows three paths of the stone thrown from the tower at different speeds. In case of path B, the launching speed of the stone is more than that of stone of path A, and therefore it travels further. The surface of the earth is curved and the stone also travels in a curved path, therefore if the stone is thrown with a sufficiently high velocity, it will follow a curved path but will never hit the earth because the earth would also be curving away from the stone(path C in the image). When the stone takes such a path it will always tend to move towards the earth due to gravity, but will always continue to move around it in a stable orbit around the earth. In real life it is almost impossible to throw a stone such that it will revolve around the earth in an orbit because it will require a very high velocity to do so. However, this launch velocity required reduces as we go higher and higher from the earth’s surface. It is therefore possible to launch a satellite from a high altitude with a manageable launching velocity which would be required to keep it in a stable orbit.

The Activity
The aim of the activity is to help the user understand the motion of a satellite in an orbit around the earth. This activity allows the user to set the height of the satellite above the earth and also the launch velocity and observe the orbit for the set parameters. For a given distance of the satellite from the centre of the earth, there is only one orbital velocity, which will keep the satellite in a perfectly circular orbit. For the height set by the user, this velocity for the circular orbit is displayed in the activity. The user is, however, encouraged to try out different launch velocities and observe the resulting orbits.

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When the initial velocity set by the user is not equal to the orbital velocity required for a circular orbit, the satellite then moves in an elliptical orbit. When the satellite moves in a elliptical orbit, its orbital velocity changes depending on its distance from the earth. The satellite moves faster when it is closer to earth and slows down as it moves away from earth. It is an interesting exercise to try out different initial velocities, especially to find out how less velocity you can set and still have the satellite miss the earth.

 

Formula & Numerical Problem


Equations for satellites in circular orbit around the earth:

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Numerical Problem

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Theory | Formula
Instructions

This exploriment requires Adobe Shockwave plug-in. Download & install only if you do not see the exploriment on the left.

This exploriment on Gravitation is aimed at understanding how the gravity affects an orbiting body. In this activity, you can change the initial velocity and the height of the satellite above the earth and see how the satellites orbits. For a particular height of the satellite, there is a unique velocity at which the satellite orbit is exactly circular. For any other velocity it is elliptical. Also, there needs to be sufficient velocity for a particular velocity to achieve orbit around the earth, or else the satellite will crash down into the earth. This activity lets you explore various parameters required for launching a satellite for a successful orbit.

Procedure
Set the altitude of the satellite, this is the height of the satellite above the surface of the earth. As you change the altitude of the satellite, the orbit radius is displayed on the right. The orbit radius is the sum of the radius of the earth and the altitude of the satellite.

The velocity required for a circular orbit for the given altitude is also displayed on the right.

For precise values you can type in the input/display* box and press ENTER to set the values.

Set the launch velocity of the satellite. You can either use the slider to set the value or you can also type in the required value in the Launch Velocity box.

Once the parameters are set as desired, click on the 'Start' button to run the activity.

Observe the orbit of the satellite. If the satellite crashes into the earth, change the settings and start again. Observe the instantaneous value of the orbit velocity displayed on the right side.

You can stop the activity any time by clicking on the 'Reset' button.Change the settings and run the activity again.

Note:
Initial Height is only in Y direction and distance is measured from the surface of the earth

Initial velocity is in X direction

* The white display box also acts as an input box where you can type in the values. The grey display box, however, is only an output box.

Tips
Try running the activity at different altitude and launch velocity for the satellite.

Set the altitude, check out the velocity required for circular orbit. Put the same value for the launch velocity. Observe the orbit. Now change to lower and higher velocities and observe the difference in the orbits.

Run the activity as many times as you want to with different inputs and try to draw your conclusions about the concept by observing the results.

For more information, check out the 'Theory' section.