Physics 30
Unit 3 ­ Circular Motion
Lesson 3.4 Outline
• Astronomical Applications
• Newton's Law of Universal Gravitation
• Examples
You will be able to:
• solve problems involving gravitational force between objects
• explain how centripetal force and gravitational force are related
Isaac Newton's three laws of motion revolutionized the study of modern physics when they were first published. However, he discovered another law that governs the motion of celestial bodies and provides an excellent approximation to calculate the force on objects due to gravity. This law is now known as Newton's Law of Universal Gravitation and became crucial to understanding the motion of the planets.
Today, Einstein's theory of relativity is now generally used to calculate the force of gravity as it is more accurate but Newton's Law still provides an excellent approximation of the force of gravity between two bodies. The math involved in Einstein's theory is also more complicated than the math in Newton's Law.
Newton's Law of Universal Gravitation
Newton stated that every mass attracts every other mass with a force that acts along the straight line distance between the centers of the objects. These forces are equal and opposite in direction. This force depends on the masses of the objects and the distance between them. In other words, anything that has mass will gravitationally attract anything else that has mass with a force. This happens between all objects that have mass and all other objects that have mass. So if you are sitting at a computer, you are gravitationally attracted to the screen and the keyboard and the mouse and they are all gravitationally attracted to you and each other and everything else around you.
So why don't we collide with other things?
Remember that the net force is what causes objects to move. So if you are only considering the force between two small objects, you can find the gravitational force. But normally the force of friction (which always opposes motion from starting or continuing) is bigger than the gravitational force so no motion can happen as a result of it.
When the masses are small compared the to Gravitation constant, the force generated due to gravity is extremely small. This is why you and I wouldn't collide when we are sitting in the same room.
Major Implications
Since Newton's equation accurately predicted the motion of most known celestial bodies at that time, it was accepted in the realm of science. However, this implied that gravity was an "instantaneous, action at a distance force" that did not have a direct cause. Up to this point the forces we have discussed require some contact or substance to transmit the force, so this was an absurd prediction (especially for the time period). Absurd, but yet it accurately predicted observable events.
Newton was very uncomfortable with this and refused to speculate on what could be causing this action at a distance force as he thought it a serious problem with his theory. Later, Einstein was able to resolve this "action at a distance" problem with the theory of general relativity.
Newton's Law of Universal Gravitation
Fg= Force of Gravity between the masses
Gm1m2
Fg = r2
G = Gravitational Constant
m1= mass of object 1
m2= mass of object 2
r = distance between the centers of the objects
G = 6.673 x 10­11
Nm2
kg2
An extremely common mistake is to forget to square the r in the bottom term.
The value of the gravitational constant was not determined by Newton. He knew that the forces between objects were related and so could only compare the gravitational attractive force between multiple objects.
G = 6.673 x 10­11
Nm2
kg2
The value of the gravitational constant was determined during the Cavendish experiment. This experiment was carried out 71 years after Newton's death and 111 years after the publication of the Law of Universal Gravitation.
What does this look like?
Fg
Fg
r
m1
For two different masses of m1 and m2. Note that the Fg is equal in magnitude, but opposite in direction. Remember that r is measured from the centers of the masses.
m2
A coffee cup with a mass of 0.75 kg gravitationally attracts a textbook with a mass of 2.0 kg. If the distance between the centers of these objects is 2.0 m, what is the force of gravity between the two objects?
Determine the force of gravitational attraction between the earth (m = 5.98 x 1024 kg) and a 70.0 kg physics student if the student is standing at sea level, a distance of 6.38 x 106 m from earth's center. Compare this with the weight of the student.
Connection to Circular Motion
When high mass objects (like planets) are travelling in circular paths and friction is negligible (like in space), the centripetal force causing the circular path results from the force of gravity between the two objects.
This situation accurately describe the motion of planets in the solar system, as the masses of planets and stars are extremely large and there are very few particles in space so friction is not an issue. We will use this to study orbits in the next lesson.
A 5400 kg satellite is rotating around the earth at 3050 m/s. If it is 6.38 x 106 m to the center of the earth and the satellite is orbiting at 36500 km. Determine the approximate mass of the earth.
Determine the velocity of the earth as it travels around the sun. (mass of the Sun = 1.99 x 1030 kg, mass of the Earth = 5.98 x 1024 kg, distance between them = 1.496 x 1011 m)