2.1 Gravitational Force Between Point Masses
Newton’s Law of Gravitation
Newton's law of universal gravitation states that any two bodies in the universe
attract each other with a force that is directly proportional to the product of
their masses and inversely proportional to the square of the distance between
them.
Mathematically, F =
where, F =
Gravitational Force
G =
6.67 x 10-11 N m2 kg-2 =
Universal Gravitational Constant
M1
and
M2 are
masses of two bodies
r =
distance between their centre
Thanks to G = 6.67 x 10-11 N m2 kg-2 gravitational forces between everyday objects of ordinary masses are negligibly small. On the other hand, gravitational forces involving massive bodies such as planets and stars are large and of huge significance.
Non-point masses can often be treated as point masses at their centres of masses. So d is the centre-to-centre distance.
Satellite Motion
An object orbits around a massive body in circular motion if the gravitational force Fg matches the required centripetal force Fc = m .
At Fg= Fc
= m . Then = . And = v2. V =
Geostationary Orbit
Geostationary satellites stay at a fixed point above the Earth, which is convenient for communication satellites because they appear stationary when viewed from Earth (if not ground antenna will have to continuously track the satellites as they move across the sky)
The GEO orbit is normally at a very high altitude which requires very powerful and expensive rockets to be launched
Unfortunately, being constrained to the equatorial plane also means geostationary satellites cannot serve the polar regions (because from the polar regions they are below the horizon and cannot be sighted.)
If a satellite were to appear stationary in the sky, it must “rotate” at the same rate as the Earth. This is achieved by parking the satellite at an altitude of 36,000 km. This so-called geostationary orbit
1. has orbital period of 24 hrs,
2. lies in the equatorial plane, and
3. orbits in a west to east direction.
Example:
A melon is taken to the moon as a food item by astronauts. Its mass is 0.65kg. The mass of the moon is 7.36 x1022 kg and the radius of the moon is 1.74 x106 m. What is the weight of the melon on the moon?
Solution:
Weight = Fg
= (6.67 x 10-11) x (0.65 x 7.36 x 1022) / (1.74 x106 )2= 1.1 N
Exercise 1: Use the data given to answer the questions that follows
1) How much would a 70.0-kg person weigh on Mercury?
2) How much would your 20.0-kg dog weigh on Neptune?
3) If Pete (mass = 90.0 kg) weighs himself and finds that he weighs 30.0 pounds, how far away from the surface of the earth is he?
4) Captain Kirk (80.0 kg) beams down to a planet that is the same size as Uranus and finds that he weighs
1250 N. What is the mass of that planet?
5) Which is greater, the force exerted by Saturn on the sun, or the force exerted by the earth on the Sun? How much greater?
6) A distance of 2.00 mm separates two objects of equal mass. If the gravitational force between them is0.0104 N, find the mass of each object.
7) Calculate the gravitational field strength (g) on the surface of Jupiter.
8) If the gravitational field strength at the top of Mount Everest is 9.772 N/kg, approximately how tall (in feet) is the mountain?
9) If you dropped a ball while standing on the surface of Mars, at what rate would it accelerate toward the ground?
10) A space probe lands on the surface of a spherical asteroid 250 miles in diameter and measures the strength of its gravitational field at that point to be 4.95 x 10-11 N/kg. What is the mass of the asteroid?
11) Determine the force the sun exerts on an object with a mass of 80.0 kg if that object is on the earth. What is the force exerted by the moon on the same object? What is the force the earth exerts on it?
12) If a person weighs 882 N on the surface of the earth, at what altitude above the earth’s surface must they be for their weight to drop to 800 N?
13) If a 50.0-kg mass weighs 554 N on the planet Saturn, calculate Saturn’s radius.
14) Calculate the distance between the center of the earth and the center of the moon at which the gravitational force exerted by the earth on an object is equal in magnitude to the force exerted by the moon on the object.