## Gravity

**Kepler's Laws**

Johannes Kepler discovered three laws that still describe the behavior of every planet and satellite. The theories behind his laws are incorrect but the laws are still valid. They are:

1. The planets travel in an elliptical orbit around the sun with the sun at one focus.

2. An imaginary line from the sun sweeps out equal areas in equal time intervals whether the planet is closer or further from the sun.

3. The ratio of the squares of the periods of any two planets revolving about the sun is equal to the ratio of the cubes of their average distances from the sun. Thus, if Ta and Tb are their periods, and ra and rb their average distances,

### (Ta/Tb)2 = (ra/rb)3

**Universal Gravitation**

Newton realized that the force that caused apples to fall toward the earth also causes the moon to fall toward the earth. The apple falls straight down because it has no horizontal motion; the moon does not hit the earth as it's falling because of its horizontal motion. If the moon was not moving horizontally it would hit the earth. The force the earth exerts on the apple and the force the earth exerts on the moon is given by the same equation, known as Newton's Law of Universal Gravitation.

where G is a proportionality constant (known as the universal constant), m stands for mass, d is the distance between the two masses, and F is the gravitational force of attraction between the two masses.

**Newton's Test of the Inverse Square Law**

Combining Newton's Law of Universal Gravity and his Second Law of Motion leads to Kepler's Third Law.

### so

### rearranged to get:

These equations describe the motion of a satellite revolving around a larger body.** **Using the example of the earth revolving around the sun, the symbols are: m1 = the mass of the sun; m2 = the mass of the earth; r = the distance between the earth and the sun; T = the period of revolution of the earth. These equations are valid for the moon revolving around the earth or the space shuttle revolving around the earth.

**Weighing the Earth**

### Using

### Cavendish was able to measure the constant G.

Determine the gravitation field strength at a distance d from the center of the earth. (This applies only to points outside the surface of the earth.)

### Combine

### solving for g;

Where g is the field strength due to the earth at a distance d from the center of the earth. Use this equation to prove that there is gravity in space at a distance of several hundreds miles (where many satellites are located). The field strength can be expressed as a ratio of its distance with a known field strength or

**Satellites**

An object becomes a satellite when it is above the atmosphere, and has a horizontally velocity great enough to prevent it from hitting the earth as it falls towards the earth.** **

### Combining

### and

### you get:

**Weight and Weightlessness**

Astronauts experience weightlessness because they are in free-fall around the earth. The space shuttle orbits at about 400 km above the surface of the earth, where g' = 8.7 m/s-s.

**The Gravitational Field**

A gravitational field is a model of our what we call gravity; the above equations explains the way our model attempts to describe the phenomenon we call gravity.

**Einstein's Theory of Gravity**

Einstein described gravity as the bending of space by matter.

**Review Questions:**

1. Consider four cases: a mass on the surface of the earth, and at a height of 1 m, 10km, and 1000 km. In which of these cases does the acceleration due to gravity on an object change significantly?

A: 1000 km.

2. The space shuttle is orbiting the earth at a height of 350 km above the surface of the earth. What must the speed of the space shuttle be in order to maintain a steady orbit?

A: 7700 m/s

3. Trevor and Derek sit about 1 meter apart (at their centers) in Physics class. They each have a mass of 48 kg. What gravitational force do they exert on each other?

A:1.54 x 10-7

4. Two balls are placed so their centers are 2.6 meters apart. The force between the two balls is 2.75 x 10-12 N. What is the mass of each ball if one ball is twice the mass of the other ball?

A: 0.37 kg and 0.75 kg

5. What is the mass of the sun and its orbital period? The Earth's average distance is 1.50 x 1011 m.

A: 2.01 x 10 30 kg

6. Halley's comet returns every 74 years. What is the average distance the comet is from the sun?

A: 19 AUwhich puts it between Saturn and Uranus

7. What is the gravitational force between two 8 kg spherical masses that are 5 m apart?

A: 1.7 x 10 -10 N

8. Two satellites of equal mass are put into orbit 30 m apart. The gravitational force between them is 2.0 x 10-7 N. (a) What is the mass of each satellite? (b) What is the initial acceleration given to each satellite by the gravitational force?

A: a. 1.6 x 10 3 kg b. 1.3 x 10 -10 m/s2

9. The mass of Earth is 6.0 x 1024 kg. If the centers of Earth and the moon are 3.9 x 108 m apart, the gravitational force between them is about 1.9 x 1020 N. What is the approximate mass of the moon?

A: 7.2 x 10 22 kg

10. The mass of an electron is 9.1 x 10-31 kg. The mass of a proton is 1.7 x 10-27 kg. They are about 1.0 x 10-10 m apart in a hydrogen atom. What gravitational force exists between the proton and the electron of a hydrogen atom?

A: 1.0 x 10 -47 N