# 8. Gravity

 Reading: Chapter 6

# Gravity

 Newton's Universal Law of Gravitation:Every particle of matter in the universe attracts every other particle with a 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.

The gravitational force between two objects with masses m1 and m2 is expressed by

 ``` G m1 m2 Force = --------- R2 ```

where R is the distance between the center of mass of each object. (1) Hold up a pencil. Where is its center of mass?

The force between a human and the Earth is felt as the human's WEIGHT. The correct metric unit for this force is appropriately the Newton. Strictly speaking, it is your MASS that is measured in kilograms.

`              G MEarth Mhuman`
`Fweight = -------------------`
`                   REarth2`

 (2a) A 180lb-astronaut weighed about 1/6 his usual weight when he was walking on the Moon. What was his mass on the moon? Same / more / less ? (b) Which of the following are ways to loose weight? (i) diet so that your mass decreases, (ii) go to a planet that has less mass than the Earth (iii) climb to the top of a very tall building (3) The Earth is not quite round--the radius at the pole is 6357 km and 6378 km at the equator. (a) Where on the surface of the Earth would you weigh the most?  (b) Where on the surface of the Earth would you weigh the least?  (c) Calculate the following ratio:    Maximum Weight --------------------  = Minimum Weight Consider an apple about to fall from an apple tree on the Earth. The full Moon is directly above. The apple will experience a downward force from the Earth's gravity--but also an upward force from the Moon's gravity. Let us compare the relative magnitude of these two forces.

G MEarth Mapple
FEarth  =  --------------------
REarth2

G MMoon Mapple
FMoon  =  --------------------
DMoon-Earth2

 (4) (a) Take the RATIO of these two forces: ```       FEarth       --------  =        FMoon``` (b) Cancel and re-arrange terms to make the expression as simple as possible. (c) Work out how many times greater the force on the apple due to the Earth is compared to the force on the apple due to the Moon using the fact that the Earth's mass is 81 times that of the Moon and that the distance to the Moon is 60 times the Earth's radius.   ```       FEarth       --------  =        FMoon``` (5) Now consider an astronaut on the Moon dropping a hammer--the Earth is shining above, pulling the hammer up, the Moon is pulling the hammer down--which wins? Follow the same procedure as the previous question--calculate the ratio of forces. (The Earth's radius is 3.7 times the Moon's radius.) ## Acceleration Due to Gravity

Newton's Second Law of motion can be written Force = Mass x Acceleration. This means that an object experiencing a force will accelerate accordingly:

acceleration = Force / Mass

The acceleration due to gravity is usually labeled 'g':

 ``` F G MEarth g = -------- = ------------ Mobject R Earth2 ```

Note! The acceleration of an object due to gravity does not depend on the object's own mass!!

 (6) Using G= 6.67 x 10-11, ME = 6 x 1024 kg and RE= 6.4 x 106 meters, you can work out the value of the acceleration due to the Earth's gravity on the Earth's surface--g. The units should be in m/s2 (meters per second per second) since an acceleration is a change in speed over time and speed is measured in m/s (ok--sometimes we use km/hr for high speeds). You should get an answer of 10 m/s2. (a) So, if you drop the text book from the top of a tall building, what will be it's speed after it has been falling for 1 second?  (b) What will be its average speed over that one second?  (c) How far will it have fallen in that one second?  (d) What if you had dropped a lead brick instead of a book?

Gravity on the Moon:

We can compare the acceleration felt on the Moon (due to the Moon's gravity) compared with the acceleration felt on the Earth (due to the Earth's gravity).

The acceleration on the Moon is

`           F            G MMoon`
`gMoon = ------- = ------------`
`          Mobject       RMoon2`

which we divide (to make a comparison) by the acceleration on the Earth

`           F            G MEarth`
```gEarth = ------ = ------------           Mobject       REarth2
```

This lets us cancel out G and Mobject and then re-arrange the terms:

`  gMoon         MMoon          REarth2`
`--------  =  ---------   x  --------`
```  gEarth          MEarth          RMoon2
```

 (7) Now plug in numbers and get the value of the acceleration due to gravity on the Moon compared with the acceleration on the surface of the Earth (using that the Moon has 1/81 times the mass of the Earth and that the Earth is 3.7 times bigger than the Moon). Hint: you only need to square a number and multiply 2 numbers to get an answer that looks like gMoon = NUMBER x gEarth

Optional Challenges

These questions are a little harder - go for it! Give them a try!

 (8) When you are standing on the Earth you experience the gravitational forces of all the objects in the universe! Luckily, the Earth's gravity wins and you do not float off into space. But the effect of the gravity of these other objects is real (it causes tides, for instance). Above you compared your weight when standing at the pole compared with the equator. Now consider the effect of the Sun's gravity pulling on you at the same time as the Earth. At what time of day will you weigh the most? And the least? (9) Above you compared the gravitational forces due to the Earth and the Moon. Close to the Earth the gravitational force of the Earth was stronger--but on the Moon the force of the Moon was stronger than that of the Earth.. This means that there must be a place between the Earth and the Moon where the gravitational forces of the Moon and Earth cancel. Where is this location? Half way between them? Closer to Earth? Closer to the Moon?

Model answers to Exercises