Universal Gravitation

Glenn Elert

Problems

practice

Verify the inverse square rule for gravitation with the following chain of calculations…
1. Determine the centripetal acceleration of the moon. (Assuming the moon is held in it's orbit by the gravitational force of the Earth, you are then also calculating the acceleration due to gravity of the Earth at the moon's orbit.)
2. Determine the ratio of the radius of the moon's orbit to the radius of the Earth.
3. Use the results of a. and b. to calculate the acceleration due to gravity on the surface of the Earth.
4. How does this value compare to the generally accepted value of g? Are the results of your calculations in close enough agreement with experimental observations to verify the inverse square rule for gravitation? Discuss briefly.
Estimate the value of the universal gravitational constant from the following approximate measurements taken during the original Cavendish experiment (and converted into SI units for us)…
- one hundred kilogram fixed and one kilogram rotating masses
- ten centimeter separation between fixed and rotating masses
- one millionth newton of force on each of the rotating masses
Check it out.
1. Determine the acceleration due to gravity (g) on the surface of the Earth from Newton's law of universal gravitation.
2. How does this value compare to the standard acceleration due to gravity (g)?
3. Are the results of your calculation close enough to the standard value to verify the distance-dependent portion of Newton's law of universal gravitation? Discuss briefly.
Jupiter is about eleven times larger in diameter and three hundred times more massive than the Earth. How does the gravitational field on Jupiter compare to that on Earth?

conceptual

What would happen to objects on the Earth's surface if…
1. the Earth's gravitational field gradually disappeared?
2. the Earth's gravitational field was fine, but the Earth slowly stopped rotating?
What effect, if any, would removing the Earth's core have on the gravitational field at its surface? (Assume the size and shape of the Earth does not change; that is, assume the Earth was partially hollowed out.)
The Earth has a radius about twice as great and a mass ten times greater than the planet Mars. How does the acceleration due to gravity on Mars compare to that on Earth?

numerical

Determine the following quantities for a 10 kg frozen turkey…
1. its mass on the surface of the Earth
2. its weight on the surface of the Earth
3. its mass in orbit one Earth radius above the surface of the Earth
4. its weight in orbit one Earth radius above the surface of the Earth
5. its mass on the surface of the moon
6. its weight on the surface of the moon
(Note: The word "weight" in these questions refer to the gravitational force, not the apparent weight.)
Astrology
1. Calculate the force of gravity between a 3.0 kg newborn baby and a 60 kg doctor standing 0.25 m away.
2. Calculate the force of gravity between a 3.0 kg newborn baby and the planet Jupiter when it is nearest to the Earth.
3. What is the ratio of the force of gravity from Jupiter on the baby compared to the force of gravity from the doctor on the baby?
4. What is the likelihood that astrology (assuming it had any validity) could be explained as a result of planetary gravitation at the moment of your birth? (Keep in mind that Jupiter is the largest planet and that it is rarely as far from the Earth as its nearest approach.)
Walking on the moon
1. Calculate the gravitational field strength on the surface of the moon.
2. How does the gravitational field on the surface of the moon compare to the gravitational field on the surface of the Earth?
3. Describe the effect that the moon's reduced gravity would have on your athletic abilities. Identify one sport or athletic event in which your abilities would get better and one in which your abilities would get worse.
Weightlessness
1. Calculate the weight of a 75 kg astronaut on the surface of the Earth.
2. Calculate the same astronaut's weight aboard a space station as it orbits 3.5 × 10⁵ m above the Earth's surface.
3. According to common wisdom, objects in outer space are "weightless". Why then isn't the answer to the second part of this question zero? What's wrong with the common wisdom?
The purpose of this problem is to determine the possible nature of the planetoid LV-426 from the 1979 science fiction horror film Alien. Begin by reading the begining of Scene 21 from the revised final screenplay. The crew of the interstellar mining ship Nostromo receive a mysterious transmission and locate the source.

INT. BRIDGE21

Dallas, Kane, Ripley and Ash stand around the illuminated map table.
Lambert sits at the radio directional console.

DALLAS

We all hear that, Lambert?

She switches on the audio system.
Hissing.
Static. Then...
An ungodly sound.
Eight seconds worth.

KANE

Good God.

RIPLEY

Doesn't sound like any radio signal I've heard.

LAMBERT

Maybe it's a voice.

DALLAS

Well we'll soon know. Can you hone in on that?

LAMBERT

What was the position?

DALLAS

6550-99

LAMBERT

Alright, I've found the quadrant. Ascension 6 minutes 20 seconds, declination 39 degrees 2 seconds.

DALLAS

Okay, put that on the screen for me.

Lambert punches buttons. One of the viewscreens flickers, and a small far off light appears.

LAMBERT

Alright, well, that's it. It's a planetoid. 1200 kilometers.

KANE

It's tiny.

DALLAS

Any rotation?

LAMBERT

About two hours.

DALLAS

What about gravity?

LAMBERT

Point eight six.

ASH

You can walk on it.

"Alien" by Walter Hill and David Giler. Based on a screenplay by Dan O'Bannon. Story by Dan O'Bannon and Ronald Shusett, 1978

Use the data given in the scene quoted above to determine the following quantities for LV-426…
1. its mass
2. its average density (assuming it's spherical)
3. the centripetal acceleration of a point on its equator
Speculate on the nature of LV-426.
1. What kind of material is it possibly made of?

The most massive exoplanet found to date is the brown dwarf HR 2562 b orbiting a star 111 light years from our Sun. Using the information in the table below, determine…

its average density…
1. in kg/m³
2. as a multiple of Earth's average density
3. as a multiple of Jupiter's average density
its surface gravity…
1. in m/s²
2. as a multiple of Earth's surface gravity
3. as a multiple of Jupiter's surface gravity

Planetary parameters
	HR 2562 b¹	Earth²	Jupiter²
mass	9535 Earth masses 30 Jupiter masses	5.97217 × 10²⁴ kg	1.89812 × 10²⁷ kg
radius	12.4 Earth radii 1.11 Jupiter radii	6,378,100 m	71,492,000 m

Note for pedants: Very massive planets like Jupiter and HR 2562 b don't really have well defined surfaces like the Earth does. The word "surface" for these objects refers to any position where the atmospheric pressure is equal to that of the Earth.

What separation between two earth-like exoplanets would result in the same gravitational force as two bowling balls in contact? Give your answer in meters and light years.

Two roughly spherical objects compared
quantity	bowling ball	earth
mass	15.00 lb 6.804 kg	1.317 × 10²⁵ lb 5.972 × 10²⁴ kg
radius, equatorial radius, polar	-	6,378,100 m 6,356,800 m
diameter	8.500 in 21.59 cm	-
gravitational force	some value	the same value
separation	in contact	?

algebraic

Determine the height h above the surface of a planet of radius r and mass m at which the gravitational field will be one-half its surface value.
The purpose of this problem is to determine the possible nature of the planet Krypton. Begin by reading the introduction from the 1950s TV series Superman.

Faster than a speeding bullet. More powerful than a locomotive. Able to leap tall buildings in a single bound.

Look, up in the sky! It's a bird! It's a plane! It's Superman!

youtu.be/Q2l4bz1FT8U

Yes, it's Superman, strange visitor from another planet who came to Earth with powers and abilities far beyond those of mortal men. Superman, who can change the course of mighty rivers, bend steel with his bare hands, and who, disguised as Clark Kent, mild-mannered reporter for a great metropolitan newspaper, fights a never-ending battle for truth, justice and the American way.

Superman's strength is partly attributed to the gravity of his home planet, Krypton. The people of Krypton evolved to stand, walk, and lift ordinary objects in Krypton's strong gravitational field. When Superman came to Earth, he found that his Kryptonian physique was sort of over-designed. He could "leap tall buildings in a single bound", for example. This is much like when humans go to the moon. They find themselves strong enough to do all sorts of things they couldn't do on Earth — like run effortlessly with long strides while wearing an 80 kg (180 lb) space suit, for example.

Think of how high a typical human can jump on Earth. Assume Superman can only jump as high as that on Krypton. Then consider how high Superman can jump on Earth. Use this knowledge to determine the physical characteristics of Krypton. (State all values on Krypton in comparison to their values on Earth. Do not state them with a number and a unit.)
1. Derive an expression that relates height jumped to the acceleration due to gravity when take off speed is constant.
  1. Use the expression derived in part a to compare g on the surface of Krypton to g on the surface of the Earth.
2. Derive an expression that relates g on the surface of a spherical planet to the density and radius of the planet (instead of the mass and radius, which is the usual way it is stated).
  1. Use the expression derived in part b to determine the radius of Krypton assuming it has the same average density as the Earth. How likely is one to find a terrestrial planet with a radius like this?
  2. Use the expression derived in part b to determine the average density of Krypton assuming it has the same radius as the Earth. How likely is one to find a terrestrial planet with a density like this?
3. Is there anything else you would like to say about Superman or Krypton?