# Universal Gravitation

## Problems

### practice

- Verify the inverse square rule for gravitation with the following chain of calculations…
- 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.)
- Determine the ratio of the radius of the moon's orbit to the radius of the Earth.
- Use the results of a. and b. to calculate the acceleration due to gravity on the surface of the Earth.
- 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.
- Determine the acceleration due to gravity (
*g*) on the surface of the Earth from Newton's law of universal gravitation. - How does this value compare to the standard acceleration due to gravity (g)?
- 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.

- Determine the acceleration due to gravity (
- 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…
- the Earth's gravitational field gradually disappeared?
- 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…
- its mass on the surface of the Earth
- its weight on the surface of the Earth
- its mass in orbit one earth radius above the surface of the Earth
- its weight in orbit one earth radius above the surface of the Earth
- its mass on the surface of the moon
- its weight on the surface of the moon

- Astrology
- Calculate the force of gravity between a 3.0 kg newborn baby and a 60 kg doctor standing 0.25 m away.
- Calculate the force of gravity between a 3.0 kg newborn baby and the planet Jupiter when it is nearest to the Earth.
- 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?
- 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
- Calculate the gravitational field strength on the surface of the moon.
- How does the gravitational field on the surface of the moon compare to the gravitational field on the surface of the Earth?
- 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
- Calculate the weight of a 75 kg astronaut on the surface of the Earth.
- Calculate the same astronaut's weight aboard a space station as it orbits 3.5 × 10
^{5}m above the Earth's surface. - 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?

### 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!

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.)

- Derive an expression that relates height jumped to the acceleration due to gravity when take off speed is constant.
- Use the expression derived in part a to compare
*g*on the surface of Krypton to*g*on the surface of the Earth.

- Use the expression derived in part a to compare
- 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).- 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? - 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?

- Use the expression derived in part b to determine the
- Is there anything else you would like to say about Superman or Krypton?

- Derive an expression that relates height jumped to the acceleration due to gravity when take off speed is constant.