practice

1. 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.
2. 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
3. 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 in close enough agreement with the standard value to verify the mass-dependent portion of Newton's law of universal gravitation? Discuss briefly.
4. 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

1. 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?
2. What effect, if any, would removing the earth's core have on the gravitational field at its surface?
3. 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

1. 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, and
   6. its weight on the surface of the moon.
2. Astrology
   1. Calculate the force of gravity between a 3.0 kg newborn baby and a 75 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.)
3. Walking on the Moon
   1. Calculate the acceleration due to gravity on the surface of the moon.
   2. What is the ratio of the acceleration due to gravity on the surface of the moon to the acceleration due to gravity on the surface of the earth?
   3. What effect would the moon's reduced gravity have on your athletic abilities?
4. Weightlessness
   1. Calculate the weight of a 75 kg astronaut on the surface of the earth.
   2. Calculate the same astronaut's weight aboard the space shuttle 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?
5. Geosynchronous, Earth-Orbiting Space Station
   
   
   [magnify]
   For a sufficiently advanced human civilization, the occasional trip into outer space may become a reality for the general population. Having large numbers of spacecraft landing and taking off from the surface of the earth would probably not be acceptable, however. One way around this would be to build a ring around the earth that rotates at the same rate as the earth. This ring would be linked to the equator by electrically powered space elevators. No more noisy, dirty rockets. Just hop on the space elevator, press the "up" button, and stare nonchalantly at the door for a couple of hours. Such a massive structure would also house a large population of full-time inhabitants. Since they're the descendants of earth-bound humans, they would probably feel most comfortable in a 1 g environment. Determine the radius of such a megastructure. State your answer in terms of
   1. multiples of earth's radius.
   2. the fraction of the distance from the earth to the moon.
   (When this question appeared in the centripetal force section of this book, the effects of gravity were assumed negligible. This was a very reasonable assumption. The pull of gravity is relatively weak at this distance and, if you have already solved the earlier version of this problem, will only affect your results slightly.)

algebraic

1. 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.
2. 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.

   You should note in this synopsis of the origins of Superman, that all his feats are those of great strength. The first appearance of the character we now recognize as Superman was in Action Comics, Vol. 1, No. 1, 1938. At that time Superman was just like any ordinary man, except he was very, very strong (and also of good character). As the comic book, then radio show, then TV show, then movie (then video game?) evolved Superman became more and more super. He learned to fly, he had x-ray vision, he could hear really quiet sounds from great distances, he could blow very cold air, and so on. In this analysis we will stick to the golden age of Superman -- back when he was just a super man.

   Superman's strength was attributed to the gravity of his home planet, Krypton. The people of Krypton evolved great strength so that they could 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. This is much like when human 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 83 kg (183 lb.) space suit, for example.

   These questions should be solved as proportions. State all answers in comparison to their values on earth ("twice as big as on earth," for example).

   1. Derive an equation that relates height jumped to the acceleration due to gravity. Use this to determine the acceleration due to gravity on the surface of Krypton from the statement, "Able to leap tall buildings in a single bound." [Hint: Think of how high a typical human can jump on earth and how high Superman can jump on earth. How much stronger then must gravity have been on Krypton in proportion to the gravity on the earth?]
   2. Derive an expression that relates the acceleration due to gravity 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 this equation to …
      1. determine the radius of Krypton assuming it has the same average density as the earth, and
      2. determine the average density of Krypton assuming it has the same radius as the earth.
   3. Comment on the physicality of these answers. (Remember, Superman and the other inhabitants of Krypton look and act like humans in nearly every way other than their unusual strength.) How likely is one to find a terrestrial planet with …
      1. the radius you just calculated or
      2. the average density you just calculated?
   4. Is there anything else you would like to say about Superman or Krypton?