8. Centripetal Forces & Gravitation

Titania, the largest moon of the planet Uranus, has 1/8 the radius of the earth and 1/1700 the mass of the earth. What is the acceleration due to gravity at the surface of Titania?

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venus?

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. Compute the acceleration due to gravity on the surface of Venus from these data.

At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 m/s² if the acceleration due to gravity at the surface has magnitude 9.80 m/s² ?

You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 395.0 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth's equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.)

Asteroid 253 Mathilde is one of several that have been visited by space probes. This asteroid is roughly spherical with a diameter of 53 km. The free-fall acceleration at the surface is 9.9 ✕ 10^-3 m/s². What is the asteroid's mass?

Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?

A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet's surface. How far above the surface are they orbiting? Your answer will be a multiple of R.

You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. What is the planet's radius?

(I) What is the weight of a 74-kg astronaut in outer space traveling with constant velocity?

(II) What is the apparent weight of a 75-kg astronaut 2800 km from the center of the Moon in a space vehicle accelerating toward the Moon at 1.8m/s²? State “direction” in each case.

(III) Two identical particles, each of mass m, are located on the x axis at x= +x₀ and x = -x₀. Determine a formula for the gravitational field due to these two particles for points on the y axis; that is, write g as a function of y, m, x₀, and so on.

(III) Two identical particles, each of mass m, are located on the x axis at x= +x₀ and x = -x₀. At what point (or points) on the y axis is the magnitude of g a maximum value, and what is its value there? [Hint: Take the derivative dg/dy.]

The rings of Saturn are composed of chunks of ice that orbit the planet. The inner radius of the rings is 73,000 km, and the outer radius is 170,000 km. Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn’s own rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 x 10²⁶.

Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter because people cannot survive more than a few g’s. Calculate the number of g’s a person would experience at Jupiter’s equator, using the following data for Jupiter: mass = 1.9 x 10²⁷ kg, equatorial radius = 7.1 x 10⁴ km, rotation period = 9 hr 55 mins. Take the centripetal acceleration into account.