8. Centripetal Forces & Gravitation

A recently discovered extrasolar planet appears to be rockier and denser than earth. It is 16 times as massive as earth, but its diameter is only twice that of earth. What is the free-fall acceleration on the surface of this planet?

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.

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?

What is the acceleration due to gravity on the surface of Mercury?

At a distance of $2R{E}_{-}$ from Earth's center, what is the acceleration due to gravity experienced by an object (neglecting atmospheric effects)?

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?

An astronaut has a mass of $60$ kg on Earth. What is her mass on the Moon, where $g=1.62m/s^2$?

What is the free-fall acceleration at the surface of (a) the moon and (b) Jupiter?

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.

(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.]

(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.

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