8. Centripetal Forces & Gravitation

You stand on the surface of a mysterious planet with a mass of 6 ×10²⁴ kg and measure the surface gravity to be 7 m/s². What must the radius of the planet be?

How far would you have to be above Earth's surface for g to be ½ of its surface value?

(I) What is the weight of a 74-kg astronaut

(d) 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

(b) accelerating toward the Moon at 1.8m/s²? State “direction” in each case.

(II) Suppose the solid wheel of Fig. 11–42 has a mass of 260 g and rotates at 85 rad/s ; it has radius 6.0 cm and is mounted at the center of a horizontal thin axle 25 cm long. At what rate does the axle precess?

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(III) Two identical particles, each of mass m, are located on the x axis at x= +x₀ and x = -x₀.

(b) 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 10⁴km , rotation period = 9 hr 55 mins . Take the centripetal acceleration into account.

A particle is released at a height r_E (radius of Earth) above the Earth’s surface. Determine its velocity when it hits the Earth. Ignore air resistance. [Hint: Use Newton’s second law, the law of universal gravitation, the chain rule, and integrate.]

The Near Earth Asteroid Rendezvous (NEAR) spacecraft, after traveling 2.1 billion km, orbited the asteroid Eros with an orbital radius of about 20 km. Eros is roughly 40km x 6km x 6km. Assume Eros has a density (mass/volume) of about 2.3 x 10³ kg/m³ .

(b) What would g be at the surface of a spherical Eros?