Estimate the density of the Earth, given that a satellite near the surface orbits with a period of 85 min. Approximate the Earth as a uniform sphere.

Three satellites orbit a planet of radius R, as shown in FIGURE EX13.24. Satellites S₁ and S₃ have mass m. Satellite S₂ has mass 2m. Satellite S₁ orbits in 250 minutes and the force on S₁ is 10,000 N. What is the kinetic-energy ratio for K₁ / K₃ for S₁ and S₃?

Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newton's third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of ≈ 1 s and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars. What is the radius of a geosynchronous orbit?

A satellite of mass 5200 kg orbits the Earth and has a period of 6800 s. Determine the magnitude of the Earth’s gravitational force on the satellite.

(II) In a remote region of space, far from any other objects, a small ball is in a 30.0-cm-radius circular orbit around a 7.0-kg bowling ball. How long does it take the small ball to complete one orbit?

A satellite circles a spherical planet of unknown mass in a circular orbit of radius 1.6 x 10⁷ m. The magnitude of the gravitational force exerted on the satellite by the planet is 120 N. If the satellite circles the planet once every 2.0 h in the larger orbit, what is the mass of the planet?

The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed around the Earth, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = 1.852km = 6076ft]. Determine the speed of each satellite.

The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using “triangulation” and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed around the Earth, allowing continuous navigational “fixes.” The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = 1.852km = 6076ft]. Determine the period of each satellite. [Originally, 1 nautical mile was defined as one minute ( 1/60 of a degree) of latitude on Earth’s surface. 1 knot is a speed of 1 nautical mile/h.]

FIGURE P13.57 shows two planets of mass m orbiting a star of mass M. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. Find an exact expression for the orbital period T. Hint: Each planet feels two forces.