8. Centripetal Forces & Gravitation

Why do satellites launched into low-Earth orbits not remain there indefinitely?

Approximately how long does it take Earth to complete one orbit around the Sun?

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.

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?

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?

A distant planet orbits a star 3 times the mass of our Sun. This planet of mass 8 × 10²⁶ kg feels a gravitational force of 2 × 10²⁶ N. What is this planet's orbital speed and how long does it take to orbit once?

How does friction with the atmosphere affect the speed of an artificial satellite in low Earth orbit?

In a two-body system consisting of a star and a planet orbiting their common center of mass, what is the period required for both objects to complete one full orbit around the center of mass?

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.

Approximately how long does it take for a spacecraft using current technology, such as a $Hohman$ transfer orbit, to travel from $Earth$ to $Mars$?

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.

You throw a baseball horizontally while on the surface of a small, spherical asteroid of mass 7×10¹⁶ kg and diameter of 22km. What is the minimum speed so that it just barely goes around the asteroid without hitting anything?

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.]

Suppose a satellite is in a circular orbit not far above Earth’s surface, and circles the Earth about once every 92 minutes. Find the centripetal acceleration of the satellite in its orbit. Express your answer in terms of g, the gravitational acceleration at the Earth’s surface.