8. Centripetal Forces & Gravitation

Suppose that you used some geometry and kinematics to estimate that the Earth goes around the Sun with an orbital speed of approximately 30,000 m/s (60,000 mph), and that the Sun is approximately 150 million kilometers away from the Earth. Use this information to estimate the mass of the Sun.

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?

A satellite orbits at an orbital period of 2 hours around the Moon. What is the satellite's orbital altitude?

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?

The Lunar Reconnaissance Orbiter (LRO) is a NASA Moon-orbiting spacecraft in an orbit with an altitude of 50 km above the Moon’s surface. What is the period of the LRO as it orbits the Moon?

A satellite of mass 5200 kg orbits the Earth and has a period of 6800 s. Determine

(b) 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?

(III) (b) 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.

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.

(b) 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].

(a) 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].

(b) 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. <IMAGE> Hint: Each planet feels two forces.

The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 x 10⁸ m/s . Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of 230 km/s . (c) The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center?