Solution: Conservation of energy: PE + KE+Wnc = PE + KE GMm RE + ||mv² +0 = GMm - +0 RE+h Cancel m's and clean up: v² = (2GM) 2 1 - 1 (2GM) (R/R+h) RE Gravitational Energy Conservation A "super cannon" launches a ball vertically from Earth's surface with a speed of v. Ignore air resistance and the Earth's spin. At its highest point, the ball is a height of h above the surface of the Earth. Earth's mass is 5.97 x 1024 kg. Earth's radius is 6.37 × 106 m. 7 ย m S 8 Type your answer... v 6,002™ S Type your answer... h 1.89×106m h x10m -

Astronomical observatrions of our Milky Way galaxy indicate that it has a mass of about 8.01011 solar masses. A star orbiting on the galaxy’s periphery is about 6.0104 light-years from its center. (a) What should the orbital period of that star be? (b) If its period is 6.0107 years instead, what is the mass of the galaxy? Such calculations are used to imply the existence of other matter, such as a very massive black hole at the center of the Milky Way.

Saturns ring system forms a relatively thin, circular disk in the equatorial plane of the planet. The inner radius of the ring system is approximately 92,000 km from the center of the planet, and the outer edge is about 137,000 km from the center of the planet. The mass of Saturn itself is 5.68 1026 kg. a. What is the period of a particle in the outer edge compared with the period of a particle in the inner edge? b. How long does it take a particle in the inner edge to move once around Saturn? c. While this inner-edge particle is completing one orbit abound Saturn, how far around Saturn does a particle on the outer edge move?

When a star dies, much of its mass may collapse into a single point known as a black hole. The gravitational force of a black hole on surrounding astronomical objects can be very great. Astronomers estimate the strength of this force by observing the orbits of such objects around a black hole. What is the gravitational force exerted by a black hole on a 1-solar-mass star whose orbit has a 1.4 1010 m radius and a period of 5.6 days?

Planetary orbits are often approximated as uniform circular motion. Figure P7.9 is a scaled representation of a planets orbit with a semimajor axis of 1.524 AU. a. Use Figure P7.9 to find the ratio of the Suns maximum gravitational field to its minimum gravitational field on the planets orbit. b. What is the ratio of the planets maximum speed to its minimum speed? c. Comment on the validity of approximating this orbit as uniform circular motion.

(a) Calculate Earth’s mass given the acceleratioln due to gravity at the North Pole is measured to be 9.832m/s2 and the radius of the Earth at the pole is 6356 km. (b) Compare this with the NASA’s Earth Fact Sheet value of 5.97261024kg .

(a) Calculate Earth's mass given the acceleration due to gravity at the North Pole is 9.830 m/s2 and the radius of the Earth is 6371 km from center to pole. (b) Compare this with the accepted value of 5.9791024 kg.

(a) Given that the period of the Moons orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84 108 m, use Equation 13.11 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large?

Consider the Earth and the Moon as a two-particle system. a. Find an expression for the gravitational field g of this two-particle system as a function of the distance r from the center of the Earth. (Do not worry about points inside either the Earth or the Moon.) b. Plot the scalar component of g as a function of distance from the center of the Earth.

Figure P7.45 shows a picture of American astronaut Clay Anderson experiencing weightlessness on board the International Space Station. a. Most people have the misconception that a person in a spacecraft is weightless because he or she is no longer affected by gravity. Show that this premise cannot be true by computing the gravitational field of the Earth at an altitude of 200 km the typical altitude of a spacecraft in orbit. Compare this result with the gravitational field on the surface of the Earth. b. Why would astronauts in orbit experience weightlessness even if they are experiencing a gravitational field (and therefore a gravitational force)?