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Problem 13
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 . (b) Our solar system was formed roughly 5 billion years ago. How many orbits has it completed?Problem 13
What is the ratio of the sun's gravitational force on you to the earth's gravitational force on you?Problem 13
Asteroid 253 Mathilde is one of several that have been visited by space probes. This asteroid is roughly spherical with a diameter of 53 km. The free-fall acceleration at the surface is 9.9 ✕ 10⁻³ m/s². What is the asteroid's mass?Problem 13
While visiting Planet Physics, you toss a rock straight up at 11 m/s and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?Problem 13
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 . (d) Assume that the sun is a typical star with a typical mass. If galactic matter is made up of stars, approximately how many stars are in the center of the galaxy?Problem 13
A small moon orbits its planet in a circular orbit at a speed of 7.5 km/s. It takes 28 hours to complete one full orbit. What is the mass of the planet?Problem 13
"A satellite orbits the sun with a period of 1.0 day. What is the radius of its orbit?"Problem 13
A new planet is discovered orbiting the star Vega in a circular orbit. The planet takes 55 earth years to complete one orbit around the star. Vega's mass is 2.1 times the sun's mass. What is the radius of the planet's orbit? Give your answer as a multiple of the radius of the earth's orbit.Problem 13
The International Space Station orbits 300 km above the surface of the earth. What is the gravitational force on a 1.0 kg sphere inside the International Space Station?Problem 13
Two 65 kg astronauts leave earth in a spacecraft, sitting 1.0 m apart. How far are they from the center of the earth when the gravitational force between them is as strong as the gravitational force of the earth on one of the astronauts?Problem 13
What is the force of attraction between a 50 kg woman and a 70 kg man sitting 1.0 m apart?Problem 13
Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is 8.00 x 10⁻⁶ N when their centers are 20 cm apart. What is the mass of each?Problem 13
(a) At what height above the earth is the free-fall acceleration 10% of its value at the surface?Problem 13
What is the free-fall acceleration at the surface of (a) the moon and (b) Jupiter?Problem 13
A recently discovered extrasolar planet appears to be rockier and denser than earth. It is 16 times as massive as earth, but its diameter is only twice that of earth. What is the free-fall acceleration on the surface of this planet?Problem 13
Suppose we could shrink the earth without changing its mass. At what fraction of its current radius would the free-fall acceleration at the surface be three times its present value?Problem 13
A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet's surface. How far above the surface are they orbiting? Your answer will be a multiple of R.Problem 13
You have been visiting a distant planet. Your measurements have determined that the planet's mass is twice that of earth but the free-fall acceleration at the surface is only one-fourth as large. (a) What is the planet's radius?Problem 13
A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius r + Δr, where Δr≪r , has the very slightly different period T + ΔT. (a) Show that ΔT/T = (3/2) (Δr/r)Problem 13
The asteroid belt circles the sun between the orbits of Mars and Jupiter. One asteroid has a period of 5.0 earth years. What are the asteroid's orbital radius and speed?Problem 13
Three stars, each with the mass of our sun, form an equilateral triangle with sides 1.0 x 10¹² m long. (This triangle would just about fit within the orbit of Jupiter.) The triangle has to rotate, because otherwise the stars would crash together in the center. What is the period of rotation?Problem 13
What is the net gravitational force on the 20.0 kg mass in FIGURE P13.36? Give your answer using unit vectors.Problem 13
A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?Problem 13.46a
A rogue band of colonists on the moon declares war and prepares to use a catapult to launch large boulders at the earth. Assume that the boulders are launched from the point on the moon nearest the earth. For this problem you can ignore the rotation of the two bodies and the orbiting of the moon. (a) What is the minimum speed with which a boulder must be launched to reach the earth?
Hint: The minimum speed is not the escape speed. You need to analyze a three-body system.
Problem 13.51
The 75,000 kg space shuttle used to fly in a 250-km-high circular orbit. It needed to reach a 610-km-high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?
Problem 13.52b
In 2000, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of 1.0 x 10¹⁶ kg and a radius of 8.8 km. (b) What is the escape speed from the asteroid?
Problem 13.54b
In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days. (b) What is the mass of the comet?
Problem 13.58d
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. (d) How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?
Problem 13.62
A 55,000 kg space capsule is in a 28,000-km-diameter circular orbit around the moon. A brief but intense firing of its engine in the forward direction suddenly decreases its speed by 50%. This causes the space capsule to go into an elliptical orbit. What are the space capsule’s (a) maximum and (b) minimum distances from the center of the moon in its new orbit?
Hint: You will need to use two conservation laws.
Problem 13.70b
Let’s look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii r₁ and r₂ , and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse. (b) Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v'₁ on the inner circular orbit and the velocity v'₁ at the low point on the elliptical orbit that spans the two circular orbits.
