Exam 12: Gravity

Suppose that the Sun were to collapse under its own gravitational pull. To what radius would it have to be reduced in order for the escape velocity to be equal the speed of light, 3.00 x 10⁸ m/s? The mass of the Sun is 2.00 x 10³⁰ kg and G = 6.67 x 10^-11 N·m²/kg².

A space vehicle is orbiting Earth in a circular orbit with a radius of 10,300,000 m. What is the minimum increase in speed that is needed for the vehicle to escape Earth's gravitational field? The mass of Earth is 5.97 × 10²⁴ kg and G = 6.67 x 10^-11 N·m²/kg².

A binary system has two stars of mass M circling their common center of mass on opposite sides of a circle of radius R. What is the potential energy of the system?

An asteroid approaches Earth so that when it is very far away it is traveling with a speed of 6500 m/s along a line that would miss the center of Earth by 8,500,000 m. Because of the gravitational attraction of Earth, it departs from this straight line and curves towards Earth, colliding with it. In what direction, below the local horizontal, is the asteroid traveling when it hits Earth? The mass of Earth is 5.97 × 10²⁴ kg and G = 6.67 x 10^-11 N·m²/kg².

An 825-kg space vehicle is circling Earth in a circular orbit with a radius of 10,300 km. Its orbit is to be changed to a larger circle with a radius of 17,400 km. How much energy is required to accomplish this? The mass of Earth is 5.97 × 10²⁴ kg and G = 6.67 x 10^-11 N·m²/kg².

In Jules Verne's From the Earth to the Moon, scientists are launched from the surface of Earth by a cannon. With what speed would their capsule have to leave the cannon if the speed of the capsule is required to be zero m/s at the point between Earth and the Moon where their gravitational pulls on the capsule are equal? The mass of Earth is 5.97 × 10²⁴ kg, the mass of the Moon is 7.35 × 10²² kg, the distance between Earth and the Moon is 3.84 × 10⁸ m, the radius of Earth is 6.37 × 10⁶ m, and G = 6.67 x 10^-11 N·m²/kg².

The acceleration due to gravity on the surface of Earth is g. Calling R the radius of Earth, at what distance from Earth's center would the acceleration due to gravity drop to g/16?

A planet has two small satellites in circular orbits around the planet. The first satellite has a period 12.0 hours and an orbital radius 6.00 × 10⁷ m. The second planet has a period 16.0 hours. What is the orbital radius of the second satellite?

Kepler's second law tells us that planets sweep out equal areas in equal times. If you compare the amount of area per time swept by Earth with the one of Jupiter, you would conclude:

How do the escape velocities for two rockets, the first weighing 20 N and the second weighing 20,000 N compare?

You have discovered a new asteroid, which has a moon in an elliptical orbit around it. From a series of observations, you establish that the closest approach occurs when the moon is 51,000 m from the asteroid, the furthest separation is 102,000 m, and the speed of the moon at the point of closest approach is 7.50 m/s. What is the mass of the asteroid? G = 6.67 x 10^-11 N·m²/kg².

There are daily variations in the barometric pressure, which occur because of tidal effects.

Four identical bodies of mass M, at the vertices of a square with a diagonal 2R, orbit their common center of mass in a circular motion with a radius R. How much energy is required to separate the four stars so they are an infinite distance apart?

A satellite is orbiting Earth in a circular orbit of radius r when it fires its rockets and acquires speed rapidly. Following this, the satellite's orbit

Our Moon revolves about Earth in the same direction as Earth's direction of rotation about its own axis. The gravitational attraction of the Moon causes tides on Earth, which gradually slow down its rotation about its own axis. What consequences does this have for the Moon?

Because Earth's orbit is slightly elliptical, Earth actually gets closer to the Sun during part of the year. When Earth is closer to the Sun its orbital speed is

The moons of Mars, Phobos (Fear) and Deimos (Terror), are very close to the planet compared to Earth's Moon. Their orbital radii are 9,378 km and 23,459 km respectively. What is the ratio of the period of revolution of Phobos to that of Deimos?

The International Space Station is orbiting at an altitude of about 370 km above the earth's surface. The mass of the earth is 5.976 × 10²⁴ kg and the radius of the earth is 6.378 × 10⁶ m. (a) Assuming a circular orbit, what is the period of the International Space Station's orbit? (b) Assuming a circular orbit, what is the speed of the International Space Station in its orbit?

How does the escape speed from Mars compare with that of the Earth? [ M_Earth=9.31∗ M_Mars ; R_Earth=1.88 ∗ R_Mars ]

What is the period of a satellite circling Mars 100 km above the planet's surface? The mass of Mars is 6.42 × 10²³ kg, its radius is 3.40 × 10⁶ m, and G = 6.67 x 10^-11 N·m²/kg²