A science-fiction author asks for your help. He wants to write about a newly discovered spherically symmetric planet that has the same average density as the earth but with a 25% larger radius. What is g on this planet?

Rhea, one of Saturn’s moons, has a radius of 764 km and an acceleration due to gravity of $0.265 \mathrm{m} / \mathrm{s}^{2}$ at its surface. Calculate its mass and average density.

Thinking About Scale. One key to success in science is finding simple ways to evaluate new ideas, and making a simple scale model is often helpful. Suppose someone tells you that the reason it is warmer during the day than at night is that the day side of Earth is closer to the Sun than the night side. Evaluate this idea by thinking about the size of Earth and its distance from the Sun in a scale model of the solar system.

Titania, the largest moon of the planet Uranus, has $\frac { 1 } { 8 }$ the radius of the earth and $\frac { 1 } { 1700 }$ the mass of the earth. What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Titania, the largest moon of the planet Uranus, has $\frac { 1 } { 8 }$ the radius of the earth and $\frac { 1 } { 1700 }$ the mass of the earth. What is the acceleration due to gravity at the surface of Titania?

In the given figure, a long straight wire carries a current i1=30.0 A and a rectangular loop carries current i2=20.0 A. Take a=1.00 cm, b=8.00 cm, and L=30.0 cm. In unitvector notation, what is the net force on the loop due to $i_1$ ?

You are exploring a distant planet. When your spaceship is in a circular orbit at a distance of 630 km above the planet’s surface, the ship’s orbital speed is 4900 m/s. By observing the planet, you determine its radius to be $4.48 \times 10^{6} \mathrm{m}$. You then land on the surface and, at a place where the ground is level, launch a small projectile with initial speed 12.6 m/s at an angle of $30.8^{\circ}$ above the horizontal. If resistance due to the planet’s atmosphere is negligible, what is the horizontal range of the projectile?

An experiment is performed in deep space with two uniform spheres, one with mass 50.0 kg and the other with mass 100.0 kg. They have equal radii, r = 0.20 m. The spheres are released from rest with their centers 40.0 m apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres. (a) Explain why linear momentum is conserved. (b) When their centers are 20.0 m apart, find (i) the speed of each sphere and (ii) the magnitude of the relative velocity with which one sphere is approaching the other. (c) How far from the initial position of the center of the 50.0-kg sphere do the surfaces of the two spheres collide?

Define the gravitational field $\vec{g}$ at some point to be equal to the gravitational force $\vec{F}_{g}$, on a small object placed at that point divided by the mass m of the object, so $\vec{g}=\vec{F}_{g} / m$. A spherical shell has mass M and radius R. What is the magnitude of the gravitational field at the following distances from the center of the shell: r<R ?

On December 25, 2004, the Huygens probe separated from the Cassini spacecraft orbiting Saturn and began a 22-day journey to Saturn’s giant moon Titan, on whose surface it landed. Besides the data in Appendix F, it is useful to know that Titan is $1.22 \times 10^{6} \mathrm{km}$ from the center of Saturn and has a mass of $1.35 \times 10^{23} \mathrm{kg}$ and a diameter of 5150 km. At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?

Find the magnitude and direction of the net gravitational force on mass A due to masses B and C in the given figure. Each mass is 2.00 kg. The distance between C and A is 10 cm and between A and B is 40 cm.

Each mass is $3.00 \mathrm{~kg}$. Find the magnitude and direction of the net gravitational force on mass $A$ due to the other masses.

The point masses m and 2 m lie along the $x$-axis, with $m$ at the origin and $2 m$ at $x=L$. A third point mass $M$ is moved along the $x$-axis. At what point is the net gravitational force on $M$ due to the other two masses equal to zero?

You are standing on the surface of a planet that has spherical symmetry and a radius of $5.00 \times 10^{6}$ m. The gravitational potential energy U of the system composed of you and the planet is $-1.20 \times 10^{+9}$ J if we choose U to be zero when you are very far from the planet. What is the magnitude of the gravity force that the planet exerts on you when you are standing on its surface?