**Part A.**

There is no net attraction between a hollow sphere and a body inside the sphere.

Therefore, a Dyson sphere of the type will be gravitationally unstable.

In case the sphere is hit by a meteor and is slightly shifted, there will be **no force** required to be exerted by the sun to bring it to its original position.

This is because the sphere will simply drift off and hit the sun.

The Dyson sphere is an hypothetical spherical structure centered around a star. Inspired by a science fiction story, physicist Freeman Dyson described such a structure for the first time in a scientific paper in 1959. His basic idea consisted of an artificial spherical structure of matter built around a star at a distance comparable to a planetary orbit, with the purpose of capturing the energy radiated by the star and reusing it for industrial purposes. Assume the mass of the sun to be 2.00×10^{30} kg.

**Part A.** Consider a solid, rigid spherical shell with a thickness of 100 m and a density of 3900 kg/m^{3}. The sphere is centered around the sun so that its inner surface is at a distance of 1.50×10^{11} m from the center of the sun. What is the net force that the sun would exert on such a Dyson sphere were it to get displaced off-center by some small amount? Express your answer numerically in newtons.

**Part B.** What is the net gravitational force Fout on a unit mass located on the outer surface of the Dyson sphere described in Part A? Express your answer in newtons.

**Part C.** What is the net gravitational force Fin on a unit mass located on the inner surface of the Dyson sphere described in Part A? Express your answer in newtons.

The gravitational attraction of the sun would make the inner surface of the Dyson sphere described in Part A uninhabitable, because everything on the inner surface would slowly accelerate toward the sun. One way to solve this problem would be to create artificial gravity through rotation. Assume that the Dyson sphere rotates at a constant angular speed around an axis through its center so that earthlike gravity is re-created along the inner equator of the Dyson sphere. Take the radius of the Earth to be 6.38×10^{6 }m and the mass of the Earth to be 5.97×10^{24} kg.**Part D. **What is the linear speed v of a unit mass located at the inner equator of such a sphere? Express your answer in meters per second.

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