A future space station has mass M distributed uniformly on a ring of radius R. (

Question

A future space station has mass M distributed uniformly on a ring of radius R. (The uniform density is p=M/2(pi)R) The ring is centered on the z-axis at z=0.

a. Find the gravitational potential on the axis.
b. Find the gravitational field along the axis.
c. Suppose M=10^8 kg, and R=1000 m. What is the weight of a 70 kg man at z=500 m?
d. Using the same values for M and R, what is the escape velocity from z=0 along the axiz?
e. For z>>R, show that the gravitational force approaches the value for a point mass M on the axis at z=0.

Explanation / Answer

p = M/(2*pi*R) a. Find the gravitational potential on the axis. the potential at z due to a small length dL is dV(z) = -G*p*dL/sqrt(R^2 + z^2) so V(z) = -G*p*(2*pi*R)/sqrt(R^2 + z^2) = -GM/sqrt(R^2 + z^2) b. Find the gravitational field along the axis. field E(z) = dV(z)/dz = GMz/sqrt(R^2 + z^2)^(3/2) c. Suppose M=10^8 kg, and R=1000 m. What is the weight of a 70 kg man at z=500 m? E(500) = 2.39*10^-9 N/kg m = 70 kg weight = m*E = 1.67*10^-7 N d. Using the same values for M and R, what is the escape velocity from z=0 along the axis? energy at z = 0 is mv^2/2 + m*V(0) = mv^2/2 - GMm/R at infinity, energy = 0 energy conservation, mv^2/2 - GMm/R = 0 v = sqrt(2GM/R) = 3.65 mm/s e. For z>>R, show that the gravitational force approaches the value for a point mass M on the axis at z=0. force F(z) = m*E(z), for z >> R, force F(infinity) = m*GMz/z^3 = 0 for z = 0, force F(0) = m*E(0) = 0 so F(infinity) = F(0)

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