To continue the sci-fi theme, we will calculate whether the \"Ringworld\" from L

Question

To continue the sci-fi theme, we will calculate whether the "Ringworld" from Larry Niven's novel of the same name will undergo SHM about it's parent star. Assume the star is the same mass as the Sun (2 times 10^30kg), now consider that an advanced civilization has taken apart every planet and created a ringworld, an artificial ring about 1,600,000 km wide, orbiting around the central star at the radius of the Earth's orbit, 1.5 times 10^11m, the gravity on inner surface is provided by centrifugal force and is equal to Earth's. (a) What is the rotation period of the ring to cause that gravity? (b) If the ring was displaced slightly vertically (along the rotation axis) above the central star it would be drawn back down by gravity. By using small angle approximations calculate the period of oscillation of the ring about the star in the vertical direction. (c) Is the ringworld stable in the plane on the ring? What would happen if the ring moved slightly close to the central star? Would it undergo SHM or something else? No detailed calculations are required but explain your reasoning.

Explanation / Answer

given, mass of the star, M = 2*10^30 kg

ring radius, R = 1.5*10^11 m

a. as acceleration due to gravity is same as that of earth

and this acceleration is provided by centrifugal force

so centrifugal acceleration = w^2*R [ w is the angular velocity of the ring]

9.81 = w^2*1.5*10^11

w = 8.087*10^-6 rad/s

so time period of rotation T = 2*pi/w = 7.776*10^5 s = 8.9924 days

b. when the ring is displaced slightly

Force on the ring due to star is given by

F = GMm*y/(y^2 + R^2)^3/2 [ where y is the diplacement of the ring in upward direction]

for y < < R

F = GMm*y/R^3

so from newtons second law

ma = GMm*y/R^3

a = -GM*y/R^3 [ where -ve sign means the force is attractive]

this is equation of SHM

so time period of osscilation, T = 2*pi*sqroot(|y/a|) = 2*pi*sqroot(R^3/GM) = 2*pi*sqroot((1.5*10^11)^3/6.67*10^-11 * 2*10^30) = 31603766.335470 s = 365.784 days

c. when the ring is brought closer to the star, the gravitational attration will increase in accordance with the inverse square law, making the equilibrium of ringworld an unstable one

so whn the ring destabilises, it will be drawn into the star

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