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A ring of mass m and a sphere of mass M (each of radius R) are placed so that the plane of the ring is perpendicular to the line joining their centres, and the distance between the centres is sqrt(8)*R. Find the gravitational force of attraction between the ring and the sphere.
- (sqrt(8)/27) * G m M / R²
- (2*sqrt(2)/3) * G m M / R²
- (1/(3*sqrt(8))) * G m M / R²
- (sqrt(8)/9) * G m M / R
Correct answer: (sqrt(8)/27) * G m M / R²
Solution
Treat the sphere as a point mass M located on the ring's axis at distance d = sqrt(8)*R. The gravitational field due to the ring (mass m, radius R) at axial distance d is E = G*m*d/(R² + d²)^(3/2). Here R² + d² = R² + 8R² = 9R², so (R² + d²)^(3/2) = (9R²)^(3/2) = 27 R³. Thus E = G*m*(sqrt(8) R)/(27 R³) = G*m*sqrt(8)/(27 R²). Force on sphere = M*E = (sqrt(8)/27) * G m M / R².
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