Two different triangular arrangements of identical point masses are constructed in…
PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences
Problem
Two different triangular arrangements of identical point masses are constructed in deep space where external forces are negligible. In both arrangements, three masses M are located at the vertices of an equilateral triangle of side length L. In Arrangement A, the fourth mass m is placed at the center of the triangle. In Arrangement B, the same mass m is placed on one of the sides’ midpoints.
The location of zero gravitational potential energy is taken at infinite separation. Ignore any masses other than those described.
(a) Derive an expression, in terms of G,M,m, and L, for the total gravitational potential energy of Arrangement A. Show all steps.
(b) Derive a corresponding expression for the total gravitational potential energy of Arrangement B.
(c) Using your symbolic results, determine which arrangement has the larger magnitude of gravitational potential energy. Explain your reasoning without substituting numerical values.
(d) Discuss physically why relocating the lighter mass from the center to the side changes the total potential energy in the direction you found. Refer to specific pairwise separations that increased or decreased.