A spacecraft is sent to study three different binary systems

PHYSICS · Multi-part · CBSE · Physics · Gravitation · Gravitation

Problem

A spacecraft is sent to study three different binary systems, each consisting of two isolated spherical bodies in deep space. The spacecraft records the mass of each body and the distance between their centers at the instant of observation, as listed below. System 1: m1=2.0×106 kgm_1 = 2.0 \times 10^{6}\,\text{kg}, m2=8.0×106 kgm_2 = 8.0 \times 10^{6}\,\text{kg}, separation r=6.0 kmr = 6.0\,\text{km} System 2: m1=5.0×106 kgm_1 = 5.0 \times 10^{6}\,\text{kg}, m2=5.0×106 kgm_2 = 5.0 \times 10^{6}\,\text{kg}, separation r=8.0 kmr = 8.0\,\text{km} System 3: m1=1.0×107 kgm_1 = 1.0 \times 10^{7}\,\text{kg}, m2=2.0×106 kgm_2 = 2.0 \times 10^{6}\,\text{kg}, separation r=4.0 kmr = 4.0\,\text{km} The location of zero gravitational potential energy is defined at infinite separation. (a) Without performing exact numerical calculations, rank the three systems according to the magnitude of their gravitational potential energy ∣Ug∣|U_g|, from greatest to least. Clearly state any equalities. (b) Justify your ranking by analyzing how gravitational potential energy depends on mass and separation. Include a brief discussion of the significance of the negative sign in UgU_g and why the ranking can be performed using magnitudes.

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