Physics

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:

m_1 = 2.0 \times 10^{6}\,\text{kg}

m_2 = 8.0 \times 10^{6}\,\text{kg}

, separation

r = 6.0\,\text{km}

System 2:

m_1 = 5.0 \times 10^{6}\,\text{kg}

m_2 = 5.0 \times 10^{6}\,\text{kg}

, separation

r = 8.0\,\text{km}

System 3:

m_1 = 1.0 \times 10^{7}\,\text{kg}

m_2 = 2.0 \times 10^{6}\,\text{kg}

, separation

r = 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

|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

U_g

and why the ranking can be performed using magnitudes.

Intuitive explanation

Gravitational potential energy depends on both how massive the objects are and how close they are—it's the product $m_1 m_2$ divided by separation $r$ , scaled by Newton's constant. To compare these systems, you need to calculate $U = -\frac{Gm_1m_2}{r}$ for each and rank them by magnitude. The system with the largest product of masses packed into the smallest distance will have the most negative (lowest) potential energy.

Intuitive explanation

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