Final answer:
To show the gravitational potential energy of a mass m at a distance r from mass M, one must integrate the gravitational force, derived from Newton's law of gravitation, over distance r, resulting in the formula PE = GMm/r.
Step-by-step explanation:
The question asks to show that the gravitational potential energy (PE) of a mass m placed at a distance r from a mass M is given by PE = GMm/r, where G is the gravitational constant. This is a physics problem involving Newton's law of gravity. Gravitational potential energy is the energy an object has due to its position in a gravitational field.
The force of gravity between two masses m and M, separated by a distance r, is given by Newton's universal law of gravitation F = GMm/r^2. To find the work done to move mass m from a reference point at infinity (where gravitational potential energy is considered to be zero) to a distance r from M, one must integrate the gravitational force over the distance r.
This yields the expression PE = -∫ F dr, which is equivalent to PE = -GMm ∫ dr/r^2. After performing the integration, we find the gravity potential energy as PE = -GMm [1/r - 1/∞], which simplifies to PE = -GMm/r, because 1/∞ is zero. The negative sign indicates that work is done against the gravitational field, but for convenience, we omit the negative sign when considering the potential energy in the field alone. Thus, we get PE = GMm/r.