Two particles, each of mass m, are a distance d apart.To bring a third particle, with mass 2m, from far away to a resting point midway between the two particles, an external agent must do work equal to: A)4Gm²/d B)-4Gm²/d C)8Gm²/d D)-8Gm²/d E)zero

Question

Quizplus · Accepted Answer

The work done by an external agent to bring a third particle of mass 2m from infinity to a point midway between two particles each of mass m, separated by a distance d, is equal to the negative of the gravitational potential energy at that point. The gravitational potential energy at the midpoint due to the two masses m is given by $U = -\frac{Gm(2m)}{d/2} = -\frac{4Gm^2}{d}$. Since the work done is equal to the change in potential energy, and bringing the mass from infinity (where potential energy is considered zero) to the midpoint increases the potential energy (in a negative sense), the work done is $W = -U = -(-\frac{4Gm^2}{d}) = \frac{4Gm^2}{d}$. However, the correct formula should account for the mass of the third particle being 2m, thus doubling the result: $W = 2 \times \frac{4Gm^2}{d} = \frac{8Gm^2}{d}$, but since we are bringing the mass in against the gravitational pull, the work done is positive in terms of energy input, but when considering the formula for gravitational potential energy, it results in a negative value, hence the correct answer is $-\frac{8Gm^2}{d}$.

Two Particles, Each of Mass M, Are a Distance D