Question

In Problem 1, What ratio m / Mgives the least gravitational potential energy for the system?

Short answer

Ratio $\frac{m}{M}$which gives least gravitational potential energy is 0.5

Step-by-step solution

The given data

1. Mass of particle is M
2. Mass of another particle is M - m

Understanding the concept of the energy

To find the ratio, use the formula for gravitational potential energy in which potential energy is proportional to the product of masses and inversely proportional to the distance between the masses.

Formula:

Potential Energy, $U=-\frac{GMm}{r}$ (i)

Calculation of ratio which gives least potential energy

Ratio of m / M which gives minimum potential energy using equation (i):

$U=-\frac{Gm\left(M-m\right)}{r}$

To find minimum potential energy,

$$\begin{gathered} \frac{dU}{dm}=0 \\ -\frac{G\left(M-2m\right)}{r}=0 \\ M-2m=0 \\ M=0.5m \\ \frac{m}{M}=0.5 \end{gathered}$$

Hence, the ratio at which we obtain least potential energy is 0.5