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Expert-verified Found in: Page 381 ### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718 # In Problem 1, What ratio m / M gives the least gravitational potential energy for the system?

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

See the step by step solution

## Step 1: The given data

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

## Step 2: 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)

## Step 3: Calculation of ratio which gives least potential energy

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

$U=-\frac{Gm\left(M-m\right)}{r}\phantom{\rule{0ex}{0ex}}$

To find minimum potential energy,

$\frac{dU}{dm}=0\phantom{\rule{0ex}{0ex}}-\frac{G\left(M-2m\right)}{r}=0\phantom{\rule{0ex}{0ex}}M-2m=0\phantom{\rule{0ex}{0ex}}M=0.5m\phantom{\rule{0ex}{0ex}}\frac{m}{M}=0.5$

Hence, the ratio at which we obtain least potential energy is 0.5 ### Want to see more solutions like these? 