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Q13-30P

Expert-verifiedFound in: Page 381

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

- Mass of particle is M
- Mass of another particle is M - m

**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)

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

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