It is possible that the Newtonian theory of gravitation may need to be modified at short-range. Let, then the potential energy between two masses M $_{1}$ and $M_{2}$ be given by $U (r) = - \frac{G M_{1} M_{2}}{r}$ (1 - Ke $^{r / r_{o}}$ ) where K is constant. For short distances, defined by $r << r_{o}$ , the force F between M $_{1}$ and M $_{2}$ will be:

F = - \frac{G M_{1} M_{2}}{r^{3}} (1 + K)

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F = - \frac{G M_{1} M_{2}}{r^{2}} (1 - 2 K)

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F = - \frac{G M_{1} M_{2}}{r^{2}} (1 - K)

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Solution

The correct option is D $F = - \frac{G M_{1} M_{2}}{r^{2}} (1 - K)$
$\frac{d U}{d r} = - \frac{G M_{1} M_{2}}{r^{2}} ⎡ ⎢ ⎣ 1 - K e^{- \frac{r}{r_{o}}} ⎤ ⎥ ⎦ - \frac{G M_{1} M_{2}}{r} (\frac{K}{r_{o}}) e^{- \frac{r}{r_{o}}}$
$= \frac{G M_{1} M_{2}}{r^{2}} ⎡ ⎢ ⎣ 1 - K e^{- \frac{r}{r_{o}}} (1 + \frac{r}{r_{o}}) ⎤ ⎥ ⎦$
$(s i n c e \frac{r}{r_{o}} << 1)$
$\Rightarrow F = - \frac{d U}{d r} ⌊_{r << r_{0}}$ $= - \frac{G M_{1} M_{2}}{r^{2}} (1 - K)$

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Similar questions

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