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In a certain region of space, the gravitational field is given by $-k/r$ , where $r$ is the distance and $k$ is a constant. If the gravitational potential at $r = r_0$ be $V_0$ , then what is the expression for the gravitational potential $(V)$ ?
$k\,log\,(r/r_0)$
$k\,log\,(r_0/r)$
$V_0\, +\, k\, log\, (r/r_0)$
$V_0\, +\, k\, log\, (r_0/r)$
Solution
We know that, the gravitational intensity is equal to the negative of the gradient of potential
i.e., $1=-\frac{d \mathrm{V}}{d r}$
Here, $\mathrm{I}=-\frac{\mathrm{K}}{r} ;$ so $\frac{d \mathrm{V}}{d r}=\frac{\mathrm{K}}{r}$Q
or, $\int_{v_0}^{v} d \mathrm{V}=\int_{r_0}^{0} \frac{\mathrm{K}}{r} d t$
or, $\mathrm{V}-\mathrm{V}_{0}=\mathrm{K} \log \frac{r}{r_{0}}$
or, $\mathrm{V}=\mathrm{V}_{0}+\mathrm{K} \log \frac{r}{r_{0}}$
