Q143 / 180 · AIIMS 2019 (25 May) · Physics

If potential energy is given by $U=\frac{a}{r^2}-\frac{b}{r}$. Then find out maximum force. (Given $a=2$, $b=4)$

  • Correct answer
    A
    $-\frac{16}{27} \mathrm{~N}$
  • B
    $-\frac{32}{27} \mathrm{~N}$
  • C
    $+\frac{32}{27} \mathrm{~N}$
  • D
    $+\frac{16}{27} \mathrm{~N}$

Solution

$\begin{aligned} & \text { } F=-\frac{d u}{d r}=-\left[-\frac{2 a}{r^3}+\frac{b}{r^2}\right]=\frac{2 a}{r^3}-\frac{b}{r^2} \\ & \frac{d F}{d r}=\frac{-6 a}{r^4}+\frac{2 b}{r^3}=0 \\ & \Rightarrow \frac{6 a}{r}=2 b \Rightarrow r=\frac{3 a}{b}=\frac{6}{4}=\frac{3}{2} \\ & \Rightarrow F_{\max }=\frac{2 \times 2}{(3 / 2)^3}-\frac{4}{(3 / 2)^2} \\ & =\frac{4 \times 8}{27}-\frac{4 \times 4}{9}=\frac{32-16 \times 3}{27}=-\frac{16}{27}\end{aligned}$

Hence, the correct answer is option: 'A'.

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