Practicing Success
A particle of mass m is kept on the axis of a fixed circular ring of mass M and radius R at a distance x from the centre of the ring. If x << R, At what value of x the force between the ring and the particle is maximum: |
$\frac{R}{\sqrt{2}}$ $\frac{R}{2 \sqrt{2}}$ $R \sqrt{2}$ $\frac{R}{2}$ |
$\frac{R}{\sqrt{2}}$ |
We see from the figure that, each portion of the ring having a mass $\delta m$ attracts the particle towards itself with a force $\delta F$ given as $\delta F=\frac{G(\delta m) m}{r^2}$ $\Rightarrow$ The axial component of this force $=\delta F_x=\delta F Cos \theta=\frac{G(\delta m) m}{r^2} . \frac{x}{r} =\frac{G m x}{r^3}(\delta m)$ $\Rightarrow$ The total axial force experienced by the particle due to the whole ring $F_x=\sum \delta F_x=\int d F_x \quad \Rightarrow \quad F_x=\frac{G m x}{r^3} \int\limits_0^M \delta m$ $\Rightarrow F_x=\frac{G M m x}{\left(R^2+x^2\right)^{3 / 2}}$ .......(1) (since r = $\sqrt{R^2+x^2}$) For $F_x$ to be maximum $\frac{dF_{x}}{dx}=0 \text { and } \frac{d^2 F_{x}}{dx}<0$ Using (1), $\frac{dF_{x}}{dx}=GM \frac{d}{dx}\left\{\frac{x}{\left(R^2+x^2\right)^{3 / 2}}\right\}$ $=G M {\frac{\left(R^2+x^2\right)^{3 / 2}-x \frac{3}{2}\left(R^2+x^2\right)^{1 / 2}(2 x)}{\left(R^2+x^2\right)^3}}$ Setting $\frac{d F_x}{d x}=0$ and simplifying the factors, we obtain, $x=\frac{R}{\sqrt{2}}$ |