A train has to negotiate a curve of radius 'r' $\mathrm{m}$, the distance between the rails is ${…

A train has to negotiate a curve of radius 'r' $\mathrm{m}$, the distance between the rails is ${ }^{\prime}{ }^{\prime} \mathrm{m}$ and outer rail is raised above inner rail by distance of ' $\mathrm{h}^{\prime} \mathrm{m}$. If the angle of banking is small, the safety speed limit on this banked road is
  1. $\sqrt{\operatorname{rg}\left(\frac{h}{\ell}\right)}$
  2. $\operatorname{rg} \frac{h}{\ell}$
  3. $\frac{\left(\frac{h}{\ell}\right)^{2}}{r g}$
  4. $\left(\operatorname{rg} \frac{h}{\ell}\right)^{2}$

Solution

$\frac{\mathrm{mv}^{2}}{\mathrm{r}}=\mu \mathrm{R}=\mu \mathrm{mg}$ $\mathrm{v}^{2}=\mu \mathrm{gr}$ $\mathrm{v}=\sqrt{\mu \mathrm{gr}}=\sqrt{\tan \theta \mathrm{gr}}=\sqrt{\sin \theta \mathrm{g} \mathrm{r}}$ [For small angle of $\theta]$ $=\sqrt{\mathrm{gr} \frac{\mathrm{h}}{\ell}}$

Asked in: MHT CET 2020 (15 Oct Shift 1)

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