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

• Correct answer
  A

  $\sqrt{\operatorname{rg}\left(\frac{h}{\ell}\right)}$
• B

  $\operatorname{rg} \frac{h}{\ell}$
• C

  $\frac{\left(\frac{h}{\ell}\right)^{2}}{r g}$
• D

  $\left(\operatorname{rg} \frac{h}{\ell}\right)^{2}$

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 $\theta]$ $=\sqrt{\mathrm{gr} \frac{\mathrm{h}}{\ell}}$

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