A railway line is taken round a circular arc of radius $1000\ m$, and is banked by raising the outer rail $h\ m$ above the inner rail. If the lateral force on the inner rail when a train travels round the curve at $10 \ ms^{-1}$ is equal to the lateral force on the outer rail when the train's speed is $20\ ms^{-1}$. The value of $4g\ tan\theta$ is equal to : (The distance between the rails is $1.5 \ m$).
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In the frame of train
$\mathrm{F}_{1}+\frac{\mathrm{mv}_{1}^{2}}{\mathrm{R}} \cos \theta=\mathrm{mg} \sin \theta$
$\frac{\mathrm{mv}_{2}^{2}}{\mathrm{R}} \cos \theta=\mathrm{mg} \sin \theta+\mathrm{F}_{2}$
$\mathrm{F}_{1}=\mathrm{F}_{2} \mathrm{given}$
$\mathrm{g} \sin \theta-\frac{\mathrm{v}_{1}^{2}}{\mathrm{R}} \cos \theta=\frac{\mathrm{v}_{2}^{2}}{\mathrm{R}} \cos \theta-\mathrm{g} \sin \theta$
$\tan \theta=\frac{\mathrm{v}_{2}^{2}+\mathrm{v}_{1}^{2}}{2 \mathrm{gR}}=\frac{100+400}{2 \times \mathrm{g} \times 1000}=\frac{1}{4 \mathrm{g}}$
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