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Question: It has been mentioned that a road having a width \(20m\) makes an arc of radius \(15m\). The outer e...

It has been mentioned that a road having a width 20m20m makes an arc of radius 15m15m. The outer edge of it is 2m2m which is higher than its inner edge. For what velocity the road should be banked?
A.10ms−1 B.14.7ms−1 C.9.8ms−1 D.none of the above \begin{aligned} & A.\sqrt{10}m{{s}^{-1}} \\\ & B.\sqrt{14.7}m{{s}^{-1}} \\\ & C.\sqrt{9.8}m{{s}^{-1}} \\\ & D.\text{none of the above} \\\ \end{aligned}

Explanation

Solution

The velocity of the banked road can be found by taking the square root of the product of the tangent of the angle of banking, acceleration due to gravity and radius of the circular path. As the banking angle is very small, the tangent of the banking angle can be approximately equivalent to the sine of the banking angle. This will help you in answering this question.

Complete answer:
The velocity of the banked road can be found by the equation given as,
tan⁡θ=V2g×r\tan \theta =\dfrac{{{V}^{2}}}{g\times r}
From this the velocity of the banked road can be found by taking the square root of the product of the tangent of the angle of banking, acceleration due to gravity and radius of the circular path. This can be written as,
V=rgtan⁡θV=\sqrt{rg\tan \theta }
Here as the banking angle is very small, the tangent of the banking angle can be approximately equivalent to the sine of the banking angle. That is we can write that,
tan⁡θ≈θ≈sin⁡θ≈220\tan \theta \approx \theta \approx \sin \theta \approx \dfrac{2}{20}
It has been already mentioned in the question that the radius of the path taken will be given as,
r=15mr=15m
Acceleration due to gravity is found to be,
g=9.8ms−2g=9.8m{{s}^{-2}}
Substituting the values in the equation can be written as,
V=15×9.8×220 V=14.7ms−1 \begin{aligned} & V=\sqrt{15\times 9.8\times \dfrac{2}{20}} \\\ & V=\sqrt{14.7}m{{s}^{-1}} \\\ \end{aligned}
Therefore the velocity of the banked road has been obtained as 14.7ms−1\sqrt{14.7}m{{s}^{-1}}.

This has been mentioned in the options as option B.

Note:
Banking of roads is the technique used in which the edges are raised for the curved roads above the inner edge in order to give the required centripetal force to the vehicles using which they take a safe turn. The angle at which the car will be inclined is referred to as the bank angle.