Question

Determine the minimum angle at which a frictionless road should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is 200.0 m.

Answer 1

Answer:

Minimum angle at which frictionless road should be banked is 11.53°

Explanation:

Consider the figure below to understand the banking curve. θ is  minimum angle at which frictionless road should be banked.

Forces acting along x-axis:

$F_{N}sin\theta=\frac{mv^{2}}{R}---(1)$

Forces acting along y-axis:

$F_{N}cos\theta-mg=0---(2)$

Dividing (1) and (2)

$\frac{F_{N}sin\theta}{F_{N}cos\theta}=\frac{\frac{mv^{2}}{R}}{mg}\\\\tan\,\theta=\frac{v^{2}}{rg}\\\\v=20\,m/s,\,r=200\,m\\\\tan\,\theta=\frac{(20)^{2}}{(200)(9.8)}\\\\tan\,\theta=0.204\\\\\theta=tan^{-1}(0.204)\\\theta=11.53^{o}$

Answer 2

The minimum angle at which a frictionless road should be banked so that a car traveling at the given speed is 11.5⁰C.

The given parameters;

• speed of the car, v = 20 m/s
• radius of the curve, r = 200 m

The normal force on the car is given as;

$Fsin \ \theta = \frac{mv^2}{r}$

The horizontal force on the car is calculated as;

$Fcos \ \theta = mg$

The ratio of the two forces is calculated as;

$\frac{Fsin\ \theta }{Fcos \ \theta } = \frac{mv^2}{rmg} \\\\tan \theta = \frac{v^2}{rg} \\\\\theta = tan^{-1}(\frac{v^2}{rg} )\\\\\theta = tan^{-1}(\frac{20^2}{200 \times 9.8} )\\\\\theta = 11.5 \ ^0 C$

Thus, the minimum angle at which a frictionless road should be banked so that a car traveling at the given speed is 11.5⁰C.

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