Problem 76 - A Complete Resource Book in Physics for JEE Main

Question

At a curved path of the road, the roadbed is raised a little on the side away from the center of the curved path. The slope of the roadbed is given by (A) \(\tan ^{-1} \frac{v^{2} g}{r}\) (B) \(\tan ^{-1} \frac{r g}{v^{2}}\) (C) \(\tan ^{-1} \frac{r}{g v^{2}}\) (D) \(\tan ^{-1} \frac{v^{2}}{r g}\)

Step-by-Step Solution

Answer

The correct formula for the slope of the roadbed in a curved path is (D) \(\tan^{-1} \frac{v^2}{rg}\).

Step 1: Identify the Forces Acting on the Vehicle

When a vehicle travels along a curved path, there are two forces acting upon it: the gravitational force (weight) and the normal force applied by the inclined road surface. The gravitational force is always acting straight downward, while the normal force is perpendicular to the sloped surface.

Step 2: Resolve Forces Into Components

To determine the slope of the inclined roadbed, we should resolve the normal force into a component parallel to the horizontal and a component that is perpendicular to the horizontal. - The horizontal component of the normal force provides the centripetal force required for the vehicle to maintain a curved path. - The vertical component of the normal force balances the gravitational force acting on the vehicle.

Step 3: Create Equation Based on Forces

Since the horizontal component of the normal force provides the centripetal force, we have: \(N_{horizontal} = \frac{mv^2}{r}\) And the vertical component of the normal force balances the gravitational force: \(N_{vertical} = mg\)

Step 4: Calculate the Slope

The slope of the roadbed is the tangent of the angle that it makes with the horizontal plane. We can determine the tangent of this angle by finding the ratio of the horizontal force component to the vertical force component. \(\tan \theta = \frac{N_{horizontal}}{N_{vertical}}\) Using the equations from Steps 3, we can substitute the expressions for the two force components: \(\tan \theta = \frac{\frac{mv^2}{r}}{mg}\) Simplifying the equation and replacing \(mg\) with \(g\) since the mass \(m\) cancels out: \(\tan \theta = \frac{v^2}{rg}\) Now, we can take the arctangent of both sides of the equation to get the angle: \(\theta = \tan^{-1} \frac{v^2}{rg}\)

Step 5: Choose the Correct Formula

Comparing the derived formula to the given options, it matches option (D): \(\theta = \tan^{-1} \frac{v^2}{rg}\) So the correct formula for the slope of the roadbed in a curved path is (D) \(\tan^{-1} \frac{v^2}{rg}\).