A circular road of radius 50 m has the angel of banking equal to `30^0`. At what speed should a vehicle go on this road so that the friction is not used?

Options

A. 25 m/s
B. 17 m/s
C. 6 m/s
D. 11 m/s

Explanation

To solve the problem of determining the speed at which a vehicle should travel on a banked circular road without relying on friction, we can follow these steps:

### Step 1: Understand the Forces Acting on the Vehicle
When a vehicle is on a banked circular road, the forces acting on it include:
- The gravitational force (mg) acting downwards.
- The normal force (N) acting perpendicular to the surface of the road.
- The centripetal force required for circular motion, which is provided by the horizontal component of the normal force.

### Step 2: Set Up the Banking Angle and Radius
Given:
- Radius of the circular road, \( r = 50 \, \text{m} \)
- Angle of banking, \( \theta = 30^\circ \)

### Step 3: Write the Equations for Forces
For the vehicle to move in a circle without friction, the following equations must hold:
1. The vertical forces must balance:
\[ N \cos \theta = mg \]
2. The horizontal forces provide the centripetal force:
\[ N \sin \theta = \frac{mv^2}{r} \]

### Step 4: Eliminate Normal Force (N)
From the first equation, we can express \( N \):
\[ N = \frac{mg}{\cos \theta} \]
Substituting \( N \) into the second equation:
\[ \frac{mg}{\cos \theta} \sin \theta = \frac{mv^2}{r} \]

### Step 5: Simplify the Equation
Cancel \( m \) from both sides:
\[ \frac{g \sin \theta}{\cos \theta} = \frac{v^2}{r} \]
This can be rewritten using the tangent function:
\[ g \tan \theta = \frac{v^2}{r} \]

### Step 6: Solve for Speed (v)
Rearranging the equation gives:
\[ v^2 = g r \tan \theta \]
Substituting \( g = 10 \, \text{m/s}^2 \), \( r = 50 \, \text{m} \), and \( \theta = 30^\circ \):
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus,
\[ v^2 = 10 \cdot 50 \cdot \frac{1}{\sqrt{3}} = \frac{500}{\sqrt{3}} \]

### Step 7: Calculate the Speed
Taking the square root:
\[ v = \sqrt{\frac{500}{\sqrt{3}}} \]
Calculating this gives:
\[ v \approx 16.99 \, \text{m/s} \approx 17 \, \text{m/s} \]

### Final Answer
The speed at which the vehicle should go on this road so that friction is not used is approximately **17 m/s**.

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