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?

Explanation

To solve the problem of determining the speed at which a vehicle should travel on a circular road of radius 50 m with a banking angle of \(30^\circ\) such that friction is not used, we can follow these steps:

### Step 1: Understand the Forces Acting on the Vehicle
When a vehicle moves on a banked circular road, the forces acting on it are:
- The gravitational force (\(mg\)) acting downwards.
- The normal force (\(N\)) acting perpendicular to the surface of the road.

### Step 2: Set Up the Force Equations
Since the vehicle is on a banked curve and friction is not used, the normal force provides the necessary centripetal force for circular motion. The components of the normal force can be expressed as:
- \(N \sin \theta\) provides the centripetal force.
- \(N \cos \theta\) balances the weight of the vehicle (\(mg\)).

### Step 3: Write the Equations
From the force balance, we have:
1. \(N \sin \theta = \frac{mv^2}{r}\) (centripetal force)
2. \(N \cos \theta = mg\) (weight balance)

### Step 4: Divide the Two Equations
Dividing the first equation by the second gives:
\[ \frac{N \sin \theta}{N \cos \theta} = \frac{mv^2/r}{mg} \]
This simplifies to:
\[ \tan \theta = \frac{v^2}{rg} \]

### Step 5: Substitute the Values
Given:
- \(\theta = 30^\circ\)
- \(r = 50 \, \text{m}\)
- \(g = 9.81 \, \text{m/s}^2\)

We know that \(\tan 30^\circ = \frac{1}{\sqrt{3}}\). Substituting this into the equation gives:
\[ \frac{1}{\sqrt{3}} = \frac{v^2}{50 \cdot 9.81} \]

### Step 6: Solve for \(v^2\)
Rearranging the equation to solve for \(v^2\):
\[ v^2 = \frac{50 \cdot 9.81}{\sqrt{3}} \]

### Step 7: Calculate \(v\)
Now, calculate \(v\):
\[ v = \sqrt{\frac{50 \cdot 9.81}{\sqrt{3}}} \]
Calculating the values:
\[ v = \sqrt{\frac{490.5}{\sqrt{3}}} \approx \sqrt{283.5} \approx 16.82 \, \text{m/s} \]

### Final Answer
The speed at which the vehicle should go on this road so that friction is not used is approximately \(16.82 \, \text{m/s}\).

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