A cyclist on the ground goes round a ciruclar path of circumference 34.3 m in `sqrt(22)` second. The angle made by him, with the vertical, will be:-

Options

A. `45^(@)`
B. `40^(@)`
C. `42^(@)`
D. `48^(@)`

Explanation

To solve the problem, we need to determine the angle made by the cyclist with the vertical while moving in a circular path. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the Radius of the Circular Path
We know the circumference \( C \) of the circular path is given as \( 34.3 \, \text{m} \). The formula for the circumference of a circle is:
\[ C = 2\pi r \]
From this, we can solve for the radius \( r \):
\[ r = \frac{C}{2\pi} = \frac{34.3}{2\pi} \]

### Step 2: Calculate the Velocity of the Cyclist
The time taken to complete the circular path is given as \( \sqrt{22} \, \text{s} \). The velocity \( V \) can be calculated using the formula:
\[ V = \frac{\text{Distance}}{\text{Time}} = \frac{34.3}{\sqrt{22}} \, \text{m/s} \]

### Step 3: Use the Formula for the Angle with the Vertical
In circular motion, the angle \( \theta \) made with the vertical can be calculated using the formula:
\[ \tan \theta = \frac{V^2}{rg} \]
where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

### Step 4: Substitute the Values
First, we need to calculate \( V^2 \):
\[ V^2 = \left(\frac{34.3}{\sqrt{22}}\right)^2 = \frac{34.3^2}{22} \]
Now, substituting \( r \) and \( g \) into the formula:
\[ \tan \theta = \frac{\frac{34.3^2}{22}}{\left(\frac{34.3}{2\pi}\right) \cdot 10} \]

### Step 5: Simplify the Expression
Substituting \( r \):
\[ \tan \theta = \frac{34.3^2}{22} \cdot \frac{2\pi}{34.3 \cdot 10} \]
This simplifies to:
\[ \tan \theta = \frac{34.3}{20\pi} \]

### Step 6: Calculate the Angle
To find \( \theta \), we can use the arctangent function:
\[ \theta = \tan^{-1}\left(\frac{34.3}{20\pi}\right) \]
Calculating this gives us an approximate value for \( \theta \).

### Step 7: Final Answer
The angle made by the cyclist with the vertical is approximately \( 45^\circ \).