A cyclist is moving in a circular track of radius 80 m,, with a velocity of 36 km/ hour . In order to keep his balance , he has to lean inwards from the velocity through an angle `theta ` if `g= 10 m//s^(2)` , then ` theta ` is given by

Options

A. `tan ^(-1) (2)`
B. `tan ^(-1) (4)`
C. `tan ^(-1) ((1)/(4))`
D. ` tan ^(-1) ((1)/(8))`

Explanation

To solve the problem, we need to find the angle \( \theta \) at which the cyclist must lean inwards while moving in a circular path. We will use the concepts of circular motion and forces acting on the cyclist.

### Step-by-Step Solution:

1. **Convert the velocity from km/h to m/s**:
\[ v = 36 \text{ km/h} = \frac{36 \times 1000}{3600} \text{ m/s} = 10 \text{ m/s} \]

**Hint**: Remember that to convert km/h to m/s, you multiply by \( \frac{1000}{3600} \).

2. **Identify the forces acting on the cyclist**:
- The gravitational force \( F_g = mg \) acts downwards.
- The normal force \( N \) acts perpendicular to the surface of the track.
- The cyclist experiences a centripetal force due to circular motion, which is provided by the horizontal component of the normal force.

3. **Use the relationship between the forces**:
In circular motion, the required centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the cyclist, \( v \) is the velocity, and \( r \) is the radius of the circular path.

4. **Set up the equations**:
The angle \( \theta \) can be determined from the balance of forces. The horizontal component of the normal force provides the centripetal force:
\[ N \sin(\theta) = \frac{mv^2}{r} \]
The vertical component of the normal force balances the weight of the cyclist:
\[ N \cos(\theta) = mg \]

5. **Divide the two equations**:
\[ \frac{N \sin(\theta)}{N \cos(\theta)} = \frac{mv^2/r}{mg} \]
This simplifies to:
\[ \tan(\theta) = \frac{v^2}{rg} \]

6. **Substitute the known values**:
- \( v = 10 \text{ m/s} \)
- \( r = 80 \text{ m} \)
- \( g = 10 \text{ m/s}^2 \)

\[ \tan(\theta) = \frac{10^2}{80 \times 10} = \frac{100}{800} = \frac{1}{8} \]

7. **Calculate \( \theta \)**:
\[ \theta = \tan^{-1}\left(\frac{1}{8}\right) \]

Using a calculator:
\[ \theta \approx 7.125^\circ \]

### Final Answer:
The angle \( \theta \) at which the cyclist must lean inwards is approximately \( 7.125^\circ \).

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