A motor cyclist is going in a vertical circle what is the necessary condition so that he may not fall down ?

Options

A. the force of gravity disappears.
B. he loses weight some how.
C. he is kept in this path due to the force exerted by surrounding air.
D. the frictional force of the wall balances his weight

Answer

To determine the necessary condition for a motorcyclist to not fall while going in a vertical circle, we need to analyze the forces acting on the motorcyclist and the motorcycle.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- When a motorcyclist is moving in a vertical circle, two main forces act on him: the gravitational force (weight) acting downwards and the centripetal force required to keep him moving in a circular path.

2. **Centripetal Force Requirement**:
- For an object to move in a circle, it must have a net inward force acting towards the center of the circle. This is known as the centripetal force. The formula for centripetal force (\(F_c\)) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \(m\) is the mass of the motorcyclist, \(v\) is the velocity, and \(r\) is the radius of the circular path.

3. **Condition at the Top of the Circle**:
- At the top of the vertical circle, the gravitational force contributes to the centripetal force. The condition for the motorcyclist to not fall at the top is:
\[ mg + N = \frac{mv^2}{r} \]
where \(N\) is the normal force. For the motorcyclist to just stay on the motorcycle without falling, \(N\) can be zero. Thus, the condition simplifies to:
\[ mg = \frac{mv^2}{r} \]
This implies:
\[ v^2 = rg \]
Therefore, the minimum speed \(v\) at the top of the circle must be:
\[ v \geq \sqrt{rg} \]

4. **Conclusion**:
- The necessary condition for the motorcyclist to not fall down while moving in a vertical circle is that he must maintain a speed greater than or equal to \(\sqrt{rg}\) at the top of the circle.

### Final Answer:
The necessary condition for a motorcyclist to not fall down while going in a vertical circle is to maintain a speed of at least \(\sqrt{rg}\) at the top of the circle.

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