A particle of mass m moves along the internal smooth surface of a vertical cylinder of the radius R. Find the force with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals `v_(0)`. And forms an angle `alpha` with the horizontal.

Options

A. `(mV_(0)^(2))/(R )sin^(2)alpha`
B. `(mV_(0)^(2))/(2R)cos^(2)alpha`
C. `(2mV_(0)^(2))/(R )cos^(2)alpha`
D. `(mV_(0)^(2))/(R )cos^(2)alpha`

Answer

To solve the problem, we need to analyze the motion of a particle of mass \( m \) moving along the internal smooth surface of a vertical cylinder of radius \( R \). The particle has an initial velocity \( v_0 \) that forms an angle \( \alpha \) with the horizontal.

### Step-by-Step Solution:

1. **Identify the Components of Velocity**:
The initial velocity \( v_0 \) can be broken down into two components:
- Horizontal component: \( v_{0x} = v_0 \cos(\alpha) \)
- Vertical component: \( v_{0y} = v_0 \sin(\alpha) \)

2. **Determine the Circular Motion Component**:
For the particle to maintain circular motion along the internal surface of the cylinder, only the horizontal component of the velocity contributes to the centripetal force. Thus, we focus on \( v_{0x} \):
\[ v_{circular} = v_0 \cos(\alpha) \]

3. **Centripetal Force Requirement**:
The centripetal force required to keep the particle in circular motion is provided by the normal force \( N \) exerted by the wall of the cylinder. The formula for centripetal force is:
\[ F_{centripetal} = \frac{m v_{circular}^2}{R} \]

4. **Substituting the Circular Velocity**:
Substitute \( v_{circular} \) into the centripetal force equation:
\[ F_{centripetal} = \frac{m (v_0 \cos(\alpha))^2}{R} \]

5. **Simplifying the Expression**:
Simplifying the expression gives:
\[ F_{centripetal} = \frac{m v_0^2 \cos^2(\alpha)}{R} \]

6. **Conclusion**:
The force with which the particle acts on the cylinder wall (which is equal to the normal force \( N \)) is:
\[ N = \frac{m v_0^2 \cos^2(\alpha)}{R} \]

### Final Answer:
The force with which the particle acts on the cylinder wall is:
\[ N = \frac{m v_0^2 \cos^2(\alpha)}{R} \]