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 the particle moving along the internal surface of a vertical cylinder. The particle has a mass \( m \) and an initial velocity \( v_0 \) forming 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. **Centripetal Acceleration:**
As the particle moves in a circular path along the cylinder, it experiences centripetal acceleration. The centripetal acceleration \( a_c \) is given by:
\[ a_c = \frac{v^2}{R} \]
where \( v \) is the tangential speed of the particle.

3. **Tangential Speed:**
Since the particle is moving with an initial velocity \( v_0 \) at an angle \( \alpha \), the tangential speed \( v \) at any point can be considered as the horizontal component of the velocity:
\[ v = v_0 \cos(\alpha) \]

4. **Calculate Centripetal Force:**
The centripetal force \( F_c \) required to keep the particle moving in a circular path is given by:
\[ F_c = m a_c = m \frac{v^2}{R} = m \frac{(v_0 \cos(\alpha))^2}{R} \]
Simplifying this, we get:
\[ F_c = m \frac{v_0^2 \cos^2(\alpha)}{R} \]

5. **Normal Force:**
The force with which the particle acts on the cylinder wall is equal to the normal force \( N \) exerted by the wall on the particle. In this case, the normal force is equal to the centripetal force required to keep the particle in circular motion:
\[ N = F_c = m \frac{v_0^2 \cos^2(\alpha)}{R} \]

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