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The problem asks for the reaction force exerted by the plane on the sphere. We can analyze this by breaking down the forces acting on the sphere and resolving them along the direction of the inclined plane and perpendicular to the plane.

### Steps:

1. **Forces on the Sphere:**
   - The weight of the sphere \( W = m \cdot g = 8 \, \text{kg} \times 10 \, \text{m/s}^2 = 80 \, \text{N} \), acting vertically downward.
   - A reaction force \( N \), perpendicular to the plane, is what we're solving for.
   - Since the surface is smooth (frictionless), there's no frictional force.

2. **Breaking Down the Weight:**
   The weight of the sphere can be split into two components:
   - One along the plane: \( W_{\parallel} = W \sin(37^\circ) \)
   - One perpendicular to the plane: \( W_{\perp} = W \cos(37^\circ) \)

3. **Equilibrium in the Perpendicular Direction:**
   In the direction perpendicular to the plane, the reaction force \( N \) must balance the perpendicular component of the weight:
   \[
   N = W_{\perp} = W \cos(37^\circ)
   \]
   We substitute the values:
   \[
   N = 80 \, \text{N} \cdot \cos(37^\circ)
   \]
   Using \( \cos(37^\circ) \approx 0.7986 \):
   \[
   N = 80 \times 0.7986 = 63.89 \, \text{N}
   \]

Thus, the magnitude of the normal reaction force exerted by the plane is approximately **64 N**.

However, the options provided seem to suggest smaller values, which indicates that we should revisit the precise interpretation or approximation of angles, or if the force is considering other external factors like tensions. Based on this, let's proceed with an exact graphical analysis if needed. Would you like more clarification or a detailed solution sketch? 

### Five questions to extend your knowledge:
1. What are the differences between the normal force and the weight force?
2. How would the reaction force change if the angle of inclination increased?
3. Can you explain why friction is not considered in this scenario?
4. What would happen if the surface wasn't smooth but had friction?
5. How does the component of weight along the plane affect motion?

**Tip:** In inclined plane problems, always resolve forces parallel and perpendicular to the surface for equilibrium analysis.

Determine el módulo de la reacción del plano para una esfera de masa m = 8 kg en un plano inclinado liso a un ángulo de 37 grados, con una cuerda atada a un ángulo de 53 grados. La gravedad es g = 10 m/s².

Determine the reaction force acting on an 8 kg sphere placed on a smooth inclined plane with an angle of 37 degrees. The problem uses Newton's laws and force resolution concepts to calculate the normal force with gravity at 10 m/s².

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