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A figura ao lado mostra dois blocos de massas m = 2,5 kg e M = 6,5 kg, 
ligados por um fio que passa sem atrito por uma roldana. Despreze as 
massas do fio e da roldana e suponha que a aceleração da gravidade vale g 
= 10 m/s2. 0 bloco de massa M está apoiado sobre a plataforma P e a força 
F aplicada sobre a roldana é suficiente apenas para manter o bloco de massa 
m em equilíbrio estático na posição indicada. Determine o valor de F e a 
intensidade da força R que a plataforma exerce sobre M.

The problem involves a pulley system with two blocks of masses \( m = 2.5 \, \text{kg} \) and \( M = 6.5 \, \text{kg} \), connected by a string that passes over a frictionless pulley. The force \( F \) is applied upward on the pulley to keep the smaller block \( m \) in static equilibrium. We are asked to find the value of \( F \) and the reaction force \( R \) that the platform exerts on the block \( M \).

### Step 1: Analyzing the forces on block \( m \)
For the block \( m \) to be in equilibrium, the net force acting on it must be zero. The forces acting on \( m \) are:
- The gravitational force \( mg = m \times g = 2.5 \times 10 = 25 \, \text{N} \) downward.
- The tension \( T \) in the string acting upward.

Since the block \( m \) is in equilibrium:
\[
T = mg = 25 \, \text{N}
\]

### Step 2: Analyzing the forces on block \( M \)
The block \( M \) has the following forces acting on it:
- The gravitational force \( Mg = M \times g = 6.5 \times 10 = 65 \, \text{N} \) downward.
- The tension \( T = 25 \, \text{N} \) upward.

The net force acting on block \( M \) due to the tension is:
\[
\text{Net force on } M = Mg - T = 65 - 25 = 40 \, \text{N}
\]

This net force is balanced by the normal force \( R \) exerted by the platform on \( M \), which is:
\[
R = 40 \, \text{N}
\]

### Step 3: Determining the force \( F \) applied to the pulley
The force \( F \) applied on the pulley must support both the weight of block \( m \) (through the tension \( T \)) and the weight of block \( M \). Therefore, the force \( F \) required to maintain the equilibrium of block \( m \) is equal to twice the tension in the string (since the pulley effectively supports the weight on both sides of the string):
\[
F = 2T = 2 \times 25 = 50 \, \text{N}
\]

### Summary of Results:
- The applied force \( F = 50 \, \text{N} \).
- The reaction force \( R = 40 \, \text{N} \).

Would you like more details on any part of this explanation? 

### Related Questions:
1. How would the force \( F \) change if the mass \( m \) was increased?
2. What would happen if the pulley had mass?
3. How does the platform's reaction force \( R \) depend on the mass \( M \)?
4. How would the system behave if friction were present on the platform?
5. What would be the acceleration of the system if \( F \) were less than \( 50 \, \text{N} \)?

### Tip:
Always consider the role of the tension in the string when analyzing pulley systems, as it directly influences the forces on the masses involved.

Explore how to determine the force F and reaction force R in a pulley system with two masses, m = 2.5 kg and M = 6.5 kg, connected by a frictionless pulley. Understand the equilibrium conditions and tension in the string. Suitable for advanced high school students studying mechanics.

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