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Duas massas M e m estão presas a um fio de massa desprezível como mostra a figura. Sabendo que 
M=2m, determine a aceleração da massa M e a tensão no fio.

This problem involves a system with two masses, \( M \) and \( m \), connected by a string over a pulley. The mass \( M \) is given as \( M = 2m \), and we are to determine the acceleration of mass \( M \) and the tension in the string.

### Step 1: Establishing the equations of motion
Let's denote the acceleration of the system by \( a \) and the tension in the string by \( T \). 

For mass \( M \):
- The forces acting on \( M \) are its weight \( Mg \) (downward) and the tension \( T \) (upward).
- The net force on \( M \) is:
\[
M \cdot a = Mg - T \quad \text{(Equation 1)}
\]

For mass \( m \):
- The forces acting on \( m \) are the tension \( T \) (upward) and its weight \( mg \) (downward).
- The net force on \( m \) is:
\[
m \cdot a = T - mg \quad \text{(Equation 2)}
\]

### Step 2: Solving the system of equations
We have the two equations:
1. \( M \cdot a = Mg - T \)
2. \( m \cdot a = T - mg \)

Substituting \( M = 2m \) into Equation 1:
\[
2m \cdot a = 2mg - T \quad \text{(Equation 3)}
\]

Now, adding Equation 2 and Equation 3 to eliminate \( T \):
\[
2m \cdot a + m \cdot a = 2mg - T + T - mg
\]
\[
3m \cdot a = mg
\]
\[
a = \frac{mg}{3m} = \frac{g}{3}
\]
Thus, the acceleration \( a = \frac{g}{3} \).

### Step 3: Finding the tension \( T \)
Substitute \( a = \frac{g}{3} \) into Equation 2:
\[
m \cdot \frac{g}{3} = T - mg
\]
\[
T = mg + \frac{mg}{3} = \frac{3mg + mg}{3} = \frac{4mg}{3}
\]

### Summary of Results:
- The acceleration of mass \( M \) is \( \frac{g}{3} \) (approximately \( 3.33 \, \text{m/s}^2 \) if \( g = 10 \, \text{m/s}^2 \)).
- The tension in the string is \( \frac{4mg}{3} \).

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

### Related Questions:
1. What would happen to the acceleration if \( M = m \)?
2. How would the tension change if the masses were equal?
3. What if the pulley had a significant mass? How would this affect the system?
4. How would you calculate the acceleration if there was friction in the pulley?
5. What would happen if the string was not massless?

### Tip:
When solving pulley problems, ensure you correctly identify the forces acting on each mass and carefully set up the equations of motion. This helps in accurately determining the system's dynamics.

Learn how to solve a physics problem involving two masses, M and m, connected by a string over a pulley. Determine the acceleration of mass M and the tension in the string using Newton's laws of motion and equilibrium equations. Suitable for high school students.

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