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Two bodies with masses M1 and M2 are connected by an inextensible rope that passes over a frictionless pulley. M1 is on the surface of a horizontal frictionless table, and M2 hangs freely as shown in the figure. Considering that M2 is equal to 2M1, the acceleration of the system is equal to

To determine the acceleration of the system, we can apply Newton's second law to each of the masses \(m_1\) and \(m_2\).

### Given:
- \( m_1 \) is on a frictionless table.
- \( m_2 \) hangs freely.
- \( m_2 = 2m_1 \).

### Forces on \( m_1 \):
- The only force acting horizontally on \( m_1 \) is the tension \( T \) in the rope.

By Newton's second law:
\[
T = m_1 a
\]

### Forces on \( m_2 \):
- The forces acting on \( m_2 \) are the gravitational force \( m_2g \) pulling it downward and the tension \( T \) pulling it upward.

By Newton's second law:
\[
m_2g - T = m_2 a
\]

### Substituting \( m_2 = 2m_1 \):
\[
2m_1g - T = 2m_1 a
\]

Now, substitute \( T = m_1 a \) from the equation for \( m_1 \):
\[
2m_1g - m_1 a = 2m_1 a
\]

Combine like terms:
\[
2m_1g = 3m_1 a
\]

Finally, solve for \( a \):
\[
a = \frac{2g}{3}
\]

### Answer:
The acceleration of the system is \(\frac{2g}{3}\).

Would you like any further details or explanations?

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Here are 5 relative questions to expand on this topic:

1. How would the acceleration change if \( m_2 = 3m_1 \)?
2. What would happen if there was friction on the table surface?
3. How does the tension in the rope change with different values of \( m_1 \) and \( m_2 \)?
4. What would be the acceleration if \( m_1 \) and \( m_2 \) were equal?
5. How would you find the tension in the rope?

**Tip:** When solving problems with pulleys, always consider the forces acting on each mass separately to apply Newton's second law correctly.

Explore how to calculate the acceleration of a system involving two masses connected by a rope over a frictionless pulley. Detailed step-by-step solution using Newton's second law of motion. Suitable for high school students studying mechanics.

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