What is the magnitude of the acceleration of the blocks?

Steps to Solve

1. Identify the Forces Acting on Each Block

For block $m_1$ (mass $3.2 , \text{kg}$), which is on a frictionless surface, the only forces acting on it are the tension $T$ in the string pulling it to the right.

For block $m_2$ (mass $5.5 , \text{kg}$), which is hanging, the forces acting on it are the gravitational force $m_2 g$ (downward) and the tension $T$ (upward).

2. Write Down the Equations Based on Newton's Second Law

For block $m_1$: $$ T = m_1 a $$ This equation states that the tension in the string provides the acceleration $a$ to block $m_1$.

For block $m_2$: $$ m_2 g - T = m_2 a $$ This equation expresses that the weight of block $m_2$ minus the tension $T$ in the string is equal to the mass times its acceleration.

3. Substitute Known Values and Rearrange the Equations

From the first equation, substitute $T$ into the second equation: $$ m_2 g - m_1 a = m_2 a $$

Rearranging this gives: $$ m_2 g = m_2 a + m_1 a $$

4. Factor and Solve for Acceleration $a$

Factor out $a$ on the right side: $$ m_2 g = (m_1 + m_2) a $$

Then, solve for $a$: $$ a = \frac{m_2 g}{m_1 + m_2} $$

5. Substitute the Values into the Equation

Substituting the known values: $$ a = \frac{5.5 , \text{kg} \times 9.80 , \text{m/s}^2}{3.2 , \text{kg} + 5.5 , \text{kg}} $$

6. Calculate the Result

Calculate the values: $$ a = \frac{53.9}{8.7} \approx 6.2 , \text{m/s}^2 $$