Let's work through the problem step-by-step to find the acceleration of the system involving two masses [tex]\( m_1 = 5 \, \text{kg} \)[/tex] and [tex]\( m_2 = 48 \, \text{kg} \)[/tex] hanging over a light, frictionless pulley. We are asked to find the acceleration of the system.
### Step 1: Analyze the forces acting on each mass
- For the mass [tex]\( m_1 \)[/tex]:
- The only force acting on [tex]\( m_1 \)[/tex] is its weight [tex]\((m_1 \cdot g)\)[/tex], which acts downward.
- For the mass [tex]\( m_2 \)[/tex]:
- The only force acting on [tex]\( m_2 \)[/tex] is its weight [tex]\((m_2 \cdot g)\)[/tex], which acts downward.
### Step 2: Set up the equations of motion
Since the pulley is frictionless, both masses will accelerate with the same magnitude of acceleration [tex]\( a \)[/tex].
- For [tex]\( m_1 \)[/tex]:
[tex]\[
T - m_1 \cdot g = m_1 \cdot a
\][/tex]
- For [tex]\( m_2 \)[/tex]:
[tex]\[
m_2 \cdot g - T = m_2 \cdot a
\][/tex]
Here, [tex]\( T \)[/tex] represents the tension in the string.
### Step 3: Combine the equations
To eliminate [tex]\( T \)[/tex], let's add the two equations together:
[tex]\[
(m_2 \cdot g - T) + (T - m_1 \cdot g) = m_2 \cdot a + m_1 \cdot a
\][/tex]
This simplifies to:
[tex]\[
m_2 \cdot g - m_1 \cdot g = (m_1 + m_2) \cdot a
\][/tex]
### Step 4: Solve for acceleration [tex]\( a \)[/tex]
Factor out [tex]\( g \)[/tex] on the left side of the equation:
[tex]\[
(m_2 - m_1) \cdot g = (m_1 + m_2) \cdot a
\][/tex]
Now, solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{(m_2 - m_1) \cdot g}{m_1 + m_2}
\][/tex]
### Step 5: Substitute the given values [tex]\( m_1 = 5 \, \text{kg} \)[/tex], [tex]\( m_2 = 48 \, \text{kg} \)[/tex], and [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
[tex]\[
a = \frac{(48 \, \text{kg} - 5 \, \text{kg}) \cdot 9.8 \, \text{m/s}^2}{5 \, \text{kg} + 48 \, \text{kg}}
\][/tex]
This simplifies to:
[tex]\[
a = \frac{43 \cdot 9.8}{53}
\][/tex]
### Conclusion
The result is an acceleration [tex]\( a \approx 7.950943396226416 \, \text{m/s}^2 \)[/tex].
Thus, the correct answer is:
[tex]\[ a \approx 7.951 \, \text{m/s}^2 \][/tex]
