The provided image outlines the solution to a problem involving Newton's Second Law to calculate the system acceleration and tensions in a system of masses. Here's a breakdown of the solution:

Steps for Solving:

(a) Calculate System Acceleration

1. The total mass of the system is computed: $M = m_1 + m_2 + m_3 = 12.0 \, \text{kg} + 24.0 \, \text{kg} + 31.0 \, \text{kg} = 67.0 \, \text{kg}.$
2. The force $T_3 = 65.0 \, \text{N}$ is applied to the system.
3. Using Newton's Second Law: $F = M \cdot a \implies a = \frac{F}{M}.$
4. Substituting values: $a = \frac{65.0}{67.0} \approx 0.97 \, \text{m/s}^2.$

(b) Calculate Tensions $T_1$ and $T_2$:

Tension $T_2$:

• Focus on blocks $m_1$ and $m_2$.
• Combined mass: $m_1 + m_2 = 12.0 \, \text{kg} + 24.0 \, \text{kg} = 36.0 \, \text{kg}$.
• Using $T_2 = (m_1 + m_2) \cdot a$: $T_2 = 36.0 \cdot 0.97 \approx 34.92 \, \text{N}.$

Tension $T_1$:

• Focus on block $m_1$.
• Using $T_1 = m_1 \cdot a$: $T_1 = 12.0 \cdot 0.97 \approx 11.64 \, \text{N}.$

Results:

• System acceleration: $a \approx 0.97 \, \text{m/s}^2$.
• Tension $T_1$: $T_1 \approx 11.64 \, \text{N}$.
• Tension $T_2$: $T_2 \approx 34.92 \, \text{N}$.

Follow-Up Questions:

1. How does the inclusion of friction change the calculation of acceleration and tension?
2. What would the tensions $T_1$ and $T_2$ be if the system's acceleration were doubled?
3. How can you verify these results experimentally?
4. What if an additional force acted on $m_1$ in the opposite direction?
5. How do these calculations change if the pulleys have mass?

Tip:

Always double-check units and assumptions (like negligible friction or ideal pulleys) to ensure they align with real-world setups.