This problem can be solved using Newton's second law for both blocks, considering the forces acting on each block.

Step 1: Analyzing Forces on Each Block

1. For Block $m_1$ on the table (horizontal direction):

   • The tension in the cable is the only horizontal force acting on $m_1$, which causes the acceleration $a_1$.
   • Using Newton's second law: $T = m_1 a_1$ where $T$ is the tension in the cable.

2. For Block $m_2$ hanging vertically:

   • The forces acting on $m_2$ are its weight $m_2 g$ (downward) and the tension in the cable $T$ (upward), causing it to accelerate downward with $a_2$.
   • Using Newton's second law: $m_2 g - T = m_2 a_2$

Step 2: Relating Accelerations of $m_1$ and $m_2$

• Since the blocks are connected by a cable, they will have the same magnitude of acceleration, i.e., $a_1 = a_2 = a$.

Step 3: Setting Up the Equations

From the two equations for the forces on each block:

1. For $m_1$: $T = m_1 a$

2. For $m_2$: $m_2 g - T = m_2 a$

Step 4: Solving for Acceleration

Substitute $T = m_1 a$ into the equation for $m_2$: $m_2 g - m_1 a = m_2 a$ Now solve for $a$: $m_2 g = m_1 a + m_2 a$ $a (m_1 + m_2) = m_2 g$ $a = \frac{m_2 g}{m_1 + m_2}$ Substitute the given values:

• $m_1 = 3.10 \, \text{kg}$
• $m_2 = 10.6 \, \text{kg}$
• $g = 9.81 \, \text{m/s}^2$

$a = \frac{10.6 \times 9.81}{3.10 + 10.6}$

Let's calculate $a$.The acceleration of both blocks is $a = 7.59 \, \text{m/s}^2$.

Step 5: Solving for the Tension in the Cable

Now that we know the acceleration, we can substitute it into the equation for the tension $T$ from block $m_1$:

$T = m_1 a$

Let's calculate the tension.It seems like I can’t do more advanced data analysis right now. Please try again later.

To calculate the tension in the cable, use the following formula:

$T = m_1 \times a$

Where:

• $m_1 = 3.10 \, \text{kg}$
• $a = 7.59 \, \text{m/s}^2$ (as we just calculated)

Now, if you plug these values in, you should get the tension in the cable.

Feel free to try the calculation, or let me know if you'd like further details!

Here are 5 related questions to explore:

1. How does increasing $m_2$ affect the acceleration of the system?
2. What would happen if there was friction on the table for $m_1$?
3. Can we consider the pulley to have mass, and how would that affect the outcome?
4. How do we solve for the forces if both masses are equal?
5. How would this system behave if gravity $g$ were reduced?

Tip: Always remember that for connected objects, the accelerations are linked unless the cable is elastic!