This is a classic physics problem involving a system of two blocks, one on a surface (block $M$) and the other hanging vertically (block $m$) through a pulley. Below is the analysis and explanation of the system:

Known Variables:

• Mass of block on the surface ($M$) = $8 \, \text{kg}$
• Mass of hanging block ($m$) = $10 \, \text{kg}$

Assumptions:

1. The surface is frictionless unless otherwise specified.
2. The pulley is ideal (no mass or friction in the pulley).
3. The string is inextensible and massless.

Forces Acting on the Blocks:

For Block $M$ (on the surface):

• Horizontal force due to tension in the string ($T$).

For Block $m$ (hanging):

• Weight force downward ($F_{\text{gravity}} = m g$).
• Upward force due to the tension in the string ($T$).

Equations of Motion:

Using Newton's Second Law ($F = ma$):

For Block $M$:

$T = M a \quad \text{(1)}$

For Block $m$:

$m g - T = m a \quad \text{(2)}$

Where:

• $T$ is the tension in the string,
• $a$ is the acceleration of the system,
• $g$ is the acceleration due to gravity ($9.8 \, \text{m/s}^2$).

Solving for Acceleration ($a$):

From Equation (1) and Equation (2), eliminate $T$:

$m g - M a = m a$

Rearranging for $a$: $a = \frac{m g}{M + m}$

Substitute the values: $a = \frac{10 \cdot 9.8}{8 + 10} = \frac{98}{18} \approx 5.44 \, \text{m/s}^2$

Solving for Tension ($T$):

Substitute $a$ into Equation (1): $T = M a = 8 \cdot 5.44 \approx 43.52 \, \text{N}$

Results:

1. Acceleration of the system ($a$): $5.44 \, \text{m/s}^2$
2. Tension in the string ($T$): $43.52 \, \text{N}$

Let me know if you'd like further details or assistance! Here are some related questions:

1. How would the result change if there were friction on the surface?
2. What is the net force acting on each block individually?
3. How would the acceleration change if $M$ and $m$ were swapped?
4. What would happen if the mass of the pulley were considered?
5. Can this setup be used to find $g$ experimentally?

Tip: Always ensure forces are balanced correctly for systems in equilibrium or constant acceleration.