To find the coefficient of kinetic friction ($\mu_k$) between mass $m_2$ and the surface, we need to analyze the forces acting on both masses.

Step 1: Analyze forces on $m_1$

For $m_1$, which is hanging vertically:

• The force due to gravity on $m_1$ is $F_{g1} = m_1 g$, where $g = 9.8 \, \text{m/s}^2$.
• The tension in the rope is $T$, which opposes the downward motion.

The net force on $m_1$ leads to its downward acceleration $a = 0.878 \, \text{m/s}^2$. Applying Newton's second law for $m_1$:

$m_1 g - T = m_1 a$

Step 2: Analyze forces on $m_2$

For $m_2$, which is on a horizontal surface:

• The gravitational force $m_2 g$ is balanced by the normal force $N$.
• The frictional force $f_k = \mu_k N$ opposes the motion.
• The tension in the rope $T$ pulls to the right, and the object experiences an acceleration $a = 0.878 \, \text{m/s}^2$.

The normal force $N$ is equal to the weight of $m_2$,