To solve this problem, let’s denote the following variables for convenience:

• $m_1 = 6 \, \text{kg}$: mass of the block on the horizontal surface.
• $m_2 = 4 \, \text{kg}$: mass of the hanging block.
• $\mu_k = 0.5$: coefficient of kinetic friction between $m_1$ and the horizontal surface.
• $g = 9.8 \, \text{m/s}^2$: acceleration due to gravity.

Step 1: Identify the Forces on Each Block

For Block $m_1$ on the Horizontal Surface:

• Gravitational force: $F_{g1} = m_1 \cdot g = 6 \cdot 9.8 = 58.8 \, \text{N}$.
• Normal force (equal to gravitational force since it's on a horizontal surface): $N = 58.8 \, \text{N}$.
• Frictional force opposing motion: $F_f = \mu_k \cdot N = 0.5 \cdot 58.8 = 29.4 \, \text{N}$.
• Tension in the cord: $T$ (this is unknown and is the same on both blocks since the pulley is light and frictionless).

For Block $m_2$ (the Hanging Block):

• Gravitational force: $F_{g2} = m_2 \cdot g = 4 \cdot 9.8 = 39.2 \, \text{N}$.
• Tension in the cord: $T$.

Step 2: Set Up the Equations of Motion

The system accelerates in the direction of the hanging block, so we can apply Newton’s Second Law for each block.

For $m_1$:

$T - F_f = m_1 \cdot a$ Substituting $F_f = 29.4 \, \text{N}$: $T - 29.4 = 6a \quad \text{(Equation 1)}$

For $m_2$:

$m_2 \cdot g - T = m_2 \cdot a$ Substituting $F_{g2} = 39.2 \, \text{N}$: $39.2 - T = 4a \quad \text{(Equation 2)}$

Step 3: Solve for Acceleration and Tension

Add Equations 1 and 2 to eliminate $T$: $(T - 29.4) + (39.2 - T) = 6a + 4a$ $9.8 = 10a$ $a = \frac{9.8}{10} = 0.98 \, \text{m/s}^2$

Now, substitute $a = 0.98 \, \text{m/s}^2$ into Equation 1 to find $T$: $T - 29.4 = 6 \times 0.98$ $T - 29.4 = 5.88$ $T = 5.88 + 29.4 = 35.28 \, \text{N}$

Final Answers:

• (a) The acceleration of each block is $0.98 \, \text{m/s}^2$.
• (b) The tension on the cord is $35.28 \, \text{N}$.

Would you like more details, or do you have any questions about this problem?

Related Questions

1. How would changing the coefficient of kinetic friction affect the acceleration?
2. What would happen if the mass of $m_2$ were increased?
3. How does the tension in the cord compare to the weight of $m_1$ and $m_2$?
4. If the pulley were not frictionless, how would it affect the system?
5. What would the acceleration be if the surface were frictionless?

Tip:

In pulley problems, always start by analyzing forces on each mass separately and set up equations for each block using Newton's Second Law.