To determine the acceleration of the hanging block, we need to apply Newton's second law of motion to both blocks and solve for the unknown acceleration. Here's how we approach the problem step by step:

Step 1: Set up the forces

• For the block on the incline (mass = 5.7 kg), there are two forces to consider:

  1. Gravitational force acting vertically downward, $F_g = m_1 g = 5.7 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 55.86 \, \text{N}$.
  2. The force of gravity along the incline: $F_{\parallel} = m_1 g \sin(\theta) = 5.7 \times 9.8 \times \sin(35^\circ)$.

• For the hanging block (mass = 4.1 kg), there are:

  1. Gravitational force acting vertically downward, $F_{g,2} = m_2 g = 4.1 \times 9.8 = 40.18 \, \text{N}$.

Step 2: Net force and tension in the string

Let $T$ be the tension in the string, and let $a$ be the acceleration of both blocks (they move together, so they have the same acceleration).

• For the block on the incline: $m_1 a = T - m_1 g \sin(\theta)$

• For the hanging block: $m_2 a = m_2 g - T$

Step 3: Combine the equations

Now we have two equations:

1. $m_1 a = T - m_1 g \sin(\theta)$
2. $m_2 a = m_2 g - T$

We can solve this system by adding the two equations to eliminate $T$: $m_1 a + m_2 a = m_2 g - m_1 g \sin(\theta)$ Factor out $a$ on the left side: $a (m_1 + m_2) = m_2 g - m_1 g \sin(\theta)$ Now, solve for $a$: $a = \frac{m_2 g - m_1 g \sin(\theta)}{m_1 + m_2}$

Step 4: Plug in the values

Substitute the known values:

• $m_1 = 5.7 \, \text{kg}$
• $m_2 = 4.1 \, \text{kg}$
• $g = 9.8 \, \text{m/s}^2$
• $\theta = 35^\circ$

$a = \frac{(4.1 \times 9.8) - (5.7 \times 9.8 \times \sin(35^\circ))}{5.7 + 4.1}$ $a = \frac{40.18 - 5.7 \times 9.8 \times 0.5736}{9.8}$ $a = \frac{40.18 - 32.05}{9.8}$ $a = \frac{8.13}{9.8}$ $a \approx 0.83 \, \text{m/s}^2$

Final Answer:

The acceleration of the hanging block is approximately $0.83 \, \text{m/s}^2$.

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Would you like further details on any of the steps or concepts? Here are some related questions for further practice:

1. How would the acceleration change if the angle of inclination were increased?
2. How would the tension in the string compare for the two blocks?
3. What would happen if the surface were not frictionless?
4. How do you solve for acceleration in a system with more than two blocks?
5. How can you find the speed of the blocks after a certain time?

Tip: To solve problems like this, always break the forces into components along the direction of motion (usually parallel and perpendicular to the incline).