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Let's analyze the problem step by step and solve it using formulas.

### Problem Analysis
We have an inclined plane with a block of weight \( w_1 = 200 \, \text{N} \) resting on it, connected by a pulley to a hanging block of unknown weight \( w_2 \). The incline is at an angle of \( 35^\circ \) with the horizontal. The problem states that there is no friction between the block and the inclined plane.

### Free-Body Diagrams and Equations
1. **Block on the Inclined Plane (\(w_1\)):**
   - The gravitational force acting on the block is \( w_1 = 200 \, \text{N} \).
   - This force can be decomposed into two components:
     - **Parallel to the plane:** \( w_1 \sin(\theta) \)
     - **Perpendicular to the plane:** \( w_1 \cos(\theta) \)

2. **Hanging Block (\(w_2\)):**
   - The gravitational force acting on the hanging block is \( w_2 \).

### Equilibrium Condition
Since there is no friction and the system is in equilibrium, the tension \( T \) in the rope is the same on both sides of the pulley.

Thus, we can write the force balance equations as:
\[
T = w_1 \sin(\theta)
\]
\[
T = w_2
\]

By setting these equal to each other, we find:
\[
w_1 \sin(\theta) = w_2
\]

### Solving for \(w_2\)
Given:
- \( w_1 = 200 \, \text{N} \)
- \( \theta = 35^\circ \)

We calculate \( w_2 \) using:
\[
w_2 = w_1 \sin(\theta)
\]

Let's compute this:

\[
w_2 = 200 \times \sin(35^\circ)
\]

Now, compute the value:

\[
w_2 \approx 200 \times 0.5736 = 114.72 \, \text{N}
\]

### Conclusion
The weight of the hanging block \( w_2 \) is approximately **114.72 N**.

Would you like further details, or do you have any questions about this solution? Here are some related questions you might consider:

1. How would the solution change if there was friction on the inclined plane?
2. What happens to the system if the angle of the incline is increased?
3. How would you find the tension in the rope if friction was present?
4. Can you solve a similar problem where the blocks are on different inclines?
5. What if the pulley had mass? How would this affect the tension in the rope?

**Tip:** Always remember to decompose forces into their components when dealing with inclined planes and other non-horizontal surfaces.

Explore how to solve a physics problem involving force analysis on an inclined plane to determine weight. Step-by-step solution includes formulas for tension in a rope and decomposing forces on inclined planes. Suitable for advanced high school students.

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