https://r2.erweima.ai/mathbot/image/172627844972878aw6vaj.jpg

This image shows a system involving a rope and pulley mechanism where forces are applied to lift or support a mass \( m \). The problem provides information about the system, including the mass \( m = 8.60 \, \text{kg} \) and the angle \( \theta = 67^\circ \), and asks for two calculations:

1. The tension force in the supporting rope.
2. The horizontal force applied to the other side of the rope.

Let’s analyze these steps:

1. **Tension Force in the Rope**: This refers to the vertical tension supporting the mass. We can calculate the tension by considering the forces acting on the mass in equilibrium:
   \[
   T = mg = 8.60 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 84.28 \, \text{N}
   \]
   This matches the result given in the image for the vertical tension force.

2. **Horizontal Force on the Other Side of the Rope**: This force can be calculated based on the horizontal component of the tension force using the given angle \( \theta = 67^\circ \). If the tension in the rope is \( T \), then the horizontal component is:
   \[
   F_{\text{horizontal}} = - T \sin(\theta)
   \]
   Let's compute this next.

Let me calculate the horizontal force for you.The tension force in the rope is \( 84.28 \, \text{N} \), which matches the correct value shown in the image. 

The horizontal force applied to the other side of the rope is approximately \( -77.58 \, \text{N} \). The negative sign indicates the direction of the force is opposite to the assumed direction.

Would you like further clarification on any of these steps?

Here are five related questions you might consider:
1. How does changing the angle \( \theta \) affect the horizontal force?
2. What would happen to the tension if the mass were increased?
3. How can we calculate the force components when the system is in motion (non-static)?
4. What other forces might act on the pulley in this system?
5. How would the results change if friction were introduced in the pulley?

**Tip:** When solving problems involving ropes and pulleys, always consider both the vertical and horizontal components of forces, as they are often not the same magnitude.

This image shows a system involving a rope and pulley mechanism where forces are applied to lift or support a mass m. The mass is m = 8.60 kg and the angle θ = 67°, and asks for two calculations: the tension force in the supporting rope and the horizontal force applied to the other side of the rope.

This problem explores a rope and pulley system used to support a mass of 8.60 kg at an angle of 67°. Learn how to calculate the tension in the supporting rope and the horizontal force using mechanics and trigonometry. Suitable for students studying advanced high school or undergraduate physics.

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