https://file.aiquickdraw.com/mathbot/file/17283941873198dodlykn.jpg

Let's break down both of the problems shown in the image.

### Problem 1:
- **Given:**
  - A 30-N bag is suspended on strings, and the angles are 60° and 50° for strings A and B, respectively.
  
  - **Objective:**
    - Find the tension forces in strings A, B, and C.

- **Approach:**
  - Start by analyzing the forces. The tension forces in strings A and B need to balance the downward force of 30 N due to the bag's weight. The vertical and horizontal components of the tensions in A and B must sum up to zero in equilibrium.
  
  1. Set up equations for vertical and horizontal equilibrium:
     - Vertical force balance: 
       \[
       T_A \sin(60^\circ) + T_B \sin(50^\circ) = 30 \, \text{N}
       \]
     - Horizontal force balance:
       \[
       T_A \cos(60^\circ) = T_B \cos(50^\circ)
       \]
  2. Solve these two equations to find \( T_A \) and \( T_B \).
  3. \( T_C \) (the vertical string) directly supports the bag, so \( T_C = 30 \, \text{N} \).

### Problem 2:
- **Given:**
  - A 75-N cart is being pulled up a frictionless incline at an angle of 30° to the horizontal, with the rope making a 20° angle with the incline.
  
  - **Objective:**
    - Find the tension in the rope and the normal force acting on the cart.

- **Approach:**
  1. The forces acting on the cart are:
     - The weight \( W = 75 \, \text{N} \) acts vertically downward.
     - The tension \( T \) in the rope acts along the rope direction.
     - The normal force \( N \) acts perpendicular to the inclined surface.
  
  2. Resolve the weight into two components: 
     - Perpendicular to the incline: \( W \cos(30^\circ) \)
     - Parallel to the incline: \( W \sin(30^\circ) \)
  
  3. Set up the force equilibrium equations:
     - Perpendicular to the incline:
       \[
       N = W \cos(30^\circ) - T \sin(20^\circ)
       \]
     - Along the incline:
       \[
       T \cos(20^\circ) = W \sin(30^\circ)
       \]
  4. Solve these equations to find the values of \( T \) and \( N \).

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Would you like a more detailed step-by-step calculation for either or both of these problems?

Here are 5 follow-up questions related to these concepts:

1. How does changing the angles of the strings in Problem 1 affect the tensions in strings A and B?
2. What would happen if the incline in Problem 2 was not frictionless? How would this affect the tension and normal force?
3. Can we use vector components to solve more complex systems with three or more tensions?
4. How do we determine the direction of the normal force when dealing with uneven or curved surfaces?
5. What role does equilibrium play in these types of force analysis?

**Tip:** When dealing with force equilibrium problems, always break forces into their components and set up equations for both vertical and horizontal (or inclined) directions. This simplifies the solution process.

Draw the free-body diagram for the 30-N bag suspended on strings as shown in the given figure. Then solve for the tension along strings A, B, and C. A 75-N cart is being pulled by a rope along a frictionless inclined surface. What are the tension along the rope and the normal force on the surface of the inclined surface?

Learn how to solve force equilibrium problems involving a 30-N bag suspended by strings and a 75-N cart on an inclined surface. This guide covers key trigonometric equations and vector components to find tension and normal force. Suitable for students in Grades 10-12.

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