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The problem involves a 75-N cart being pulled by a rope along a frictionless inclined surface. The forces acting on the cart need to be analyzed to determine the tension in the rope and the normal force exerted by the surface.

### Step 1: Define the Given Data
1. **Weight of the Cart (W)**: 75 N
2. **Inclination Angle of the Surface (θ)**: 30°
3. **Angle of the Rope with the Surface (α)**: 20°

### Step 2: Identify Forces Acting on the Cart
1. **Weight (W)**: Acts vertically downward.
2. **Tension (T)**: Acts along the direction of the rope, at an angle of 20° to the inclined surface.
3. **Normal Force (N)**: Acts perpendicular to the inclined surface.

### Step 3: Decompose Forces into Components
1. **Weight Components**:
   - Parallel to the incline: \( W \sin 30^\circ \)
   - Perpendicular to the incline: \( W \cos 30^\circ \)

2. **Tension Components**:
   - Along the incline: \( T \cos 20^\circ \)
   - Perpendicular to the incline: \( T \sin 20^\circ \)

### Step 4: Set Up Equations Using Newton's Second Law
1. **Force Balance Along the Incline**:
   \[
   T \cos 20^\circ = W \sin 30^\circ
   \]
   Solving for \( T \):
   \[
   T = \frac{W \sin 30^\circ}{\cos 20^\circ}
   \]
   Substituting \( W = 75 \, \text{N} \):
   \[
   T = \frac{75 \times 0.5}{\cos 20^\circ}
   \]
   \[
   T = \frac{37.5}{0.9397} \approx 39.9 \, \text{N}
   \]

2. **Force Balance Perpendicular to the Incline**:
   \[
   N + T \sin 20^\circ = W \cos 30^\circ
   \]
   Solving for \( N \):
   \[
   N = W \cos 30^\circ - T \sin 20^\circ
   \]
   Substituting values:
   \[
   N = 75 \times 0.866 - 39.9 \times 0.342
   \]
   \[
   N \approx 64.95 - 13.65 = 51.3 \, \text{N}
   \]

### Final Results
1. **Tension along the Rope**: Approximately **39.9 N**.
2. **Normal Force on the Surface**: Approximately **51.3 N**.

Would you like further details or have any questions? Here are five related questions:

1. How would the tension change if the inclination angle of the surface increased?
2. What would happen to the normal force if the angle of the rope increased?
3. How would adding friction affect the results?
4. Can the problem be solved using energy conservation methods?
5. What would be the acceleration of the cart if it were not in equilibrium?

**Tip**: Always resolve forces into components parallel and perpendicular to the surface to simplify calculations involving inclined planes.

A 75-N cart is being pulled by a rope along a frictionless inclined surface. The angles provided are 30° for the incline and 20° for the rope with respect to the incline. What are the tension along the rope and the normal force on the surface of the inclined surface?

Determine the tension in a rope and the normal force on a 75-N cart being pulled along a frictionless incline. Includes force analysis using trigonometric functions and Newton's Second Law. Suitable for students in Grades 11-12.

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