Activity 2 forcos Ivo forces act on an object. The first is a horizontal 50 N force. The second is a 40 N force acting at an angle to the horlzontal. Use a graphical method drawn to a scale of 1 mm : I N to find the resultant if the 40 N force is incilned at: \( 1.1 .130^{\circ} \) to the horizontal \( 1.1 .260^{\circ} \) to the horizontal.

To determine the resultant force when combining a horizontal 50 N force with a 40 N force inclined at different angles (130° and 260°) to the horizontal, we'll use both **graphical** and **analytical** methods. Since a graphical method typically involves drawing vectors to scale and measuring the resultant with a protractor and ruler, I'll provide a step-by-step guide for the graphical approach, followed by an analytical solution for verification. --- ## **Graphical Method** ### **Tools Needed:** - Graph paper - Ruler - Protractor - Pencil ### **Scale:** 1 mm : 1 N ### **Steps:** 1. **Draw the Horizontal 50 N Force:** - On your graph paper, draw a horizontal line 50 mm long to represent the 50 N force. Label this vector **F₁**. 2. **Draw the Second 40 N Force at the Given Angle:** - **Case 1:** **130° to the Horizontal** - Place the protractor's center at the tail of **F₁**. - Measure an angle of 130° from the positive x-axis (horizontal) upward. - Draw a line 40 mm long at this angle to represent the 40 N force. Label this vector **F₂₁**. - **Case 2:** **260° to the Horizontal** - Similarly, place the protractor's center at the tail of **F₁**. - Measure an angle of 260° from the positive x-axis downward. - Draw a line 40 mm long at this angle to represent the 40 N force. Label this vector **F₂₂**. 3. **Construct the Resultant Vector:** - Using the **head-to-tail** method: - For each case, place the tail of **F₂₁** and **F₂₂** at the head of **F₁**. - Draw a straight line from the tail of **F₁** to the head of **F₂₁** (for Case 1) and to the head of **F₂₂** (for Case 2). These lines represent the resultant vectors **R₁** and **R₂**, respectively. 4. **Measure the Resultant:** - Use the ruler to measure the length of **R₁** and **R₂** in millimeters. This length corresponds to the magnitude of the resultant force in newtons. - Use the protractor to measure the angle of each resultant with respect to the horizontal. --- ## **Analytical Method (For Verification)** While the graphical method provides a visual understanding, the analytical method will give precise values for the magnitude and direction of the resultant forces. ### **Given:** - **F₁ = 50 N** (Horizontal) - **F₂₁ = 40 N** at **130°** - **F₂₂ = 40 N** at **260°** ### **Steps:** 1. **Resolve Each Force into Components:** - **Case 1:** **130° to the Horizontal** - \( F_{2x₁} = F₂₁ \cdot \cos(130°) = 40 \cdot \cos(130°) \approx -25.71 \, \text{N} \) - \( F_{2y₁} = F₂₁ \cdot \sin(130°) = 40 \cdot \sin(130°) \approx 30.64 \, \text{N} \) - **Case 2:** **260° to the Horizontal** - \( F_{2x₂} = F₂₂ \cdot \cos(260°) = 40 \cdot \cos(260°) \approx -6.94 \, \text{N} \) - \( F_{2y₂} = F₂₂ \cdot \sin(260°) = 40 \cdot \sin(260°) \approx -39.39 \, \text{N} \) 2. **Sum the Components:** - **Case 1:** - \( R_x₁ = F₁ + F_{2x₁} = 50 + (-25.71) = 24.29 \, \text{N} \) - \( R_y₁ = 0 + F_{2y₁} = 30.64 \, \text{N} \) - **Case 2:** - \( R_x₂ = F₁ + F_{2x₂} = 50 + (-6.94) = 43.06 \, \text{N} \) - \( R_y₂ = 0 + F_{2y₂} = -39.39 \, \text{N} \) 3. **Calculate the Magnitude of the Resultant:** - **Case 1:** - \( |R₁| = \sqrt{R_x₁^2 + R_y₁^2} = \sqrt{24.29^2 + 30.64^2} \approx 39.09 \, \text{N} \) - **Case 2:** - \( |R₂| = \sqrt{R_x₂^2 + R_y₂^2} = \sqrt{43.06^2 + (-39.39)^2} \approx 58.36 \, \text{N} \) 4. **Determine the Direction of the Resultant:** - **Case 1:** - \( \theta₁ = \arctan\left(\frac{R_y₁}{R_x₁}\right) = \arctan\left(\frac{30.64}{24.29}\right) \approx 51.5° \) above the horizontal. - **Case 2:** - \( \theta₂ = \arctan\left(\frac{|R_y₂|}{R_x₂}\right) = \arctan\left(\frac{39.39}{43.06}\right) \approx 42.0° \) below the horizontal. --- ## **Results:** 1. **When the 40 N force is inclined at 130°:** - **Graphical Method:** Resultant vector **R₁** is approximately **39 N** at an angle of **51.5° above the horizontal**. - **Analytical Confirmation:** Magnitude ≈ **39.09 N**, Direction ≈ **51.5° above horizontal**. 2. **When the 40 N force is inclined at 260°:** - **Graphical Method:** Resultant vector **R₂** is approximately **58 N** at an angle of **42.0° below the horizontal**. - **Analytical Confirmation:** Magnitude ≈ **58.36 N**, Direction ≈ **42.0° below horizontal**. --- **Note:** While the graphical method provides a visual approximation, the analytical method offers precise calculations. In practice, both methods are valuable: graphical for intuitive understanding and analytical for accuracy.