Let's break down the solution step by step. Here’s the **problem restated**: --- **Given:** - Two forces applied at point A: - \( P = 60 \) lb at \( 20^\circ \) to the vertical (left side) - \( Q = 25 \) lb at \( 35^\circ \) to the vertical (right side) - **Required:** Determine graphically the magnitude and direction of their resultant using: - (a) the **parallelogram law** - (b) the **triangle rule** --- ## 1. **Find the Angle Between the Forces** The angle between \( P \) and \( Q \) is: \[ \theta = 20^\circ + 35^\circ = 55^\circ \] --- ## 2. **Parallelogram Law** The parallelogram law states: \[ R = \sqrt{P^2 + Q^2 + 2PQ\cos\theta} \] Where: - \( P = 60 \) lb - \( Q = 25 \) lb - \( \theta = 55^\circ \) ### **Calculate:** \[ R = \sqrt{60^2 + 25^2 + 2 \times 60 \times 25 \times \cos(55^\circ)} \] First, calculate \( \cos(55^\circ) \): \[ \cos(55^\circ) \approx 0.5736 \] Plug values in: \[ R = \sqrt{3600 + 625 + 2 \times 60 \times 25 \times 0.5736} \] \[ 2 \times 60 \times 25 = 3000 \] \[ 3000 \times 0.5736 = 1720.8 \] \[ R = \sqrt{3600 + 625 + 1720.8} \] \[ R = \sqrt{5945.8} \] \[ R \approx 77.1 \text{ lb} \] --- ### **Direction of Resultant (\(\alpha\))** Let \(\alpha\) be the angle between \(P\) and \(R\): \[ \tan\alpha = \frac{Q \sin\theta}{P + Q \cos\theta} \] Plug in values: - \(Q = 25\) - \(P = 60\) - \(\sin(55^\circ) \approx 0.8192\) - \(\cos(55^\circ) \approx 0.5736\) \[ \tan\alpha = \frac{25 \times 0.8192}{60 + 25 \times 0.5736} \] \(25 \times 0.8192 = 20.48\) \(25 \times 0.5736 = 14.34\) \[ \tan\alpha = \frac{20.48}{74.34} \] \[ \tan\alpha \approx 0.2756 \] \[ \alpha \approx \arctan(0.2756) \approx 15.4^\circ \] **This angle is measured from \(P\) towards the resultant \(R\).** --- ## 3. **Triangle Rule** The triangle rule gives the same result as parallelogram law since it is just rearranging the vectors head-to-tail. The magnitude and direction will be identical: - **Magnitude:** \(77.1\) lb - **Direction:** \(15.4^\circ\) from \(P\) towards \(Q\) --- ## **Summary Table** | Method | Magnitude (lb) | Direction (from \(P\)) | |--------------------|----------------|------------------------| | Parallelogram Law | 77.1 | \(15.4^\circ\) | | Triangle Rule | 77.1 | \(15.4^\circ\) | --- ## **Graphical Representation** (Explanation) - **Parallelogram law:** Draw vectors \(P\) and \(Q\) from point \(A\) at their given angles. Complete the parallelogram. The diagonal from \(A\) gives the resultant. - **Triangle rule:** Draw \(P\) from \(A\), then \(Q\) from the tip of \(P\) at the correct angle. The vector from \(A\) to the tip of \(Q\) is the resultant. --- If you need a graphical drawing or more details, let me know!