1. Calculate the resultant of the forces for each of the following. **a)** Forces: 3 N ← and 5 N → - Step 1: Define right as positive and left as negative. - Step 2: Calculate resultant force: $$R = -3 + 5 = 2\,\text{N}$$ to the right. **b)** Forces: 4 N ← and 8 N → - Step 1: Define right as positive and left as negative. - Step 2: Calculate resultant force: $$R = -4 + 8 = 4\,\text{N}$$ to the right. **c)** Forces: 6 N ←, 4 N →, 3 N ↑, and 3 N ↓ - Step 1: Calculate horizontal resultant: $$R_x = -6 + 4 = -2\,\text{N}$$ (left). - Step 2: Calculate vertical resultant: $$R_y = 3 - 3 = 0\,\text{N}$$. - Step 3: Resultant force magnitude: $$R = \sqrt{(-2)^2 + 0^2} = 2\,\text{N}$$ to the left. **d)** Forces: 3 N ←, 7 N →, 4 N →, 3 N ↑, and 3 N ↓ - Step 1: Calculate horizontal resultant: $$R_x = -3 + 7 + 4 = 8\,\text{N}$$ to the right. - Step 2: Calculate vertical resultant: $$R_y = 3 - 3 = 0\,\text{N}$$. - Step 3: Resultant force magnitude: $$R = \sqrt{8^2 + 0^2} = 8\,\text{N}$$ to the right. 2. Calculate the resultant of the forces for each of the following. **a)** Forces: 8 N → and 6 N ↓ - Step 1: Calculate resultant magnitude: $$R = \sqrt{8^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\,\text{N}$$. - Step 2: Calculate angle with horizontal: $$\theta = \tan^{-1}\left(\frac{6}{8}\right) = 36.87^\circ$$ below horizontal. **b)** Forces: 4 N →, 5 N →, and 12 N ↓ - Step 1: Calculate horizontal resultant: $$R_x = 4 + 5 = 9\,\text{N}$$. - Step 2: Vertical resultant: $$R_y = -12\,\text{N}$$. - Step 3: Resultant magnitude: $$R = \sqrt{9^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15\,\text{N}$$. - Step 4: Angle: $$\theta = \tan^{-1}\left(\frac{12}{9}\right) = 53.13^\circ$$ below horizontal. **c)** Forces: 12 N ↑ and 5 N → - Step 1: Resultant magnitude: $$R = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13\,\text{N}$$. - Step 2: Angle: $$\theta = \tan^{-1}\left(\frac{12}{5}\right) = 67.38^\circ$$ above horizontal. **d)** Forces: 15 N ↑ and 25 N → - Step 1: Resultant magnitude: $$R = \sqrt{15^2 + 25^2} = \sqrt{225 + 625} = \sqrt{850} \approx 29.15\,\text{N}$$. - Step 2: Angle: $$\theta = \tan^{-1}\left(\frac{15}{25}\right) = 30.96^\circ$$ above horizontal.