In this problem, three forces , , and are acting on point in such a way that the system is in equilibrium. In equilibrium, the vector sum of all forces must be zero, meaning both the horizontal and vertical components of the forces must balance.
Given the directions and angles, we can resolve the forces into horizontal and vertical components.
Step 1: Resolve the forces and into components.
- has an angle of 30° with the horizontal, so its components are: - Horizontal component: - Vertical component: - has an angle of 45° with the horizontal, so its components are: - Horizontal component: - Vertical component: - is acting vertically downward with a magnitude of 100 N, so its components are: - Horizontal component: 0 N (since it is vertically downward), - Vertical component: Step 2: Apply the equilibrium conditions.
For horizontal equilibrium, the sum of the horizontal components of the forces must be zero: For vertical equilibrium, the sum of the vertical components of the forces must be zero: Step 3: Solve the system of equations.
From the horizontal equilibrium equation: Simplifying: Now, using the vertical equilibrium equation: Substituting the known values: Substitute into the equation: Simplifying: Solving for : Now, substituting into the equation for : Thus, the magnitude of is .
