Three forces act on a bracket: $F_1 = 100\text{ N}$ at $30^\circ$, $F_2 = 150\text{ N}$ at $120^\circ$, and $F_3 = 200\text{ N}$ at $270^\circ$ (measured counter-clockwise from the positive x-axis). Determine the magnitude and direction of the resultant.

A force of $F = 400 \text{ N}$ acts on the end of a lever of length $L = 2.00 \text{ m}$ at an angle of $60.0^\circ$ to the lever. Calculate the moment of the force about the pivot point.

Two forces act on a point: $\mathbf{F}_1 = \{30.0\mathbf{i} + 40.0\mathbf{j} - 50.0\mathbf{k}\} \text{ N}$ and $\mathbf{F}_2 = \{-20.0\mathbf{i} + 10.0\mathbf{j} + 30.0\mathbf{k}\} \text{ N}$. Find the resultant force vector and its magnitude.

Four vertical forces act on a horizontal beam of length $10.0\text{ m}$. The forces are $F_1 = 20.0\text{ kN}$ downward at $x = 0.00\text{ m}$, $F_2 = 40.0\text{ kN}$ downward at $x = 3.00\text{ m}$, $F_3 = 10.0\text{ kN}$ upward at $x = 6.00\text{ m}$, and $F_4 = 30.0\text{ kN}$ downward at $x = 10.0\text{ m}$. Determine the magnitude, direction, and line of action of the resultant force.

A force $\mathbf{F} = \{10.0\mathbf{i} + 20.0\mathbf{j} - 15.0\mathbf{k}\}\text{ N}$ is applied at a point $P$ with position vector $\mathbf{r} = \{2.00\mathbf{i} + 3.00\mathbf{j} + 4.00\mathbf{k}\}\text{ m}$ relative to the origin $O$. Calculate the moment of the force about the origin.

A force $F = 500\text{ N}$ acts downward on a cantilever beam at a distance of $3.00\text{ m}$ from the wall (point $A$). Replace this force with an equivalent force-couple system at point $A$.

A simply supported beam is subjected to a uniform distributed load of $w = 2.50\text{ kN/m}$ over its entire length of $L = 8.00\text{ m}$. Determine the magnitude and location of the equivalent concentrated force.

A weight $W = 1000\text{ N}$ is suspended by two cables, $A$ and $B$. Cable $A$ is at an angle of $30.0^\circ$ to the horizontal, and Cable $B$ is at $45.0^\circ$ to the horizontal. Find the tension in both cables.

When a cable is used to pull a car out of a ditch, the force exerted by the winch is applied to the bumper of the car. According to the Principle of Transmissibility, we can treat this force as if it were applied anywhere along its line of action. Discuss why this is valid for the car as a whole, but not for the bumper itself.

When a driver turns a steering wheel using both hands, they apply equal and opposite forces on opposite sides of the wheel. Explain this action in terms of force systems and why it is an ideal way to turn the wheel.

When designing a bridge, engineers must account for both concentrated loads (like a heavy truck parked on the deck) and distributed loads (like the weight of the concrete deck itself or a steady stream of traffic). Discuss why modeling loads as concentrated forces is often a simplification of distributed forces, and when it is appropriate to do so.

A mechanic is using a long wrench to loosen a stuck bolt. They can either push perpendicular to the handle at the very end, or they can push at an angle closer to the middle. Using Varignon's Theorem, explain how the total moment on the bolt relates to the components of the applied force.