Equilibrium of Particles - Examples & Applications

A $200\text{ kg}$ crate is suspended by two cables, AB and AC, attached to a ceiling. Cable AB makes an angle of $45^\circ$ with the horizontal, and cable AC makes an angle of $30^\circ$ with the horizontal. Determine the tension in each cable to keep the crate in equilibrium.

A particle is acted upon by three forces:

• $\mathbf{F}_1 = \{10\mathbf{i} - 20\mathbf{j} + 15\mathbf{k}\}\text{ N}$
• $\mathbf{F}_2 = \{-5\mathbf{i} + 10\mathbf{j} - 10\mathbf{k}\}\text{ N}$
• $\mathbf{F}_3 = \{F_{3x}\mathbf{i} + F_{3y}\mathbf{j} + F_{3z}\mathbf{k}\}\text{ N}$

Determine the components of $\mathbf{F}_3$ required for the particle to be in equilibrium.

A particle is suspended by three cables anchored to walls. The forces acting on the particle in equilibrium are $W = \{-200\mathbf{j}\}\text{ N}$ (weight), $\mathbf{F}_A = F_A\{0.8\mathbf{i} + 0.6\mathbf{j}\}\text{ N}$, and $\mathbf{F}_B = \{-50\mathbf{i} + F_{By}\mathbf{j}\}\text{ N}$. Find $F_A$ and $F_{By}$.

A spring with a stiffness $k = 500\text{ N/m}$ is used to suspend a $10\text{ kg}$ mass vertically. Determine the displacement (stretch) of the spring when the system is in equilibrium. Assume $g = 9.81\text{ m/s}^2$.

A block of weight $W = 500\text{ N}$ rests on a smooth inclined plane angled at $30^\circ$ to the horizontal. A force $P$ is applied parallel to the incline, directed upwards, to keep the block in equilibrium. Find the magnitude of $P$ and the normal force $N$ exerted by the plane.

A cable passes over a small frictionless pulley and supports a $400\text{ N}$ weight at one end. The other end is pulled downwards at an angle of $45^\circ$ from the vertical to maintain equilibrium. What is the tension in the cable, and what are the vertical and horizontal components of the reaction force at the pulley's axle?

A balloon is held in place by three cables. The upward buoyant force on the balloon is $F_B = 800\text{ N}$. Cable 1 pulls with force $T_1$ entirely along the negative x-axis. Cable 2 pulls with force $T_2$ entirely along the negative y-axis. Cable 3 pulls with force $T_3$ having coordinate direction angles $\alpha = 60^\circ$, $\beta = 60^\circ$, and $\gamma = 135^\circ$. Determine $T_3$.

A ring is subjected to three concurrent forces: $F_1 = 300\text{ N}$ at $0^\circ$ (positive x-axis), $F_2 = 400\text{ N}$ at $90^\circ$ (positive y-axis), and an unknown force $F_3$. If the ring is in equilibrium, what are the magnitude and angle of $F_3$ (measured counterclockwise from the positive x-axis)?

Two spring scales are hooked end-to-end. One end is attached to a wall, and a person pulls the other end with a force of $50 \text{ N}$. What will each spring scale read, assuming they are light enough to be considered massless? Explain your reasoning.

A person hangs a very heavy wet blanket exactly in the middle of a tightly strung horizontal clothesline. The clothesline sags slightly. Why is it impossible to pull the clothesline perfectly horizontal again without breaking it, no matter how hard you pull? Explain using the equilibrium equations for the point where the blanket hangs.

A builder needs to lift a heavy bucket. They can either lift it straight up, or they can route the lifting rope over a frictionless pulley attached to a beam above. Does using the pulley reduce the required pulling force to maintain equilibrium? Why or why not?

A skydiver is falling at terminal velocity, where their downward speed is perfectly constant. Is the skydiver in a state of particle equilibrium? Why or why not?