Solved Examples of Static Equilibrium Problems

📚 Quick Study Guide

• ⚖️ Static equilibrium means an object is not moving (translational equilibrium) and not rotating (rotational equilibrium).
• ➕ For translational equilibrium, the vector sum of all forces acting on the object must be zero: $\sum \vec{F} = 0$. This means $\sum F_x = 0$ and $\sum F_y = 0$.
• 🌀 For rotational equilibrium, the net torque about any axis must be zero: $\sum \tau = 0$. Torque is calculated as $\tau = rF\sin(\theta)$, where $r$ is the distance from the axis of rotation to the point where the force is applied, $F$ is the magnitude of the force, and $\theta$ is the angle between the force vector and the lever arm.
• 📐 Choose a convenient axis of rotation to simplify torque calculations. Often, choosing an axis where an unknown force acts eliminates that force from the torque equation.
• ✏️ Free-body diagrams are essential for visualizing forces and their directions.
• 💡 Remember to consider the weight of the object, which acts at its center of gravity. For uniform objects, the center of gravity is at the geometric center.

Practice Quiz

1. A uniform beam of length $L$ and weight $W$ is pinned to a wall and supported by a cable attached to the end of the beam. The cable makes an angle of $\theta$ with the beam. What is the tension in the cable?
   1. $\frac{W}{2\sin(\theta)}$
   2. $\frac{W}{\sin(\theta)}$
   3. $\frac{2W}{\sin(\theta)}$
   4. $\frac{W}{\cos(\theta)}$
2. A ladder leans against a smooth wall (no friction). The ladder's weight is $W$ and its length is $L$. The angle between the ladder and the ground is $\theta$. What is the normal force exerted by the wall on the ladder?
   1. $\frac{W}{2\tan(\theta)}$
   2. $W\tan(\theta)$
   3. $W\cos(\theta)$
   4. $\frac{W}{\cos(\theta)}$
3. A sign of weight $W$ is hung from two cables that make angles $\theta_1$ and $\theta_2$ with the horizontal. What is the vertical component of the tension in the first cable?
   1. $\frac{W\sin(\theta_2)}{\sin(\theta_1 + \theta_2)}$
   2. $\frac{W\cos(\theta_2)}{\sin(\theta_1 + \theta_2)}$
   3. $W\sin(\theta_1)$
   4. $\frac{W}{\sin(\theta_1 + \theta_2)}$
4. A block of weight $W$ rests on an inclined plane that makes an angle $\theta$ with the horizontal. What is the magnitude of the static friction force required to keep the block from sliding down the plane?
   1. $W\sin(\theta)$
   2. $W\cos(\theta)$
   3. $W\tan(\theta)$
   4. $W$
5. A bridge of weight $W$ and length $L$ is supported at both ends. A truck of weight $W_t$ is located a distance $L/4$ from one end. What is the support force at the other end?
   1. $\frac{W}{2} + \frac{3W_t}{4}$
   2. $\frac{W}{2} + \frac{W_t}{4}$
   3. $\frac{W}{2} + \frac{W_t}{2}$
   4. $\frac{W}{4} + \frac{3W_t}{4}$
6. A meter stick of mass $m$ is pivoted at the 20 cm mark. Where should a mass of $2m$ be hung to balance the meter stick horizontally?
   1. 30 cm mark
   2. 50 cm mark
   3. 60 cm mark
   4. 80 cm mark
7. A square sign is supported by a hinge at its upper left corner and a cable running from the upper right corner to a point on the wall directly above the hinge. If the sign has side length $s$ and weight $W$, what is the horizontal component of the force exerted by the hinge?
   1. $\frac{W}{2}$
   2. $W$
   3. $2W$
   4. $\frac{W}{\sqrt{2}}$