AP Physics questions on Rotational Equilibrium

📚 Quick Study Guide

• ⚖️ Equilibrium Condition: For an object to be in rotational equilibrium, the net torque acting on it must be zero. Mathematically, $\sum \tau = 0$.
• 🔄 Torque: Torque ($\tau$) is the rotational equivalent of force and 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.
• 📍 Center of Gravity: The point where the entire weight of an object can be considered to act. For an object to be in stable equilibrium, its center of gravity must be above its support base.
• 📏 Lever Arm: The perpendicular distance from the axis of rotation to the line of action of the force.
• 📐 Sign Convention: Torque is considered positive if it tends to cause a counterclockwise rotation and negative if it tends to cause a clockwise rotation.
• 💡 Problem Solving Tips:
  • Draw a free-body diagram.
  • Choose a convenient axis of rotation (often at a point where several forces act).
  • Apply the equilibrium conditions: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum \tau = 0$.

Practice Quiz

1. A uniform beam of length $L$ and weight $W$ is supported by two ropes at its ends. What is the tension in each rope when a weight $W$ is placed at the center of the beam?
   1. $W/2$
   2. $W$
   3. $3W/4$
   4. $W/4$
2. A seesaw is balanced with a child on each end. If one child leans inward, closer to the pivot, what happens to the seesaw?
   1. The seesaw tips toward the child who leaned inward.
   2. The seesaw tips toward the other child.
   3. The seesaw remains balanced.
   4. The seesaw starts oscillating.
3. A ladder leans against a wall. Which of the following forces contribute to the torque about the base of the ladder?
   1. The normal force from the ground.
   2. The weight of the ladder.
   3. The normal force from the wall.
   4. All of the above.
4. A meter stick is pivoted at the 50 cm mark. A mass of 100g is hung at the 20 cm mark. Where should a mass of 50g be hung to balance the meter stick?
   1. 80 cm mark
   2. 20 cm mark
   3. 60 cm mark
   4. 90 cm mark
5. A door requires a torque of 10 Nm to open. If you apply a force of 5 N at the doorknob, which is 1 m from the hinges, will the door open?
   1. Yes
   2. No
   3. Only if the force is applied perpendicularly.
   4. Only if the force is applied parallel.
6. A wheel is in rotational equilibrium. If a force is applied that creates a clockwise torque, what must happen to maintain equilibrium?
   1. Another clockwise torque must be applied.
   2. A counterclockwise torque of equal magnitude must be applied.
   3. The wheel must be allowed to rotate.
   4. The force must be removed.
7. A sign is hanging from a single pivot. To ensure it remains in rotational equilibrium, where should the center of gravity of the sign be located?
   1. Directly above the pivot.
   2. Directly below the pivot.
   3. Far away from the pivot.
   4. It doesn't matter where the center of gravity is.