AP Physics – Torque and Rotational Equilibrium

AP Physics – Torque and Rotational Equilibrium

AP Physics: Torque and Rotational Equilibrium – Exam-Ready Study Guide

What This Is

Torque is the "rotational equivalent of force"—it’s what causes objects to spin or twist. Rotational equilibrium means an object isn’t rotating (or is rotating at a constant speed), so the net torque on it is zero. This topic is heavily tested on the AP Physics exam, especially in FRQs involving seesaws, wrenches, or balanced beams. Real-world example: When you use a wrench to loosen a bolt, you apply torque. The longer the wrench (or the harder you push perpendicular to it), the more torque you generate—this is why mechanics use long-handled wrenches for stubborn bolts!

Key Terms & Concepts

• Torque (?): A measure of how effectively a force causes rotation. Formula: ? = r × F = rF sin?, where:
• ? = torque (N·m)
• r = lever arm (distance from pivot to force application, in meters)
• F = applied force (N)

• ? = angle between r and F (max torque when ? = 90°).

• Lever arm (r): The perpendicular distance from the pivot point to the line of action of the force. Key idea: Torque depends on both force and lever arm.

• Rotational equilibrium: When the net torque on an object is zero ( = 0), so it doesn’t rotate (or rotates at constant speed). Example: A balanced seesaw.

• Pivot point (fulcrum): The fixed point around which rotation occurs. AP trap: The pivot isn’t always at the center of mass!

• Clockwise (CW) vs. counterclockwise (CCW) torque: Conventionally, CW torque is negative, CCW is positive (or vice versa—always define your sign convention).

• Center of mass (COM): The average position of an object’s mass. For uniform objects, it’s at the geometric center. Why it matters: If the COM is directly above the pivot, the object is balanced.

• Static equilibrium: When an object is at rest (no translation and no rotation). Requires ?F = 0 and = 0.

• Moment of inertia (I): The rotational equivalent of mass—resistance to rotational acceleration. Formula: ?_net = I? (where ? = angular acceleration). AP note: You won’t calculate I from scratch, but you’ll use given values.

• Angular momentum (L): L = I? (where ? = angular velocity). Conservation: If net torque is zero, L is conserved (e.g., figure skaters pulling in their arms to spin faster).

• Right-hand rule (for torque): Point fingers in direction of r, curl toward F—thumb points in direction of ?. AP loves testing this!

Step-by-Step: Solving Torque Problems

1. Identify the pivot point.
2. If not given, choose the point where the most forces act (simplifies calculations).

3. Example: For a seesaw, the pivot is the fulcrum.

4. Draw a free-body diagram (FBD).

5. Label all forces (gravity, normal, applied forces) and their distances from the pivot.

6. Pro tip: Break forces into components if they’re not perpendicular to the lever arm.

7. Assign a sign convention.

8. Decide which direction (CW or CCW) is positive. Stick to it!

9. Calculate torques.

10. For each force: ? = rF sin?.

11. Shortcut: If F is perpendicular to r, ? = rF (since sin90° = 1).

12. Set up equilibrium condition.

13. For rotational equilibrium: = 0.

14. For static equilibrium: ?F_x = 0, ?F_y = 0, and = 0.

15. Solve for the unknown.

16. Use algebra to isolate the variable (e.g., force, distance, or angle).

Example problem: A 50 N child sits 2 m from the pivot of a seesaw. Where should a 75 N child sit to balance it? - Solution: ?_CW = ?_CCW-(50 N)(2 m) = (75 N)(x)-x = 1.33 m.

Common Mistakes

• Mistake: Forgetting that torque depends on r sin?, not just r.

• Correction: Only the perpendicular component of F contributes to torque. If F is at an angle, use ? = rF sin?.

• Mistake: Mixing up CW and CCW signs.

• Correction: Always define your sign convention (e.g., "CCW is positive") and label torques accordingly.

• Mistake: Ignoring the weight of the object itself.

• Correction: For beams or planks, include the torque from their own weight (acting at the COM).

• Mistake: Assuming the pivot is at the center of mass.

• Correction: The pivot can be anywhere—don’t assume symmetry!

• Mistake: Using F = ma instead of ? = I? for rotational problems.

• Correction: Linear and rotational dynamics are analogous but separate. Use torque for rotation.

AP Exam Insights

• FRQs often test:
• Balancing torques (e.g., seesaws, ladders leaning against walls).
• Static equilibrium (e.g., a sign hanging from a pole).

• Tricky twist: A problem might give you a force at an angle—you’ll need to break it into components.

• Multiple-choice traps:

• Lever arm confusion: A force applied at the pivot (r = 0) produces zero torque, no matter how large the force.
• Sign errors: Forgetting to assign CW/CCW signs can lead to wrong answers.

• Unit errors: Torque is in N·m, not J (though the units are equivalent, J is for energy).

• Key distinction: Torque vs. force—torque is about rotation, force is about translation. A force can cause both!

Quick Check Questions

1. Multiple Choice: A 10 N force is applied at the end of a 0.5 m wrench at a 30° angle to the wrench. What is the torque? (A) 2.5 N·m (B) 4.3 N·m (C) 5.0 N·m (D) 10 N·m Answer: (A) ? = rF sin? = (0.5 m)(10 N)(sin30°) = 2.5 N·m.

2. Short FRQ: A uniform 2 kg meterstick is balanced on a fulcrum at the 30 cm mark. A 1 kg mass hangs from the 10 cm mark. Where should a 0.5 kg mass be placed to balance the stick? Answer: At the 90 cm mark. = 0-(1 kg)(9.8 m/s²)(0.2 m) = (0.5 kg)(9.8 m/s²)(x)-x = 0.4 m from the pivot-30 cm + 40 cm = 70 cm (but the stick’s weight acts at 50 cm, so adjust: 0.5 kg must be at 90 cm).

Last-Minute Cram Sheet

1. Torque formula: ? = rF sin? (max when ? = 90°).
2. Rotational equilibrium: = 0.
3. Static equilibrium: ?F = 0 and = 0.
4. Lever arm: Perpendicular distance from pivot to force.
5. Sign convention: Define CW/CCW (e.g., CCW = +).
6. Pivot at center? Not always—check the problem!
7. Weight of object: Acts at COM (e.g., middle of a uniform beam).
8. Right-hand rule: Fingers = r, curl = F, thumb = ?.
9. Force at pivot? ? = 0 (no lever arm).
10. Units: Torque is N·m, not J (even though 1 N·m = 1 J).