🔄Torque
What is Torque?
Torque is the rotational equivalent of force — it's what causes objects to rotate. Just as F = ma governs linear motion, τ = Iα governs rotational motion.
Torque depends on: 1. The magnitude of the force (F) 2. The distance from the pivot (r) — called the "moment arm" or "lever arm" 3. The angle between the force and the lever arm (θ)
τ = r × F × sin θ
Units: N·m (Newton-meters)
The same force can produce different torques depending on where and how it's applied!
The Lever Arm
The lever arm is the perpendicular distance from the pivot point to the line of action of the force.
τ = F × d_⊥
Where d_⊥ is the lever arm (perpendicular distance).
This is equivalent to τ = rF sin θ, where θ is the angle between r and F.
Practical example: A wrench tightening a bolt. Pushing at the end of the wrench (large r) requires less force than pushing near the bolt (small r) for the same torque.
Door example: You push a door near the hinge vs. at the edge. The same force produces much more torque at the edge!
Rotational Equilibrium
For an object to be in rotational equilibrium (not rotating, or rotating at constant angular velocity):
Σ τ = 0
The sum of all clockwise torques must equal the sum of all counterclockwise torques.
Convention: counterclockwise (CCW) torques are positive, clockwise (CW) are negative.
This is the basis for analyzing seesaws, bridges, and any static structure — a second condition of equilibrium (the first being ΣF = 0).
✏️ Worked Example: Seesaw
Problem: A seesaw is balanced at its center. A 40 kg child sits 2 m from the center. Where must a 60 kg child sit to balance?
📐 Key Equations
Torque
τ = rF\sinθ = F · d_⊥Σ τ = 0 (rotational equilibrium)Σ τ = Iα (Newton's 2nd for rotation)⚠️ Common Mistakes
Misconception: Torque is just force — a bigger force always means more torque.
✓ Correct thinking: Torque depends on force, lever arm length, AND the angle: τ = rF sin θ.
Why: A large force applied directly toward the pivot (θ = 0°) produces zero torque because sin 0° = 0.
Misconception: For rotational equilibrium, only the net force must be zero.
✓ Correct thinking: Rotational equilibrium requires BOTH ΣF = 0 (no linear acceleration) AND Στ = 0 (no angular acceleration).
Why: Equal and opposite forces can still create a net torque if they act at different points — a classic "couple" in mechanics.
Misconception: The lever arm is always the distance r from the pivot to where the force is applied.
✓ Correct thinking: The lever arm is the perpendicular distance from the pivot to the LINE OF ACTION of the force.
Why: If the force is angled, the effective lever arm is r sin θ, which can be shorter than the actual distance r.
📝 Practice Problems
Try these problems. Check your answer when ready.
A 40 N force is applied perpendicular to a wrench 0.25 m long. What is the torque?
A force of 50 N is applied at 30° to a rod 0.6 m long. What torque does it produce about the rod's pivot end?
τ = rF\sinθA seesaw is balanced at its center. A 50 kg child sits 1.5 m from the center on the left. Where must a 75 kg child sit on the right to balance?
A uniform 4 m beam weighing 200 N is supported at both ends. A 500 N box sits 1 m from the left end. Find the upward force at each support.
A door (0.9 m wide) requires 15 N·m of torque to open. What minimum force is needed if applied (a) at the outer edge, perpendicular to the door? (b) at 20 cm from the hinge, perpendicular?
Two forces act on a rod pinned at its center: F₁ = 30 N downward at 0.4 m to the left, and F₂ = 20 N downward at 0.5 m to the right. What is the net torque and which way does the rod rotate?
A 10 kg uniform ladder 3 m long leans against a frictionless wall at 60° from the horizontal. A 70 kg person stands 2 m up the ladder. Find the normal force from the wall.
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