The Memory Trick
💡 τ = r × F × sin θ
Torque is the rotational equivalent of force — it's what causes angular acceleration, just as force causes linear acceleration. Torque depends on three things: the applied force (F), the distance from the pivot point to where the force is applied (r, the lever arm), and the angle between the force and the lever arm (θ).
Why It Works
Only the component of force perpendicular to the lever arm actually contributes to rotation — that's what sin θ captures. A force applied straight along the lever arm (θ=0°) produces zero torque no matter how strong it is.
Step by Step
Applying Torque
1
Longer lever arm means more torque
For the same applied force, increasing the distance from the pivot point increases torque proportionally.
Opening a door by pushing near the hinges takes much more force than pushing at the handle — the handle has a much longer lever arm.
2
Direction matters — sign convention
By convention, counterclockwise torque is positive and clockwise torque is negative — useful for combining multiple torques acting on the same object.
Two people pushing opposite sides of a revolving door create torques of opposite sign, which partially or fully cancel depending on their magnitudes.
3
Only the perpendicular force component counts
τ = rF sin θ means force applied exactly along the lever arm (parallel to it) produces zero torque — only the component perpendicular to the arm contributes.
Pushing straight into a door (along the door's plane, toward the hinge line) does nothing to open it — sin 0° = 0.
🏥 Worked Example
A 30 N force is applied at the end of a 0.6 m wrench handle, perpendicular to the handle. What torque is produced?
1
Identify the angle: the force is perpendicular to the handle, so θ = 90°, and sin 90° = 1.
2
Apply τ = rF sin θ: τ = 0.6 × 30 × 1 = 18 N·m.
3
Compare to a shorter lever arm: the same 30 N force applied at only 0.3 m from the pivot would produce just 9 N·m — half the torque, for the same force.
📌 Exam Application
Exams test correctly applying τ = rF sin θ, understanding why lever arm length matters as much as force magnitude, and correctly signing multiple torques (clockwise vs. counterclockwise) when combining them.
⚠️ Most Common Torque Mistakes
The most common trap is forgetting the sin θ term entirely and just multiplying force by distance, regardless of the angle between them — a force applied parallel to the lever arm produces zero torque, no matter how large it is.
✓ Quick Self-Test
1) Write the formula for torque, including the angle term. τ = r × F × sin θ. 2) Why does pushing a door open near the hinges take more force than pushing at the handle? The lever arm (r) is shorter near the hinges, so more force is needed to produce the same torque. 3) What torque is produced by a force applied exactly parallel to the lever arm? Zero — sin 0° = 0. 4) By convention, which rotational direction is considered positive torque? Counterclockwise. 5) If lever arm length doubles while force stays the same, what happens to torque? It doubles.
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