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Torque

Source lecture(s): SC133 Lec 13

Intuition

Why do door handles live far from the hinges? Because turning effectiveness isn't just force — it's force times leverage. Push near the hinge and the door ignores you; the same push at the handle swings it easily. Torque is the turning power of a force: how hard, how far from the axis, and how perpendicular.

Definition

\[\boxed{\,\vec\tau = \vec r\times\vec F\,} \qquad \tau = rF\sin\phi = F\cdot\underbrace{r\sin\phi}_{\text{moment arm}}\]
  • \(\vec r\): from the axis (or pivot) to the point where the force acts
  • \(\phi\): angle between \(\vec r\) and \(\vec F\)
  • Moment arm \(r_\perp = r\sin\phi\): the perpendicular distance from the axis to the force's line of action — often the fastest way to compute \(\tau\)
  • Direction by the right-hand rule (cross product); in 2-D problems, just call counterclockwise positive

Maximum torque: push perpendicular to the lever (\(\phi = 90°\)). Zero torque: push along the lever, however hard.

Newton's second law for rotation

\[\sum \tau = I\alpha\]

Torque is to angular acceleration what force is to acceleration, with moment of inertia as the stubbornness. Torque also does work (\(W = \tau\,\theta\)) and delivers power (\(P = \tau\omega\)) — the reason engine specs quote both torque and rpm.

Worked example: the falling rod

A uniform rod (mass \(M\), length \(L\)) pivots at one end and is released horizontal. Initial angular acceleration?

Gravity acts at the CM, moment arm \(L/2\): \(\tau = Mg\frac{L}{2}\); with \(I_\text{end} = \tfrac13 ML^2\),

\[\alpha = \frac{\tau}{I} = \frac{MgL/2}{ML^2/3} = \frac{3g}{2L}\]

The tip then accelerates at \(a = L\alpha = \tfrac32 g\) — faster than free fall! (Stack a coin on the tip of a falling ruler: the ruler drops out from under it.)

Worked example: the wrench and the pipe

A 20 N·m bolt, a 25 cm wrench: minimum force \(F = \tau / r = 20/0.25 = 80\) N applied perpendicular. Slip a pipe over the wrench to double \(r\) and the required force halves — leverage is free torque (the reason "cheater bars" both work and snap bolts).

Static equilibrium preview

A body at rest needs both \(\sum\vec F = 0\) and \(\sum\tau = 0\) (about any point) — the twin conditions that power every ladder, bridge, and seesaw problem in equilibrium & elasticity.

Common mistakes

  • Using the full distance instead of the moment arm. Only the perpendicular component of \(\vec r\) (or of \(\vec F\) — your choice) contributes.
  • Forgetting torque depends on the chosen axis. State the pivot; in equilibrium problems, choose the axis that kills the most unknowns.
  • Sign chaos. Fix a positive rotation sense first and audit each torque against it.
  • Confusing torque (N·m) with energy (J). Same units, different beasts — torque is a vector, and its "metre" is a lever arm, not a displacement.

Knowledge graph position

Prerequisites: Vectors, Newton's laws, Rotation. Leads to: Equilibrium, Angular momentum.

Quiz

Q1 (computational). A 3 N force acts at \((2\,\text{m}, 0)\) pointing in the \(+y\) direction. Torque about the origin?

Answer

\(\vec\tau = \vec r\times\vec F = (2\hat\imath)\times(3\hat\jmath) = 6\hat k\) — 6 N·m counterclockwise.

Q2 (conceptual). Why does a tightrope walker carry a long pole?

Answer

The pole hugely increases the system's moment of inertia about the wire, so a given gravitational torque produces a much smaller \(\alpha = \tau/I\) — the tip starts slowly, buying time to correct. (Drooping ends even lower the CM.)

Q3 (multiple choice). You push on a door with fixed force. Torque about the hinges is greatest when you push: (a) near the hinge, ⊥ door (b) at the handle, ⊥ door (c) at the handle, along the door's plane

Answer

(b). Maximize both the distance and the perpendicularity; (c) has zero moment arm no matter the distance.