Torque

Definition

Torque is the rotational analog of force. It's the cross product of position vector r (from pivot to point of force application) and force F:

τ = r × F

Magnitude:

|τ| = r · F · sin θ

where θ is the angle between r and F. Maximum torque when F is perpendicular to r (sin 90° = 1); zero when F is along r (sin 0° = 0).

Direction is given by right-hand rule — curl fingers from r to F; thumb points along τ.

Units: N·m. (Same units as energy J, but they're different physical quantities.)

Lever arm — the geometric view

Equivalent way to think about torque: F times perpendicular distance from pivot to force's line of action.

The "lever arm" or "moment arm" is the shortest perpendicular distance from the pivot to where the force's line passes:

τ = F × (lever arm)

This is why:

• Longer wrenches give more torque for the same force.
• Door handles are placed far from hinges (long lever arm = easy to push).
• Push perpendicular to a door for maximum torque.

Newton's 2nd for rotation

The rotational equivalent of F = ma:

τ = I · α

where α is angular acceleration (rad/s²). Or equivalently:

τ = dL/dt

(Torque is rate of change of angular momentum, just as force is rate of change of linear momentum.)

Linear	Rotational
Mass m	Moment of inertia I
Velocity v	Angular velocity ω
Acceleration a	Angular acceleration α
Force F	Torque τ
Momentum p = mv	Angular momentum L = Iω
F = ma	τ = Iα
F = dp/dt	τ = dL/dt
KE = ½mv²	KE = ½Iω²
p conserved if F = 0	L conserved if τ = 0

Real-world torque problems

Situation	τ calculation
Tightening bolt with 30 cm wrench, 50 N force perpendicular	τ = 0.30 × 50 = 15 N·m
Opening door, push 20 N at 70 cm from hinge, perpendicular	τ = 0.70 × 20 = 14 N·m
Same door, push at angle 45°	τ = 0.70 × 20 × sin 45° = 9.9 N·m (less effective)
Car engine — 200 hp at 6000 rpm	τ = P/ω = (149 kW)/(628 rad/s) = 237 N·m
Earth's rotation slowing (tidal)	τ ≈ -4.5 × 10¹⁶ N·m → day lengthens by ~2 ms/century
Helicopter main rotor (medium helo)	~10,000 N·m torque needed; counter-rotor balances

Static equilibrium — Στ = 0

For a body at rotational equilibrium (no angular acceleration), the sum of torques about ANY axis is zero.

Στ = 0  (for any chosen axis)

This is hugely useful for solving statics problems — bridges, beams, see-saws, ladders. Choose your pivot cleverly to eliminate unknowns.

Example: balanced see-saw

A 30 kg child sits 1.5 m from a fulcrum. Where must a 60 kg adult sit on the other side to balance?

About the fulcrum:

τ_child = m_child · g · d_child = 30 × 9.81 × 1.5 = 441 N·m
τ_adult = m_adult · g · d_adult = 60 × 9.81 × d_adult
Set equal: 441 = 60 × 9.81 × d
d_adult = 441 / 588.6 = 0.75 m

The adult sits half the distance because they're double the mass.

JavaScript — torque calculations

// 2D torque about origin (z-component only)
function torque2D(rX, rY, fX, fY) {
  return rX * fY - rY * fX;  // z-component of cross product
}

// Magnitude form: τ = r · F · sin θ
function torqueFromAngle(r, F, thetaRad) {
  return r * F * Math.sin(thetaRad);
}

// Apply Newton's 2nd for rotation: τ = Iα
function angularAcceleration(torque, momentOfInertia) {
  return torque / momentOfInertia;
}

// Door analysis: 70 cm handle, 20 N push
console.log(`Perpendicular: ${torqueFromAngle(0.70, 20, Math.PI/2)} N·m`); // 14
console.log(`At 45°: ${torqueFromAngle(0.70, 20, Math.PI/4).toFixed(2)} N·m`); // 9.9
console.log(`Along door (parallel): ${torqueFromAngle(0.70, 20, 0)} N·m`); // 0

// Power-torque relationship for engines
function torqueFromPower(power_W, omega_rad_per_sec) {
  return power_W / omega_rad_per_sec;
}

// 200 hp = 149 kW, at 6000 rpm = 628 rad/s
const engineTorque = torqueFromPower(149000, 6000 * 2 * Math.PI / 60);
console.log(`Engine torque: ${engineTorque.toFixed(0)} N·m`);  // ~237

// See-saw balance: where to put person of mass m to balance another?
function balancePoint(m1, d1, m2) {
  // m1·d1 = m2·d2 (about pivot)
  return (m1 * d1) / m2;
}

console.log(`60 kg adult balances 30 kg child at 1.5 m: ${balancePoint(30, 1.5, 60)} m from fulcrum`);
// 0.75 m

// Rotational kinematics: starts at rest, applies torque
function spinUp(torque, momentOfInertia, time) {
  const alpha = torque / momentOfInertia;
  const omega = alpha * time;
  const theta = 0.5 * alpha * time * time;
  return { omega, theta, alpha };
}

console.log(spinUp(10, 0.5, 5)); // 10 N·m on I=0.5 kg·m² for 5s
// alpha = 20 rad/s², omega = 100 rad/s after 5s, θ = 250 rad

Where torque shows up

• Engineering and mechanics. Engines (rated by torque), turbines, motors, gearboxes — all spec'd in N·m or lb·ft.
• Tool design. Wrenches, screwdrivers (torque-limited or torque-tracking), torque wrenches for precise bolt tightening.
• Statics — buildings and bridges. Beams, trusses, foundations — Στ = 0 for static equilibrium.
• Robotics and prosthetics. Joint torques for arm/leg movement; servos rated by torque.
• Sports. Golf swing torque, baseball pitch spin (torque about wrist), gymnastics rotations.
• Spacecraft attitude control. Reaction wheels apply torque to change orientation; gyroscopes maintain stability.
• Auto industry. Engines marketed by peak torque (acceleration feel); transmissions multiply torque at lower gears.

Common mistakes

• Forgetting the cross product (or sin θ). Pushing along the lever arm gives zero torque (parallel forces don't rotate). Always include the perpendicular component.
• Confusing torque with force. Torque has units N·m; force has units N. They're related but different. A 100 N force at 1 m gives 100 N·m of torque; same force at 0.5 m gives 50 N·m.
• Confusing N·m with joules. Same units, but torque (N·m) and energy (J) are different physics. Torque × angle = work; force × distance = work.
• Wrong sign convention. Counterclockwise torque is conventionally positive (right-hand rule). Mixing signs gives wrong angular acceleration direction.
• Choosing pivot incorrectly. For Στ = 0, you can choose ANY pivot. Smart pivot (through unknowns or central) eliminates terms.
• Ignoring angle in lever-arm calculation. τ = r × F means perpendicular distance from pivot to force line, not r itself. Get the geometry right.