Rotation and torque

Why this matters

Many of the things engineers build turn: shafts, gears, wheels, turbines, motors, doors and wrenches. Even structures that do not move, such as beams, cantilevers and cranes, must not start turning, so their designers balance moments (another name for torques) as carefully as forces. This lesson covers the rotational versions of the ideas you already know: position, velocity, acceleration and force.

Describing rotation

• Angle θ, in radians. One full turn is 2π rad (360°).
• Angular velocity ω, in rad/s: how fast the angle changes.
• Angular acceleration α, in rad/s².

Machines are usually rated in revolutions per minute (rpm). To convert:

ω (rad/s) = rpm × 2π / 60
3000 rpm  = 3000 × 2π / 60 = 314 rad/s

A point at distance r from the axis moves along its circle at speed v = r·ω. The rim of a 0.30 m radius wheel turning at 10 rad/s moves at 3.0 m/s. The constant-acceleration equations from the kinematics lesson carry over directly, with θ, ω and α in place of x, v and a.

Torque

A force makes something rotate only if it acts at a distance from the pivot and has a part perpendicular to the lever arm. Torque measures that turning effect:

τ = r · F · sin θ

• r is the distance from the pivot to where the force acts.
• θ is the angle between the lever arm and the force.
• The units are newton-metres (N·m). (Joules are also N·m, but torque is not energy; always write torque as N·m.)

Equivalently, τ = F × (perpendicular distance from the pivot to the force's line of action). That perpendicular distance is the moment arm.

By convention, counterclockwise torques are positive and clockwise torques negative.

Worked example 1: a wrench

You pull with 40 N on a wrench, 0.25 m from the bolt.

pulling at 90° to the handle:  τ = 0.25 × 40 × sin 90° = 10 N·m
pulling at 30° to the handle:  τ = 0.25 × 40 × sin 30° =  5 N·m
pulling along the handle:      τ = 0.25 × 40 × sin 0°  =  0 N·m

A longer wrench gives more torque for the same pull, which is the whole point of a breaker bar. Bolts on real equipment have specified tightening torques from the manufacturer and are tightened with a calibrated torque wrench; do not estimate them by feel.

Rotational equilibrium

An object is in static equilibrium when it is not accelerating in any direction and not starting to rotate:

ΣF = 0     and     Στ = 0   (about any pivot you choose)

Worked example 2: a seesaw

A 30 kg child sits 2.0 m from the pivot of a seesaw. Where must a 40 kg child sit on the other side to balance it?

counterclockwise:  30 × 9.81 × 2.0 = 588.6 N·m
clockwise:         40 × 9.81 × d
balance:           40 × 9.81 × d = 588.6   →   d = 1.5 m

The g cancels, so in practice you can balance mass × distance: 30 × 2.0 = 40 × 1.5. The heavier child sits closer.

Choosing the pivot at an unknown force removes that force from the torque equation, because its moment arm is zero. This trick makes beam and bracket problems much easier, and you will use it constantly in statics.

Newton's second law for rotation

The rotational version of ΣF = m·a is:

Στ = I · α

I is the moment of inertia (kg·m²): rotational mass. It depends not only on the mass but on how far that mass sits from the axis. A solid disc of mass M and radius R has I = ½·M·R²; a thin ring of the same mass and radius has I = M·R², twice as much, because all of its mass is at the rim. That is why flywheels put their mass far from the axis.

Rotational power

For a rotating shaft, power is torque times angular velocity:

P = τ · ω

A motor delivering 10 N·m at 3000 rpm (314 rad/s) supplies 10 × 314 ≈ 3,140 W, about 3.1 kW. This relation is how motor and engine data sheets link torque, speed and power.

In code

import math

def rpm_to_rad_s(rpm):
    return rpm * 2 * math.pi / 60

print(round(10 * rpm_to_rad_s(3000)))   # 3142 (watts)

Common mistakes

• Using the full length when the force is at an angle. Only the perpendicular part counts: multiply by sin θ, or use the moment arm.
• Forgetting sign conventions. Decide which way is positive before you add torques.
• Mixing rpm and rad/s. Formulas such as v = r·ω and P = τ·ω need rad/s.
• Writing torque in joules. Use N·m.

Engineering note

Balancing moments on a seesaw is physics. Sizing a real beam, crane, shaft or bolted joint involves materials, fatigue, dynamic loads, codes and safety factors, and anything that affects public safety must be reviewed and signed off by a licensed engineer.

Quick recap

• τ = r·F·sin θ, in N·m.
• Equilibrium: ΣF = 0 and Στ = 0.
• Στ = I·α; mass far from the axis means a large I.
• P = τ·ω, with ω in rad/s.

Try it

In the Do tab, write the torque function and use it to balance a seesaw.