PHYSICS · ROTATIONAL MECHANICS
Moments & Torque
Forces don't only move objects. They can turn them.
Explore moments, torque, lever arms, couples, rotational equilibrium, and the physics of forces acting around a pivot.
Moment and torque describe the rotational effect of a force.
Single Force
τ = rF sin θ
Couple
M_couple = Fd
The crucial difference: a single force has a net force, while an ideal couple has zero net force but a nonzero moment. That is why a couple produces a pure turning effect.
SECTION 2
Key Definitions
SECTION 3
Moment vs. Torque
Moment
Torque
There is not a fundamentally different equation for moment and torque. Both are based on the vector relationship:
The terminology usually reflects the physical situation being discussed. Do not teach students that moment and torque are completely different physical quantities.
SECTION 4
Ordinary Torque vs. Torque of a Couple
Single Force
Imagine pushing horizontally on the handle of a door. The door rotates because the force acts some distance from its hinges.
Result
Net Force ≠ 0 + Moment ≠ 0
The calculated moment depends on the reference point chosen.
Couple
Imagine turning a steering wheel with two hands. One hand pushes upward while the other pushes downward with equal force on the opposite side.
Result
ΣF = 0 but ΣM ≠ 0
The result is a pure turning effect.
The crucial difference: A single force can produce both translation and rotation. A couple produces rotation without a net translational force.
SECTION 5
Moment & Torque Calculator
Drag the force arrow or application point to update the torque in real time.
τ = rF sin θ = (0.50)(80)(sin 90°) = 40.00 N·m
EXAMPLE
Force: 80 N, Distance: 0.50 m, Angle: 90°
τ = (0.50)(80)(sin 90°) = 40 N·m
Because the force is perpendicular to the lever arm, the torque is at its maximum possible value.
SECTION 6
Angle Explorer
Keep F and r constant while changing θ. See how torque varies with angle.
0°
τ = 0
30°
τ = 0.5rF
90°
τ = max
150°
τ = 0.5rF
180°
τ = 0
A large force does not necessarily produce a large torque. The direction of the force matters. Only the component of force perpendicular to the position vector contributes to the torque.
SECTION 7
Torque of a Couple
M_couple = Fd = (25)(0.40) = 10.00 N·m
Because this is a couple, its moment is independent of the reference point used to calculate it.
COUPLE EXAMPLE
Two equal and opposite forces of 25 N act on a steering wheel. Perpendicular distance: 0.40 m.
M_couple = Fd = (25)(0.40) = 10 N·m
You do not multiply by two — the separation distance already accounts for both forces.
SECTION 8
Why Is a Couple Different?
Single Force
Net force ≠ 0
Net moment ≠ 0
The object both moves and rotates.
Couple
Net force = 0
Net moment ≠ 0
The object rotates without its center of mass accelerating sideways.
A couple creates pure rotational loading.
SECTIONS 9–11
Sign Convention, Multiple Forces & Equilibrium
Sign Convention
Counterclockwise = Positive (+)
M = +12 N·m
Clockwise = Negative (−)
M = −7 N·m
A different sign convention may be used if it is clearly defined and used consistently.
Multiple Forces
When several forces act on an object: ΣM = M₁ + M₂ + M₃ + ...Clockwise and counterclockwise moments must be assigned opposite signs.
Net counterclockwise moment of 5 N·m.
Rotational Equilibrium
An object is in rotational equilibrium when ΣM = 0.
ΣF = 0
No translational acceleration
ΣM = 0
No angular acceleration
For complete static equilibrium, both conditions must be satisfied simultaneously.
SECTION 12
Real-World Examples
Opening a Door
Pushing near the doorknob produces much more torque than pushing near the hinges. The farther the force is from the hinge, the larger the torque.
Using a Wrench
A longer wrench produces more torque for the same applied force. A force applied perpendicular to the wrench is most effective.
Riding a Bicycle
Force applied to a bicycle pedal creates torque about the crankshaft. The amount of torque changes as the pedal rotates because the angle between the crank arm and the applied force changes.
Seesaw
The effect of each person's weight depends on both their weight and their distance from the pivot. A lighter person can balance a heavier person by sitting farther from the pivot.
Steering Wheel
Two hands applying approximately equal and opposite tangential forces can produce a couple. The net force can be nearly zero while a substantial torque remains.
Turning a Jar Lid
Forces from the fingers and thumb can combine to create a couple that twists the lid.
Electric Motor
Electromagnetic forces generate torque on a rotor. This torque causes the motor shaft to rotate.
Car Engine
Combustion forces act through connecting rods and the crankshaft to generate torque. Engine torque describes the rotational output available at the crankshaft.
SECTION 13
Zul'Elkoen Applications
Starship Airlock
A crew member applies force to a manual emergency airlock handle. Users calculate how much torque is produced about the handle's pivot.
Gulliver Maintenance
A technician uses a long tool to loosen a seized fastener in the Gulliver's machinery bay. Users determine whether increasing force or increasing lever-arm length is more effective.
Attitude Control
Opposing thrusters positioned on different sides of a spacecraft can create a turning effect resembling a couple. The net linear force can be minimized while the spacecraft rotates.
SECTION 14
Common Mistakes
SECTIONS 15–16
Try It Yourself
A force of 49 N is applied at 0.3 m from a pivot, at 61° to the lever arm, acting clockwise. Calculate the torque.
SECTION 17
Quick Knowledge Check
A force acts directly toward a pivot. What torque does it produce?
SECTION 18
Advanced View
τ = rF sin θ
Basic mode uses the scalar relationship between force, distance, and angle. This is sufficient for most two-dimensional problems involving a single pivot.
