The moment of a force
A force can make a body turn as well as move. The moment of a force measures that turning tendency about a point or axis.
Turning effect
Pushing a door near its handle opens it easily; pushing near the hinge barely moves it. The turning effect of a force, its moment or torque, depends on both the force and how far its line of action is from the pivot. Statics problems with rigid bodies require moments as well as forces.
Scalar definition
The moment of a force F about point O is \(M_O = Fd\), where d is the perpendicular distance from O to the force's line of action, called the moment arm. Moments are measured in newton-metres (N·m). An 80 N force applied perpendicular to a wrench 0.3 m from the bolt produces 24 N·m.
Direction and sign
In 2D, a moment tends to rotate a body either anticlockwise or clockwise about O. A common convention takes anticlockwise as positive. In 3D, a moment is a vector along the axis of rotation, given by the right-hand rule: curl your fingers in the turning direction, and your thumb points along the moment vector.
Angled forces
If the force is not perpendicular to the wrench, only its perpendicular component turns it. Applying 80 N at 60° to a 0.3 m wrench gives \(M = 80 \times 0.3 \times \sin 60° \approx 20.8\) N·m. A force whose line of action passes through O has zero moment about O, however large.
Resolving into components
It is often easiest to resolve a force into x and y components and add their moments. For a force with components \(F_x\) and \(F_y\) acting at a point (x, y) relative to O, the anticlockwise moment is \(M_O = xF_y - yF_x\).
Resultant moment
When several forces act, the resultant moment about O is the sum of their individual moments, with signs. A beam with 100 N acting clockwise 2 m from O and 150 N acting anticlockwise 1 m from O has a net moment of -200 + 150 = -50 N·m, that is, 50 N·m clockwise.
Moments in everyday tools
Long-handled tools multiply moments: bolt cutters, wheel braces, and pry bars all let a modest force produce a large moment. Torque wrenches let mechanics tighten bolts to a specified moment, for example around 100 to 140 N·m for many car wheel nuts, to avoid stripping threads or loose wheels.
Worked example
A gardener lifts a wheelbarrow whose 600 N load acts 0.5 m behind the wheel axle, while the handles are 1.5 m behind the axle. Taking moments about the axle, the lifting force F satisfies F x 1.5 = 600 x 0.5, so F = 200 N, one third of the load. The wheelbarrow is a lever with a mechanical advantage of 3.
Key takeaways
A moment measures a force's turning tendency: M = Fd, with d the perpendicular distance from the point to the line of action, in N·m. Anticlockwise is usually positive in 2D; in 3D, moments are vectors given by the right-hand rule. Only the perpendicular part of a force turns, so M = F r sin theta, and forces through the point give zero moment. With components, M = x Fy - y Fx. Add moments with signs. Long handles and levers trade distance for force.
Flashcard
Turning effect of a force
Check your understanding
An 80 N force acts perpendicular to a wrench 0.3 m from the bolt. What moment does it produce?