48610 Chap.9 Moments of a Force and Moment Equilibrium
Moments of a Force and Moment Equilibrium
A ring has no size, so every force on it passes through one point and the only question is whether they cancel. Real components have size, and two equal and opposite forces applied at different places do not cancel at all: they spin the body.
The quantity that measures that turning effect is the moment, equal to the force multiplied by the perpendicular distance from the chosen point to the line of action of the force. Two features of the definition carry most of the marks. A moment is always about a stated point, because the same force has a different moment about every point.
And a force whose line of action passes through that point has no moment about it, however large it is.
Week 7 asks for every moment two ways. The geometric route finds the perpendicular distance and multiplies.
The component route splits the force at its point of application and adds the moments of the components, which is Varignon's theorem, and it is usually faster because one component often points straight at the pivot and drops out. Where a position is given by coordinates rather than a length and an angle, the same idea becomes a formula in the position and force components.
Adding the moment equation to the two force equations gives three equations for a rigid body in a plane, and choosing the point about which to take moments is a free choice that can remove two unknowns at a stroke.
What this chapter covers
- 01
The moment as force times perpendicular distance
- 02
Always about a stated point, and zero when the line passes through it
- 03
Newton metres, and why they are not joules
- 04
Sense and sign, and declaring the convention
- 05
Sliding a force along its own line without changing its moment
- 06
Varignon's theorem and the component route
- 07
Resolving at the point of application so one component drops out
- 08
The coordinate form and why the two terms oppose
- 09
Couples: equal and opposite forces with no resultant
- 10
Rigid body equilibrium as three equations
- 11
Choosing the moment point to remove unknowns
The counterweight that balances a jib crane
- +1Declare the sense and measure every distance horizontally from the tower axis, because all the weights act vertically and a vertical force's lever arm about A is its horizontal offset.
- +1Write the moment sum: the load at 14.0 m and the jib at 12.0 m on one side, the counter jib at 4.00 m and the counterweight at 5.00 m on the other, each weight being its mass times g.
- +1Divide the whole equation by g. Gravity multiplies every term, so it cancels and the equation becomes one in megagrams and metres: 2.00 times 14.0 plus 1.50 times 12.0 minus 0.50 times 4.00 minus 5.00 m equals zero.
- +1Evaluate: 28.0 plus 18.0 minus 2.00 is 44.0, so the counterweight mass is 44.0 divided by 5.00, that is 8.80 Mg. Check it by re-taking moments about a different point, which must give the same mass.
Key terms
- Moment
- The turning effect of a force about a point, equal to the force multiplied by the perpendicular distance from the point to the force's line of action, with units of newton metres.
- Lever arm
- The perpendicular distance from the chosen point to the line of action of a force. It is not the length of the bar unless the force happens to act square to the bar.
- Line of action
- The infinite straight line along which a force acts. A force may be slid anywhere along it without changing its moment about any point, which is why it is extended in both directions when drawing.
- Varignon's theorem
- The rule that the moment of a force about a point equals the sum of the moments of its components about the same point, which licenses the component route to any moment.
- Sense
- The direction in which a moment turns the body, clockwise or anticlockwise on a page. A moment quoted without its sense is only half an answer.
- Couple
- Two equal and opposite parallel forces separated by a distance, producing a pure turning effect with no resultant force. Its moment is the same about every point, so it needs no location.
- Rigid body equilibrium
- The condition that the two force sums and the moment sum are all zero, giving three equations in a plane and therefore three solvable unknowns from one free body diagram.
Moments of a Force and Moment Equilibrium FAQ
Why is the lever arm not just the length of the bar?
Because the lever arm is measured perpendicular to the line of action of the force, and only a force acting square to the bar has a perpendicular distance equal to the bar length. A force at 55 degrees to an 80 mm toggle has a lever arm of 80 sin 55, which is 65.5 mm. The equivalent way to see it is that only the component of the force across the bar turns it; the component along the bar points at the pivot and does nothing.
When should I use the coordinate form instead of the perpendicular distance?
When the geometry makes a perpendicular distance awkward to find, which is usually when the point is given by coordinates rather than by a length and an angle. The coordinate form gives each force component its own lever arm, the vertical component acting at the horizontal offset and the horizontal component at the vertical offset, with the two opposing.
Students who have met the vector cross product elsewhere will recognise it as the scalar result of that product, and the tutorial suggests using it as an independent check.
Does it matter which point I take moments about?
Not for correctness, but enormously for effort. Any point gives a valid equation, so taking moments about a point through which an unknown force passes removes that unknown entirely, because its lever arm is zero. Choosing a pin that carries two unknown components can turn three simultaneous equations into one equation with one unknown.
It also gives you a genuinely independent check: re-take moments about a different point and the answer must be the same.
What makes a couple different from an ordinary moment?
A couple has no resultant force, only a turning effect, and its moment is the same about every point on the body. That is why an applied couple in a problem does not need a location. The Week 8 exercises introduce it as an applied couple and note that you can think of it as an applied moment or torque; the torque a motor delivers to a shaft is exactly this.
Assessment move
Do every moment problem twice, by perpendicular distance and by components, until the two routes agree without thinking. That is not extra work: the tutorial explicitly asks you to check each answer by a different method, and the second route takes a fraction of the time of the first.
Get into the habit of extending the line of action in both directions before measuring anything, because measuring to the arrowhead instead of to the line is the most common error in the topic. And always write the sense next to the magnitude, as a curved arrow on the diagram and as a word in the answer.
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