System of Particles and Rotational Motion · Lesson 9 of 15
Equilibrium, Couples and Levers
“Use force and moment balance as independent conditions for rigid-body equilibrium.”
• Distinguish translational and rotational equilibrium. • Explain a couple and its origin-independent moment. • Derive the effect of shifting the torque origin. • Apply the principle of moments and mechanical advantage.
Two balances must be checked
For a body at rest to remain at rest in the rigid-body problems here, both the resultant force and resultant torque must vanish. Force balance controls CM acceleration; moment balance controls rotational change. A particle model only needs the concurrent force balance, but an extended body can turn even with zero resultant force. Mechanical equilibrium need not mean rest if the body already has a steady allowed motion; static equilibrium is the at-rest case used in our support calculations.
Two equal parallel forces in the same direction, applied symmetrically at opposite rod ends, can have cancelling moments about the CM but a nonzero resultant force: translation changes. Reverse one force and the resultant force becomes zero, while the moments reinforce: rotation changes. This shows why the conditions are independent. Do not treat torque balance alone as proof of complete equilibrium.
A couple
A couple consists of two equal, opposite forces with distinct parallel lines of action. The net force vanishes but the torque magnitude is Fd, where d is the perpendicular separation. Turning a bottle lid with opposite finger forces is a familiar example. A compass needle in a uniform magnetic field also experiences a turning effect, represented at this level by a force-pair model, until aligned.
This relation follows by replacing every rᵢ with rᵢ − a and distributing the cross products in the torque sum. It explains why, once force balance holds, any convenient pivot can be chosen for the moment equation. Choosing a support point often removes an unknown support force from that equation. It does not remove the need to check the force equations afterward.
The principle of moments
An ideal lever is a light rod pivoted at a fulcrum. A load F₁ at perpendicular arm d₁ and an opposing effort F₂ at arm d₂ balance when their moments agree. A support reaction at the fulcrum supplies force balance but has zero moment about the fulcrum. If the rod’s own weight is not negligible, include it at its CM.
Problem
A 120 N load acts 0.20 m from a fulcrum. The effort acts 0.80 m on the opposite side, with both forces perpendicular to the lever.
- 1.The required effort is 120(0.20)/0.80 = 30 N. Mechanical advantage is 120/30 = 4.
- 2.If both applied forces point downward, the upward fulcrum reaction is 120 + 30 = 150 N.
- 3.The lever reduces required effort by using a longer arm; it does not imply creating energy.
Quiz
Zero resultant force alone guarantees what?
Rotation may still change if a couple acts.
A couple has which net force?
Equal opposite forces cancel as vectors.
The moment of a couple depends on what?
Its magnitude is Fd.
Why take moments about a support?
The support force acts through that point.
A larger effort arm gives what ideal mechanical advantage?
MA equals effort arm divided by load arm.
Key Takeaways
• Force and torque equilibrium are independent requirements. • A couple has zero resultant force and a nonzero moment. • Torque becomes origin independent when the resultant force vanishes. • Use perpendicular arms in lever calculations.