The Two Conditions for Equilibrium
The Two Conditions for Equilibrium
- Lesson 3.01 said equilibrium means zero resultant force. That was only half of the rule.
- Forces can cancel as a push and still spin the object. Full equilibrium needs two separate checks.
The two conditions
- The resultant force on the object is zero (it does not accelerate).
- The resultant moment (torque) about any point is zero (it does not start to turn).
- A couple is the standard example of why condition 1 alone is not enough: its two forces add to zero, but the object spins.
- The multiple-choice paper tests this with three-force diagrams where the forces balance but the moments do not, so none of the objects is in equilibrium (9702/13/O/N/24 Q15).
- Two forces alone can hold equilibrium only if they are equal, opposite and along the same line of action: otherwise they form a couple (9702/13/M/J/24 Q12).
Your turn— tap to reveal the worked answer (9702/11/M/J/22 Q12)
When is an object in equilibrium? (9702/11/M/J/22 Q12)
Answer: when there is no resultant force and no resultant torque. Naming only one condition is a standard wrong option.
Full problems: moments plus force balance
- Structured questions use both conditions in order. First take moments to find one force; then balance up and down forces to find another.
- The pivot choice matters. Take moments about the hinge (or any point where an unknown force acts). That force's distance is zero, so it disappears from the moment equation, and you solve for the other unknown first.
Worked example
Exam version: a hinged rod
A uniform rod AC is hinged to a wall at A and held up by a vertical cord at C, 2.6 m from A, with tension 22 N. The rod's weight is 19 N, acting 1.3 m from A. Two equal wheels of weight W hang at 0.45 m and 1.85 m from A. Find W, then find the force of the hinge on the rod. (9702/23/M/J/20 Q3)
Find W by moments about A
- About A, the unknown hinge force has zero distance, so it is not in the equation.
- Clockwise (loads) = anticlockwise (cord): .
- , so .
Answer
Find the hinge force by force balance
- Condition 1: up = down. Down: . Up so far: the cord, 22 N.
- The hinge supplies the rest: , vertically upward.
Answer
Take moments about the hinge first: the unknown hinge force disappears, and one equation gives one unknown. Then use up = down to find the hinge force. This two-step order is the standard route in every 2024/25 beam question.
- Ways the exam changes it: the support is a cable at an angle (use the perpendicular distance, or the cable's vertical component) (9702/22/O/N/24 Q3(c)); the beam rests on a floating cylinder, so moments feed into an upthrust calculation (lesson 4.12) (9702/22/F/M/25 Q2); a trapdoor with a changing force asks which way the resultant moment now turns (9702/22/M/J/24 Q4(c)).