1. 12. Static Equilibrium

2. Conditions for Equilibrium
A bridge is an example
of a system in static
equilibrium. The bridge
undergoes neither
linear nor rotational
motion!

3. Conditions for Equilibrium
A system is in static equilibrium if:
the net external force is zero
the net external torque is zero

4. Problem Solving Guideline
All static equilibrium problems are solved the same
way:
Find all external forces
Choose a pivot
Find all external torques
Set net force to zero
Set net torque to zero
Solve for unknown quantities

5. Problem Solving Guideline
It is generally simpler to choose the pivot at the
point of application of the force for which you
have the least information.

6. Example – A Drawbridge What is the tension in the
supporting cable of a 14 m,
11,000 kg drawbridge?
Forces: 1. Force at pivot 2. Tension in cable
3. Weight of bridge

7. Example – A Drawbridge Pivot: A sensible choice is the
hinge since we do not
know the exact direction
of the hinge force, nor do
we care about it!

8. Example – A Drawbridge Torques Due to the weight τg = –(L/2) mg sinθ1
(This torque is into the page. Why?)
Due to the tension
τT = LT sinθ2

9. Examples of Static Equilibrium

10. Example – A Leaning Ladder
At what minimum angle can the ladder lean
without slipping?
The wall is frictionless and there
is friction between the floor and
the ladder.

11. Example – A Leaning Ladder
Forces:
1. Normal force at bottom of ladder
2. Friction force at bottom of ladder
3. Ladder’s weight
4. Normal force at top of ladder
Pivot:
Choose bottom of ladder
Why?

12. Example – A Leaning Ladder
Torques:
1. Due to ladder’s weight
2. Due to normal force at top of
ladder
Solve:
Force, x: μ n1 – n2 = 0
Force, y: n1 – mg = 0
Torque:
Ln2sinϕ – (L/2) mg cosϕ = 0

13. Example – A Leaning Ladder
From the force equations we get n2 = μmg.
Therefore,
μ sinϕ – (1/2)cosϕ = 0
and so,
tanϕ = 1/(2μ)

14. Example – Standing on a Plank

15. Example – Standing on a Plank
Net force:
Net torque:

16. Example – Standing on a Plank
Force on right scale
Force on left scale
Do these make sense?

17. Example – Force on Elbow
What is the force on the elbow?
m = 6 kg
Assume biceps force acts
3.4 cm from pivot point O.

18. Example – Force on Elbow
Model forearm as a horizontal
rod

19. Example – Force on Elbow
The force we know least about is
the force on the elbow. So, let’s
take the elbow (O) as the pivot.
Net torque:

20. Example – Force on Elbow
Net force x: y:

21. Summary
For a system to be in static equilibrium, both the net force and the net torque must be zero.
When solving static equilibrium problems, it often simplifies things to choose the pivot so that the torque from unknown forces is zero.