12. Static Equilibrium

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12. Static Equilibrium
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Conditions for Equilibrium
A bridge is an example of a system in static
equilibrium. The bridge undergoes neither linear
nor rotational motion!
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Conditions for Equilibrium
A system is in static equilibrium if
the net external force is zero
the net external torque is zero
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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

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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.
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Example A Drawbridge
What is the tension in the supporting cable of a
14 m, 11,000 kg drawbridge? Forces1. Force at
pivot2. Tension in cable 3. Weight of bridge
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Example A Drawbridge
PivotA sensible choice is the hinge since we
do not know the exact direction of the hinge
force, nor do we care about it!
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Example A Drawbridge
Torques Due to the weight tg (L/2) mg
sinq1 (This torque is into the
page. Why?) Due to the tension tT LT
sinq2
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Examples of Static Equilibrium
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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.
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Example A Leaning Ladder
Forces 1. Normal force at bottom of ladder 2.
Friction force at bottom of ladder 3. Ladders
weight 4. Normal force at top of
ladder Pivot Choose bottom of ladder Why?
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Example A Leaning Ladder
Torques 1. Due to ladders weight 2. Due to
normal force at top of ladder Solve Force,
x m n1 n2 0 Force, y n1 mg
0 Torque Ln2sinf (L/2) mg cosf 0
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Example A Leaning Ladder
From the force equations we get n2 m
mg. Therefore, msinf (1/2)cosf 0 and so,
tanf 1/(2m)
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Example Standing on a Plank
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Example Standing on a Plank
Net force
Net torque
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Example Standing on a Plank
Force on right scale
Force on left scale
Do these make sense?
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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.
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Example Force on Elbow
Model forearm as a horizontal rod
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Example Force on Elbow
The force we know least about is the force on
the elbow. So, lets take the elbow (O) as the
pivot. Net torque
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Example Force on Elbow
Net force
x y
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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.