Rotational Equilibrium

Practice

practice problem 1

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solution

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practice problem 2

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solution

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practice problem 3

The photographs below show a counterweighted, steel drawbridge in the closed (span down) and open (span up) positions.
Span Down Span Down
Bridge Closed — Span Down [magnify] Bridge Open — Span Up [magnify]

Determine the following quantities in terms of the weight of the bridge span.

  1. The weight of the counterweight and the tension when the bridge is closed.
  2. The tension when the bridge is open and the torque needed to keep the span open.

solution

The elements used to solve this problem are highlighted in color: pivot points in red, lever arms in yellow, and forces in green.

Span Down with Annotations Span Up with Annotations
Bridge Closed — Span Down [magnify] Bridge Open — Span Up [magnify]
  1. Answer the first two parts by stating the equilibrium conditions when the span is down.
         
    left axis   right axis
    ∑τcounterclockwise   =   ∑τclockwise   ∑τcounterclockwise   =   ∑τclockwise
    (1 L)(Wc)(sin 90°)   =   (√2 L)(T1)(sin 90°)   (√2 L)(T1)(sin 90°)   =   (4 L)(Wb)(sin 90°)
    1 Wc   =   √2 T1   √2 T1   =   4 Wb
     
    combine equations
    Wc   =   4 Wb
    T1   =   2√2 Wb  =  2.83Wb
         
  2. Answer the second two parts by stating the equilibrium conditions when the span is up.
         
    left axis   right axis
    ∑τcounterclockwise   =   ∑τclockwise   ∑τcounterclockwise   =   ∑τclockwise
    (1 L)(4 Wb)(sin 45°)   =   (√2 L)(T2)(sin 135°)   (√2 L)(T2)(sin 45°) + τ   =   (4 L)(Wb)(sin 135°)
    2√2 LWb   =   LT2   LT2 + τ   =   2√2 LWb
     
    combine equations
    T2   =   2√2 Wb  =  2.83Wb
     τ   =   0 
         

Because of the clever way the linkage folds, the bridge is maintained in balance throughout operation. There is no change in the tension and no extra torque is needed to keep it open. In accordance with Newton's second law of motion, some extra torque is needed to start it moving, but not much. The bridge and counterweight, which together weigh more than a million kilograms, is opened an closed with a relatively small electric motor. Something like 50 or 60 kW (75 hp) is powerful enough.

The bridge in this photo is a part of the La Salle Causeway, which spans the Cataraqui River in Kingston, Ontario. It is an example of a "Strauss heel trunnion bascule bridge".

practice problem 4

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solution

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The Physics Hypertextbook

  • No condition is permanent.