The Physics
Hypertextbook
Opus in profectus

# Rotational Equilibrium

## Practice

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### practice problem 2

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### practice problem 3

The photographs below show a counterweighted, steel drawbridge when the span is down and when the span is up. Determine the following quantities in terms of the weight of the bridge span.
1. The weight of the counterweight and the tension when the span is down.
2. The tension when the bridge is open and the torque needed to keep the span up.

#### solution

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

• The red triangles represent the pivot points for…
• the counterweight assembly on the left and
• the bridge span on the right.
• The yellow lines highlight the lever arms.
• The line running along the roadbed is divided into intervals of equal length, L.
• The other lines have lengths that can be determined with geometric reasoning if we assume that the angles between beams are all multiples of 45° (that is, 0°, 45°, 90°, 135°, 180°).
• The green arrows show the relevant forces.
• Wb and Wc are the weights of the moveable bridge span and counterweight, respectively.
• T1 and T2 are the tensions in the linkage when the bridge is closed and open, respectively.
• When the bridge is closed the moveable span is balanced so that there is no normal force on the far end.
1. Answer the first two parts by stating the equilibrium conditions when the span is down.
 left axis ∑τcounterclockwise = ∑τclockwise (1 L)(Wc)(sin 90°) = (√2 L)(T1)(sin 90°) 1 Wc = √2 T1 right axis ∑τcounterclockwise = ∑τclockwise (√2 L)(T1)(sin 90°) = (4 L)(Wb)(sin 90°) √2 T1 = 4 Wb

Combine equations.

 Wc = 4 Wb T1 = 2√2 Wb
2. Answer the second two parts by stating the equilibrium conditions when the span is up.
 left axis ∑τcounterclockwise = ∑τclockwise (1 L)(4 Wb)(sin 45°) = (√2 L)(T2)(sin 135°) 2√2 LWb = LT2 right axis ∑τcounterclockwise = ∑τclockwise (√2 L)(T2)(sin 45°) + τ = (4 L)(Wb)(sin 135°) LT2 + τ = 2√2 LWb

Combine equations.

 T2 = 2√2 Wb 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".

• "Strauss" for Joseph B. Strauss, chief engineer and owner of the Strauss Bascule Bridge Company in Chicago
• "Heel" since it tips back like a foot balanced on its heel
• "Trunnion" for the two large, weight-bearing axles (the term originally referred to the axle upon which cannons and artillery were balanced)
• "Bascule" from the French word for seesaw (when one side goes up, the other goes down)

### practice problem 4

Write something completely different.