# Rotational Equilibrium

## Practice

### practice problem 1

#### solution

Answer it.

### practice problem 2

#### solution

Answer it.

### practice problem 3

Span down | Span up |

- The weight of the counterweight
*and*the tension when the span is down. - 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 line running along the roadbed is divided into intervals of equal length,
- The green arrows show the relevant forces.
*W*and_{b}*W*are the weights of the moveable bridge span and counterweight, respectively._{c}*T*and_{1}*T*are the tensions in the linkage when the bridge is closed and open, respectively._{2}- When the bridge is closed the moveable span is balanced so that there is no normal force on the far end.

- Answer the first two parts by stating the equilibrium conditions when the span is down.
*left axis*∑τ _{counterclockwise}= ∑τ _{clockwise}(1 *L*)(*W*)(sin 90°)_{c}= (√2 *L*)(*T*_{1})(sin 90°)1 *W*_{c}= √2 *T*_{1}*right axis*∑τ _{counterclockwise}= ∑τ _{clockwise}(√2 *L*)(*T*_{1})(sin 90°)= (4 *L*)(*W*)(sin 90°)_{b}√2 *T*_{1}= 4 *W*_{b}Combine equations.

*W*=_{c}4 *W*_{b}*T*_{1}=2√2 *W*_{b} - Answer the second two parts by stating the equilibrium conditions when the span is up.
*left axis*∑τ _{counterclockwise}= ∑τ _{clockwise}(1 *L*)(4*W*)(sin 45°)_{b}= (√2 *L*)(*T*_{2})(sin 135°)2√2 *LW*_{b}= *LT*_{2}*right axis*∑τ _{counterclockwise}= ∑τ _{clockwise}(√2 *L*)(*T*_{2})(sin 45°) + τ= (4 *L*)(*W*)(sin 135°)_{b}*LT*_{2}+ τ= 2√2 *LW*_{b}Combine equations.

*T*_{2}=2√2 *W*_{b}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

#### solution

Answer it.