Question 7.7: Lifting a drawbridge A drawbridge across the mouth of an inl......
Lifting a drawbridge
A drawbridge across the mouth of an inlet on the coastal highway is lifted by a cable to allow sailboats to enter the inlet. You are driving across the 16-m-long drawbridge when the bridge attendant accidentally activates the bridge. You abruptly stop the car 4.0 m from the end of the bridge. The cable makes a 53° angle with the horizontal bridge. The mass of your car is 1000 kg and the mass of the bridge is 4000 kg. Estimate the tension force that the cable exerts on the bridge as it slowly starts to lift the bridge.
Sketch and translate We sketch the situation below and choose the bridge as the system. We place the axis of rotation where the drawbridge connects by a hinge to the roadway at the left side of the bridge— a good choice, as we have no information about that force.
Simplify and diagram We model the car as a pointlike object and the bridge as a rigid body with uniform mass distribution. The latter assumption means that Earth exerts a gravitational force on the center of the bridge. The bridge has just started to rise, so it is still approximately horizontal. Since it moves very slowly, we will assume that it is in static equilibrium. As we can see from the force diagram, four objects exert forces on the bridge. (1) The hinges on the left side exert a force \vec{F}_{\mathrm{H} \text { on } \mathrm{B}} that is unknown in magnitude and direction.
(2) Earth exerts a (4000 kg)(9.8 N/kg) = 39,200 N gravitational force \vec{F}_{\mathrm{E} \text { on B }} on the center of the bridge. (3) The car pushes down on the bridge, exerting a
(1000 \mathrm{~kg})(9.8 \mathrm{~N} / \mathrm{kg})=9800 \mathrm{~N} \text { force } \vec{F}_{\text {Car on B }} 4.0 \mathrm{~m} from the right side of the bridge. (4) The cable exerts an unknown force \vec{T}_{\text {Cable on } B} on B on the right edge of the bridge at a
53° angle above the horizontal.
Represent mathematically Since four objects exert forces on the bridge, the torque condition of equilibrium will include four torques produced by these forces
F_{\text {H on B }}(0)+\left(-F_{\text {E on B }} L_{\mathrm{CM}} \sin 90^{\circ}\right)+\left(-F_{\text {Car on B }} L_{\text {Car }} \sin 90^{\circ}\right)
+T_{\text {Cable on B }} L_{\text {Cable }} \sin 53^{\circ}=0 \text {. }
Substitute sin 90° = 1.0 and sin 53° = 0.80 and rearrange the above to find an expression to determine the unknown tension force that the cable exerts on the bridge
T_{\text {Cable on B }}=\frac{F_{\mathrm{E} \text { on B }} L_{\mathrm{CM}}+F_{\text {Car on } \mathrm{B}} L_{\text {Car }}}{L_{\text {Cable }} \sin 53^{\circ}}
Solve and evaluate Substitute the following values into the above equation: L_{\mathrm{CM}}=8.0 \mathrm{~m}, L_{\mathrm{Car}}=12.0 \mathrm{~m} L_{\text {Cable }}=16.0 \mathrm{~m}, F_{\text {Eon B }}=3.92 \times 10^4 \mathrm{~N} \text {, and } F_{\text {Caron B }}= 9800 \mathrm{~N} \text {. This yields }
T_{\text {Cable on B }}=\frac{\left(3.92 \times 10^4 \mathrm{~N}\right)(8.0 \mathrm{~m})+(9800 \mathrm{~N})(12.0 \mathrm{~m})}{(16 \mathrm{~m}) \sin 53^{\circ}}
= 34,000 N
The unit is correct. The value of 34,000 N is reasonable given that the bridge holds the 9800-N car near the free end and that Earth exerts a force of 39,000 N on the bridge in its middle at its center of mass.
Try it yourself: What force would the cable have to exert on the bridge if your car was 4.0 m from the hinged end of the bridge instead of 4.0 m from the free end?
Answer: 28,000 N. This makes sense since the car is exerting a smaller torque when it is closer to the axis of rotation.