Question 2.SP.4: In a ship-unloading operation, a 3500-lb automobile is suppo......
In a ship-unloading operation, a 3500-lb automobile is supported by a cable. A worker ties a rope to the cable at A and pulls on it in order to center the automobile over its intended position on the dock. At the moment illustrated, the automobile is stationary, the angle between the cable and the vertical is 2°, and the angle between the rope and the horizontal is 30°. What are the tensions in the rope and cable?
STRATEGY: This is a problem of equilibrium under three coplanar forces. You can treat point A as a particle and solve the problem using a force triangle.
- Weight of the automobile: 3500 lb
- Angle between the cable and the vertical: 2°
- Angle between the rope and the horizontal: 30°
Final Answer
MODELING and ANALYSIS:
Free-Body Diagram. Choose point A as the particle and draw the complete free-body diagram (Fig. 1). T_{A B} is the tension in the cable AB, and T_{A C} is the tension in the rope.
Equilibrium Condition. Since only three forces act on point A, draw a force triangle to express that it is in equilibrium (Fig. 2). Using the law of sines,
\frac{T_{A B}}{\sin 120^{\circ}}=\frac{T_{A C}}{\sin 2^{\circ}}=\frac{3500 ~\mathrm{lb}}{\sin 58^{\circ}}
With a calculator, compute and store the value of the last quotient. Multiplying this value successively by sin 120° and sin 2°, you obtain
T_{A B}=3570 ~\mathrm{lb} \quad T_{A C}=144 ~\mathrm{lb}
REFLECT and THINK: This is a common problem of knowing one force in a three-force equilibrium problem and calculating the other forces from the given geometry. This basic type of problem will occur often as part of more complicated situations in this text.
