A Truck on a Drawbridge. A loaded cement mixer drives onto an old drawbridge, where it stalls with its center of gravity three-quarters of the way across the span. The truck driver radios for help, sets the handbrake, and waits. Meanwhile, a boat approaches, so the drawbridge is raised by means of a cable attached to the end opposite the hinge (?Fig. P11.52?). The drawbridge is 40.0 m long and has a mass of 18,000 kg; its center of gravity is at its midpoint. The cement mixer, with driver, has mass 30,000 kg. When the drawbridge has been raised to an angle of 30o above the horizontal, the cable makes an angle of 70o with the surface of the bridge. (a) What is the tension T in the cable when the drawbridge is held in this position? (b) What are the horizontal and vertical components of the force the hinge exerts on the span?
Solution 56P Introduction To calculate the tension of the bridge we will consider the moment of the bridge with respect to the hinge. Then we have to calculate the horizontal and vertical component of force exerted by the hinge. Step 1 The free body diagram of the system is shown below. Now the angle of the tension with respect to the horizontal plane is = 70° 30° = 40° Sum of the moment about the hinge is = w cos × (20 m) + w cos × (30 m) T × (40 m) = 0 b t 2 2 (18000 kg)(cos30°)(9.8 m/s ) × (20 m) + (30000 kg)(cos30°)(9.8 m/s ) × (30 m) T × (40 m) = 0 T = 2.67 × 10 N5 Hence the tension of the cable is 2.67 × 10 N. 5