15. Rotational Equilibrium

A 70 kg, 1.90 m man doing push-ups holds himself in place making 20° with the floor. His feet and arms are, respectively, 1.15 m below and 0.4 m above his center of mass. You may model him as a thin, long board, and assume his arms and feet are perpendicular to the floor. How much force does the floor apply to each of his hands? (Use g=10 m/s².) 

(II) Three children are trying to balance on a seesaw, which includes a fulcrum rock acting as a pivot at the center, and a very light board 3.2 m long (Fig. 12–60). Two playmates are already on either end. Boy A has a mass of 45 kg, and boy B a mass of 35 kg. Where should girl C, whose mass is 25 kg, place herself so as to balance the seesaw?

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(II) A 172-cm-tall person lies on a light (massless) board which is supported by two scales, one under the top of her head and one beneath the bottom of her feet (Fig. 12–65). The two scales read, respectively, 35.1 and 31.6 kg. What distance is the center of gravity of this person from the top of her head?

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(II) The Leaning Tower of Pisa is 55 m tall and about 7.7 m in radius. The top is 4.5 m off center. Is the tower in stable equilibrium? If so, how much farther can it lean before it becomes unstable? Assume the tower is of uniform composition.

(II) The center of gravity of a loaded truck depends on how the truck is packed. If it is 4.0 m high and 2.4 m wide, and its cg is 2.2 m above the ground, on how steep a slope can the truck be parked without tipping over (Fig. 12–77)?

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A 50-story building is being planned. It is to be 180.0 m high with a base 46.0 m by 76.0 m. Its total mass will be about 1.8 x 10⁷ kg, and its weight therefore about 1.8 x 10⁸ N. Suppose a 200-km/h wind exerts a force of 950N/m² over the 76.0-m-wide face (Fig. 12–86). Calculate the torque about the potential pivot point, the rear edge of the building (where F→_E acts in Fig. 12–86), and determine whether the building will topple. Assume the total force of the wind acts at the midpoint of the building’s face, and that the building is not anchored in bedrock. [Hint: F→_E in Fig. 12–86 represents the force that the Earth would exert on the building in the case where the building would just begin to tip.]

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The forces acting on an 85,000-kg aircraft flying at constant velocity are shown in Fig. 12–108. The engine thrust, F_T = 5.0 x 10⁵ N, acts on a line 1.6 m below the cm. Determine the drag force F_D and the distance above the cm that it acts. Assume F→_D and F→_T are horizontal. ( F→_L is the “lift” force on the wing.)

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