Pyramid Builders. Ancient pyramid builders are balancing a uniform rectangular slab of stone tipped at an angle ? above the horizontal using a rope (?Fig. P11.80?). The rope is held by five workers who share the force equally. (a) If ? = 20.0o, what force does each worker exert on the rope? (b) As ? increases, does each worker have to exert more or less force than in part (a), assuming they do not change the angle of the rope? Why? (c) At what angle do the workers need to exert ?no force to balance the slab? What happens if ? exceeds this value?
Solution 86P Introduction First we have to calculate the tension of the rope. From the tension we can calculate the force exerted by each person. Then we have to discuss whether tension will increase or decrease with increase of angle . Then we have to find out at which angle at which the slab will balance and what happens to the tension if the angle increases further. Step 1 The free body diagram of the stone is shown below. The rock is rotating about the corner of the stone which is in contact with the ground (A, shown as big dot in the figure). To find the tension we have to find the moment of force about the axis of rotation for both the tension and the weight. Then, as we know that for equilibrium the total moment will be zero, we have to make total moment as zero and then find out the tension. Step 2 Let us now first calculate the angle . From the triangle ABC we can have the following figure, In the figure BC is the line drawn from the center of mass to the center of the bottom edge. Now as the stone is rectangular, the center of mass will be at the center of the rectangle. Hence the line BC is the half of the length of the longer side of the stone. Hence the angel is = tan1 1.875 m= 65.0° ( 0.875 m Now from the triangle ABD, we have + + = 90° = 90° 20° = 90° 20° 65° = 5° Hence force perpendicular to the line joining the point A and B is w = mgsin = mgsin5° = 0.087mg Now if T is the tension then the force perpendicular to the line joining the aixis and the point where the force is applied (AE) is T = T sin52° = 0.79T Now the distance AB is AB = 1875 + 0.875 = 2.07 m Hence the total moment about A is AB × w AE × T = (2.07 m)(0.087mg) (3.75 m)(0.79T) = 0 (2.07 m)(0.087mg) T = (3.75 m)(0.79)0.035mg Hence the force exerted by each person is F = T = 0.035mg= 0.007mg 5 5 That is each person have to apply 0.7% of the total weight of the stone. Step 3 If we increase the angle , the angle will decrease. So the perpendicular component of weight mgsin will also decrease. Hence the moment of force about the point A will also decrease. Hence the required tension to hold it at that position will also decrease and hence the required force will also decrease.