A cylindrical concrete silo is 4 m in diameter and 35 m high. It consists of a 6000 kg concrete base and 38000 kg cylindrical concrete walls. Neglect the thickness of the silo walls and base. (a) How high off the ground is the center of mass of the silo when it is empty? __________ m (b) How high off the ground is the center of mass of the silo when it is two-thirds full of silage whose density is 840 kg/m3? ___________ m
Obviously the silo has a CM halfway up when empty. So it is h= 35 / 2 = 17.5 m
To find the CM of the empty silo you must take into account the mass of the base and the mass of the cylindrical sides. The CM of the base is at the center of the 6000 kg disk shaped base. The CM of the cylindrical walls is 17.5 m up from the CM of the base. Now the CM of the composite figure is just the sum of the CM's of the two. So using the base as our origin and measuring y upward to the CM of the composite figure 6000y = 38000(17.5-y) 6y =38(17.5-y) 44y = 665 y = 15.1 meters b) How high off the ground is the center of mass of the silo when it is two-thirds full of silage whose density is 840 kg/m3? 2/3 full means the height is (2/3)(35m)= 23.3 m V = (pi)(2)^2(23.3) = 293 mass= ( 293 cubic meters)(840 kg/m3) =246120 kg Since this cylinder has a CM at 1/2 its height, its CM is at y = (1/2)(23.3) = 11.65 m So the composite CM of the silo with silage is found by adding the CM of the empty silo with the silage.
