259 Worksheet 6.5. Work and Energy 1. Find the work in joules to compress a spring 4 cm past equilibrium, assuming that the spring constant is k = 150 kg/s2 . 3. Calculate the work in joules required to pump all the water out of the conical tank pictured. Assume the lengths shown are measured in meters. Water exits here 5 10 260 Solutions to Worksheet 6.5 1. Find the work in joules to compress a spring 4 cm past equilibrium, assuming that the spring constant is k = 150 kg/s2 . The work required to compress the spring 4 cm past equilibrium is Z −.04 −.04 150x dx = 75x2 0 = 0.12 J 0 3. Calculate the work in joules required to pump all the water out of the conical tank pictured. Assume the lengths shown are measured in meters. Water exits here 5 10 Place the origin at the vertex of the inverted cone, and let the positive y-axis point upward. Consider a layer of water at a height of y meters. From similar triangles, the area of the layer is y 2 π m2 2 so the volume is y 2 π ∆y m3 2 Thus the weight of one layer is y 2 ∆y N 9800π 2 261 The layer must be lifted 10 − y meters, so the total work needed to empty the tank is Z 10 y 2 (10 − y) dy ≈ 6.414 × 107 J 9800π 2 0
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