Homework 12 – Work and Energy 1) Compute the work (in Joules) required to stretch a spring from equilibrium to 12 cm past equilibrium, assuming a spring constant of k = 800 N/m. 2) If 5 J of work are required to stretch a spring 10 cm past equilibrium, how much work is required to stretch it 15 cm past equilibrium? 3) Calculate the work against gravity to build a right circular cone of height 4 m and base radius 1.2 m out of a lightweight material of density 600 kg/m3. 4) Built around 2600 BC, the Great Pyramid of Giza in Egypt is 146 m high and has a square base of side 230 m. Find the work (against gravity) required to build the pyramid if the density of the stone is estimated at 2000 kg/m3. 5) a) Using an integral, calculate the work required to pump all of the water out of the tank, which is initially full. Distances are in meters, and the water exits the spigot shown. b) Set up an integral to calculate the work required to pump the water out of the tank if it is initially full to a depth of 6 m. 6) a) Using an integral, calculate the work required to pump all of the water out of the tank, which is initially full. The radius, r, is 5 m. The length of the tank is 7 m. Assume that the water exits from a small hole at the top. Hint: evaluate the integral by interpreting it as the area of a circle. b) Set up an integral to calculate the work required to pump the water out of the tank if it is initially full to a depth of 3 m, and the water exits through a spigot that is 1 m above the tank. 7) How much work is done lifting a 12‐m chain that has mass de nsity 3 kg/m (initially coiled on the ground) so that its top end is 10 m above the ground? 8) a) The gravitational force between two objects of mass m and M, separated by a distance r, has magnitude GMm/r2, where G = 6.67 × 10−11 m3kg−1s−1. Show that if two objects of mass M and m are separated by a distance r1, then the work required to increase the separation to a distance r2 is equal to ܹൌܩܯ݉ቀଵభെଵమቁ. b) Use the result of part
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