22. Trough in Figure 12: water exits by pouring over the sides. FIGURE 12 23. Find the work W required to empty the tank in Figure 8 through die hole at the top if the tank is half full of water. 24. Assume die tank in Figure 8 is full of water and let W be die work required to pump out half of die water through die hole at the top. Do you expect W to equal the work computed in Exercise 23'? Explain and then compute W. 25. Assume the tank in Figure 10 is full. Find the work required to pump out half of the water. Hint: First, determine the level H at which the water remaining in the tank is equal to one~lialf the total capacity of the tank. 26. Assume that the tank in Figtlre 10 is full. (2) Calculate the work Fty} required to pump out water until the water level has reached level _\\-. tb} CF: '5 Plot Ftvt. (c) E What is the signicance of F'ty} as a rate of change? (d) 3:153 5 lfyour goal is to pump out all of the water. at which water level ."tJ will half of the work be done? 27. Calculate the work required to lift a 10m chain over die side of a building (Figure 13} Assume that the chain has a density of 8 kgim. Hiii t: Break up die chai n into N segments. estimate die work performed on a segment. and compute the limit as N i~ oe as an integral. T i Segment of length Av FIGURE 13 The small segment of the chain of length Ay located 3' meters from the top is lifted through a vertical distance _\\I. 28. How much work is done lifting a 3m chain over the side of a building if the chain has mass density 4 kgim? 29. AISm chain has mass 18 kg. Find the work required to lift the chain over the side of a building. 30. A 10~m chain with mass density 4 kgim is initially coiled on the ground. How much work is performed in lifting the chain so that it is fully extended (and one end touches the grou nd}? 3]. How much work is done lifting a l2m chain that has mass density 3 kgim (initially coiled on the ground} so that its top end is 10 m above the ground? 32. A500kg wrecking ball hangs from a 12121 cable ofdensity 15 kgim attached to a crane. Calculate the work done if the crane lifts the ball from ground level to ]2 min the air by drawing in the cable. 33. Calculate the work required to lift a 3ni chain over the side of a building if the chain has variable density of p {.r} = .12 3x + 10 kgim for 0 5 x 5 3. 34. A 3431 chain with linear mass density ,0th = 2xt4 i x] kgim lies on the ground. Calculate die work required to lift the chain so dial its bottom is 2 m above ground. Exercises 3537: Tire gravitational force between two objects (farms in and M, separated by a distance r. has magnitude G Mm ,1 r2. where G = 6.67 x 101'm3kgls1. 35. Show that if two objects of mass M and m are separated by a dis tance :1. then the work required to increase the separation to a distance I"; is equal to W = GMiii{rl_l r34}. 36. Use the result of Exercise 35 to calculate the work required to place a 2000kg satellite in an orbit 1200 km above the surface of the earth. Assume that the earth is a sphere of radius Re 2 6.37 x 105 m and mass Me = 5.93 x 1024 kg. Treat the satellite as a point mass. 37. Use the result of Exercise 3 5 to compute the work required to move a lSDDkg satellite from an orbit 1000 to an orbit 1500 [cm above the surface of the earth. 38. The pressure P and volume V of the gas in a cylinder of length 0.8 meters and radius 0.2 meters. with a movable piston. are related by PV "4 = k. wherek is acoustant (Figure 14}. When the piston is fully extended. die gas pressure is 2000 kilopascals (one kilopascal is 103 newtons per square meter}. (a) Calculate k. {b} The force on the piston is PA. where A is the piston's area. Calcu late the force as a function of the length .\\' of the column of gas. (c) Calculate the work required to compress the gas column from 1.5 m to 1.2 m. 1' FIGURE 14 Gas in a cylinder with a piston