A 10-foot-long chain hangs from the ceiling. It weighs 45 lb. Find the total amount of work done (in ft-lb) when you raise the lower end of the chain up to the ceiling so that it is level with the upper end. Round your answer to two decimal places.

112.50 ft-lb

Explanation:

The linear density of the rope λ = 45 lb/10 ft = 4.5 lb/ft

Since the length of the rope is 10 ft and the bottom is raised to the ceiling, its length now becomes 10/2 = 5 ft. So, Let y = 0 define the position of the bottom of the chain and y = 10 define the position of the top of the chain. When the bottom is raised to the ceiling, its length 2y. Its position is thus 10 - 2y. So we define a function f(y) such that

Workdone = ∫₀¹⁰λf(y)dy = ∫₀⁵λf(y)dy + ∫₅¹⁰λf(y)dy

                 = ∫₀⁵λf(y)dy + ∫₅¹⁰λf(y)dy

                 =  λ∫₀⁵f(y)dy + λ∫₅¹⁰f(y)dy

                 = 4.5∫₀⁵(10 -2y)dy + 4.5∫₅¹⁰0dy

                 = 4.5[10y - y²]₀⁵

                 = 4.5[10(5) - 5² - (10(0) - 0²)]

                 = 4.5(50 - 25 - 0)

                 = 4.5(25)

                 = 112.50 ft-lb

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