Math 1132 Worksheet 6.4 Name: Discussion Section: 6.4 Work

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1 Mth 1132 Worksheet 6.4 Nme: Discussion Section: 6.4 Work Force formul for springs. By Hooke s Lw, the force required to mintin spring stretched x units beyond its nturl length is f(x) = kx where k is positive constnt clled the spring constnt. Work. The work done in moving n object from to b is W = F d if the force begin pplied is constnt. The work done in moving n object from to b is W = f(x)dx if the force begin pplied is vrible. Work formul for tnks. The integrl for work required to empty tnk of liquid from height to b is ρ g D(y) A(y) dy where D(y) is the depth of y slice, A(y) is the re of the cross section t y, ρ is the density of the liquid, g is the ccelertion of the liquid (e.g., grvity), nd nd b represent the wter levels in the tnk.

2 1. Exmple: An inverted conicl tnk with rdius 6 ft nd height 20 ft is full of wter. Find the work required to pump the wter out of the top of the tnk. Let y = 0 t the bottom of the tnk. Use the fct tht wter weighs 62.5 lb/ft 2. Thinking bout the problem: How should I pproch this problem? Hve I seen problem similr to it before? If so, how did I pproch it? First, I know wht n inverted conicl tnk would look like, so I drw it. Then I cn lbel the rdius nd the height s stted in the problem Now I know wht my tnk looks like. Using the dimensions given in the problem, I will need to find, b, ρ, g, D(y), nd A(y) s the integrl for work required to empty tnk

3 is ρ g D(y) A(y) dy. Doing the problem: The problem sks me for the work required to pump wter out of tnk. So I strt with the formul ρ g D(y) A(y) dy. Since wter weighs 62.5 lb/ft 2, I know tht ρ g = 62.5 (the grvity is incorported in the unit lb). I cn situte the cone such tht y = 0 is t the bottom of the tnk, so I will lbel my cone s follows: 6 20 y r y 20 I know tht I will empty the entire tnk, so I know tht I will integrte from the bottom of the tnk (y = 0) to the top (y = 20). So I deduce tht = 0 nd b = 20. Let y be the height of the slice so my D(y) = 20 y. To find A(y), I will need to find the re of the slice. I know tht re of circle is πr 2, so I need to find r s function of y. However, I notice tht this cn be considered s problem with similr tringles, so I drw the following figure.

4 6 20 r y By lw of similr tringles, I know tht Therefore the re of the slice r y = 6 20 r = y A(y) = πr ( ) 2 3 = π y 10 = πy Now I hve found ll the prts of my integrl so I find tht the work to pump the wter

5 out of the top of the tnk is W = = 20 0 = 62.5 π = π ρ g D(y) A(y) dy 62.5 (20 y) πy dy (20 y) y 2 dy 20y 2 y 3 dy ( ) 20y 3 = π y = π = 75000π ft-lbs.

6 Solutions should show ll of your work, not just single finl nswer. 2. A cble tht weighs 4 lb/ft is used to lift 300 lb lump of col up from the bottom of mineshft tht is 1000 ft deep. Determine the work needed to bring the col to the top of the mineshft using the cble. (Hint: Compute the work done in lifting the cble nd the col seprtely.) () Drw nd lbel picture of the sitution. (b) Compute the work done in lifting the col to the top of the mineshft by itself. (c) Wht is the weight of x slice of the rope?

7 3. A circulr swimming pool with dimeter of 8 m. nd height of 1.5 m. contins wter to depth of 1 m. Compute the mount of work required to pump ll the wter out of the pool over the side, giving your finl nswer in joules to the nerest integer. Consider the density of wter to be 1000 kg/m 3 nd the ccelertion due to grvity to be g = 9.8 m/sec 2. Let y = 0 t the top of the tnk () Wht is A(y)? (b) Wht is D(y)? (c) Wht is ρ g? (d) Wht should the bounds nd b be? (e) Set up the integrl to find work needed to empty the swimming pool.

8 4. Suppose the following tnk is full of oil with density of 900 kg/m 3. Set up but do not evlute the integrl to find the mount work required to pump ll the oil out of the tnk. 3 m 5 m 1 m 10 m 5. T/F (with justifiction): The work required to stretch spring hving spring constnt k distnce x from its equilibrium (rest) position is kx.

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