Cylindrical drum
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Rates of change apply to many different situations, not just distance-time or motion. While we have not discussed these specific types of problems, try and apply what you have learned about rates of change to answer the questions.
a. Water pours out of the bottom of a cylinder and out of a cylindrical cone (see diagram) at a constant rate. Sketch a graph that might show the height of the water as a function of time (height-time graph) first for the cylinder and then for the cylindrical cone.
b. There is a hole in a tank of water. A significant amount of water is pouring out of the hole. A special sealant is put into the tank. After a half an hour it starts to gradually clog up the hole until no more water leaks out. Sketch a graph of the amount of water in the tank as a function of time (amount of water-time graph) that would reasonably match the given description.
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Cylindrical drum is depicted.
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a) Cylindrical drum Area of c/s A
dh
h
v
Area of c/s a
We know that at any instant water coming out of the hole at the bottom will have a speed
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v = √2gh .......(1)
Let area of cross section of the hole be a. Then rate at which water flows out of the hole (m3/sec) = av .........(2)
Let the water level in the tank be h at any given instant and let it go down by dh in time dt. Let area of cross section of the tank be A. Then, volume of water flowing out in time dt = Adh
Volume of water flowing out per sec = Adh/dt .......(3)
As both (2) and (3) represent the flow of water out of the tank per sec., we can equate them. Hence,
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av = a√2gh = - A dh/dt [-ve sign signifies reducing volume]
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dh/√h = - (a/A)√2g dt
Integrating both the sides:
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∫dh/√h = - (a/A)√2g ∫dt
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2√h = - [(a/A)√2g] t + C where C is the constant of ...
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