a cylindrical tank of height 0.4 m is open at the top and has a diameter of 0.16m. water is filled in it up to a height of 0.16 m. the gime taken to empty the tank through a hole of radius 5×10^-3m at the bottom is
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Let r is the radius of hole e.g., r = 5 × 10⁻³ m
R is the radius of tank e.g., R = 0.08 m = 8 × 10⁻² m
According to Torricelli theorem,
speed of water flow through hole if height of water is h from the ground, v = √{2gh}
now, change in volume of water through hole = change in volume of tank
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⇒![\bold{\pi r^2v}=\bold{-\pi R^2 \frac{dh}{dt}}\\\\\frac{r^2\sqrt{2gh}}{R^2}=\frac{-dh}{dt}\\\\\frac{5\times10^{-3}\times\sqrt{2g}dt}{8\times10^{-2}}=\frac{-dh}{\sqrt{h}}\\\\\int{0.0173}\,dt=-\int\limits^0_{0.16}{\frac{dh}{\sqrt{h}}}\\\\0.0173t=-2[h^{1/2}]^0_{0.16}=2(0.16)^{1/2}=0.8\\\\t=\frac{0.8}{0.0173}=46.24sec](https://tex.z-dn.net/?f=%5Cbold%7B%5Cpi+r%5E2v%7D%3D%5Cbold%7B-%5Cpi+R%5E2+%5Cfrac%7Bdh%7D%7Bdt%7D%7D%5C%5C%5C%5C%5Cfrac%7Br%5E2%5Csqrt%7B2gh%7D%7D%7BR%5E2%7D%3D%5Cfrac%7B-dh%7D%7Bdt%7D%5C%5C%5C%5C%5Cfrac%7B5%5Ctimes10%5E%7B-3%7D%5Ctimes%5Csqrt%7B2g%7Ddt%7D%7B8%5Ctimes10%5E%7B-2%7D%7D%3D%5Cfrac%7B-dh%7D%7B%5Csqrt%7Bh%7D%7D%5C%5C%5C%5C%5Cint%7B0.0173%7D%5C%2Cdt%3D-%5Cint%5Climits%5E0_%7B0.16%7D%7B%5Cfrac%7Bdh%7D%7B%5Csqrt%7Bh%7D%7D%7D%5C%5C%5C%5C0.0173t%3D-2%5Bh%5E%7B1%2F2%7D%5D%5E0_%7B0.16%7D%3D2%280.16%29%5E%7B1%2F2%7D%3D0.8%5C%5C%5C%5Ct%3D%5Cfrac%7B0.8%7D%7B0.0173%7D%3D46.24sec "\bold{\pi r^2v}=\bold{-\pi R^2 \frac{dh}{dt}}\\\\\frac{r^2\sqrt{2gh}}{R^2}=\frac{-dh}{dt}\\\\\frac{5\times10^{-3}\times\sqrt{2g}dt}{8\times10^{-2}}=\frac{-dh}{\sqrt{h}}\\\\\int{0.0173}\,dt=-\int\limits^0_{0.16}{\frac{dh}{\sqrt{h}}}\\\\0.0173t=-2[h^{1/2}]^0_{0.16}=2(0.16)^{1/2}=0.8\\\\t=\frac{0.8}{0.0173}=46.24sec")
R is the radius of tank e.g., R = 0.08 m = 8 × 10⁻² m
According to Torricelli theorem,
speed of water flow through hole if height of water is h from the ground, v = √{2gh}
now, change in volume of water through hole = change in volume of tank
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