- #1

songoku

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- Homework Statement
- A tank has the shape of an inverted circular cone with height 10 m and base radius 4 m. It is filled with water to a height of 8 m. Find the work required to empty the tank by pumping all of the water to the top of the tank. (The density of water is 1000 kg/m^3)

- Relevant Equations
- W = ##\int F dx##

I take the origin to be at the apex of the cone. Using the similarity of the triangle, where ##r## is radius of water and ##y## is height of water from the apex of cone:

$$\frac{r}{y}=\frac{4}{10}$$

$$r=\frac{2}{5}y$$

The mass of water = ##\rho .V## = ##\rho . \pi r^2~\Delta y## = ##\rho . \pi \frac{4}{25}y^2~\Delta y##

The weight of water = ##\rho . \pi \frac{4}{25}y^2~\Delta y. g##

The distance needed to move the water to the top of the tank = 10 - y

The work needed:

$$W=\int_{2}^{10} \rho . \pi \frac{4}{25}y^2. g (10-y) ~dy$$

But I got wrong answer and the teacher said my origin was wrong. I still don't understand why I can't take origin to be at the apex.

Thanks

$$\frac{r}{y}=\frac{4}{10}$$

$$r=\frac{2}{5}y$$

The mass of water = ##\rho .V## = ##\rho . \pi r^2~\Delta y## = ##\rho . \pi \frac{4}{25}y^2~\Delta y##

The weight of water = ##\rho . \pi \frac{4}{25}y^2~\Delta y. g##

The distance needed to move the water to the top of the tank = 10 - y

The work needed:

$$W=\int_{2}^{10} \rho . \pi \frac{4}{25}y^2. g (10-y) ~dy$$

But I got wrong answer and the teacher said my origin was wrong. I still don't understand why I can't take origin to be at the apex.

Thanks