Answer to Question #108763 in Calculus for Edward

Question

Find the force on one side of a cubical container 6.0 cm on an edge if the container is filled with
mercury. The weight density of mercury is 250 kN/m³.

Answer 1

The total force on the plate is given by

$$F=w\displaystyle\int_{a}^bxydy$$

where

$x$  is the length (in m) of the element of area (expressed in terms of y)

$y$  is the depth (in m) of the element of area

$w$  is the density of the liquid (in N m^-3)

(for water, this is w  = 9800 N m^-3)

$a$  is the depth at the top of the area in question (in m)

$b$  is the depth at the bottom of the area in question (in m)

We have:

$x=0.06\ m$

$y$  is the variable of integration

$w=250000\ N/m^3$

$a=0$

$b=0.06$

$$F=250000\displaystyle\int_{0}^{0.06}0.06ydy=15000\bigg[{y^2\over2}\bigg]\begin{matrix} 0.06 \\ 0 \end{matrix}=27(N)$$

$F=27N$