Q.47

Expert-verifiedFound in: Page 386

Book edition
4th

Author(s)
Randall D. Knight

Pages
1240 pages

ISBN
9780133942651

An aquarium of length , width (front to back) , and depth is filled to the top with liquid of density

a. Find an expression for the force of the liquid on the bottom of the aquarium.

b. Find an expression for the force of the liquid on the front window of the aquarium.

c. Evaluate the forces for a-long, deep aquarium filled with water.

Force acting on the aquarium in plane is

Expression for hydrostatic pressure at depth is given by

Here is the surface pressure where is the density of the liquid.

Let force acted upon by the liquid on bottom of the aquarium, Force acted upon by the liquid on the bottom of the aquarium equals to the gravity of the liquid because the bottom of the aquarium is the only support in vertical direction:

The volume equals to length times width times depth,

Density of the liquid is given by,

Re arrange equation for

Substitute in the equation

Substitute in the equation

Therefore force acted upon by the liquid on bottom of the aquarium is

The hydrostatic pressure at depth is given by

So in the front window, at spot depth , there is the pressure indicated above pushing the windows outward. There is also a pressure due the atmosphere that pushes the window inward, which cancels the first term in the above equation. So the net pressure is the one only related to the liquid:

Here is the density of the liquid; is the distance from the point of interest to the surface of the liquid. And is the gravitational acceleration. The total force on the window is to integrate the pressure in the plane,

At depth in our coordinates is actually so we have,

The function does not depend on .So we can integrate the above equation

We can continue to do the integration over

The result of inetgration of

Conversion of units of length of the aquarium from cm to

Conversion of units of breadth of aquarium from to

Conversion of units of depth of the aquarium from

Substitute

for in equation

Therefore the force acting on aquarium is

Using part (b) expression for force acting on aquarium in plane

Substitute equation

plane is

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