Applications to Physics and Engineering

This section gives additional applications of definite integrals, including hydrostatic pressure and the total force on an object immersed in a liquid to a given depth and centers of mass (blance points) of two dimensional objects.

Concepts to Master

1. Calculate the force on a vertical plane die to hydrostatic pressure; the pressure at a given depth
2. Moments on inertia about the x-axis and y-axis; centroid (center of mass)
3. Theorem of Pappus

Summary and Focus Questions

A. The pressure on a plate suspended horizontally in a liquid with density  at a depth d is
                                                                      
where g is the gravitational constant (9.8 meters/seconds²).  P is in units of newtons per meter² (= pascals) or in pounds per ft².

An important principle of fluid pressure in the experimentally verified fact that at a point in a liquid the pressure in the same in all directions.

1.  A circular plate of radius 3 is suspended horizontally at a depth of 12 m in an oil having density 1500 kg/m³.  What is the pressure on the plate?

Suppose a plate is suspended vertically in a liquid with mass density  the the total depth of the liquid is H.  Suppose the durface of the plate can be described as bounded by x = f(y), x = k(y), y = c, y = d with f(y) > k(y) for all y [c, d].  Then the force due to liquid pressure on a section of the plate is approximately

                                      (density)(gravitational constant)(depth)(area of section) =

                                          (   )(           g                     )(H – y_i*)(f(y_i*) – k(y_i*)) y_i

 

2. An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water.  Find the (a) hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.

B. Centroid

Usually we are looking at a region that lies between two curves y = f(x) and y = g(x).  First find the area of the region, A. 

Then the center of mass is

             

 C. Theorem of Pappus

And a surprising connection between centroids and volumes of revolutions.  Basically, if you have a region that lies on one side of a line (like a triangle in quadrant 1 and does not overlap into other quadrants) and you rotate the region around that line, the

Volume = (the area of the region)(distance traveled by the centroid)

3.  A torus is formed by rotating a circle of radius 2 at center (2, 5) about the x-axis.  Find the volume of the torus.