1. Chapter 8 – Further Applications of Integration
8.3 Applications to Physics and Engineering
8.3 Applications to Physics and Engineering
Erickson

2. Applications to Physics and Engineering
Among the many applications of integral calculus to physics and engineering, we will consider two today:
Force due to water pressure
Center of mass
Our strategy is to break up the physical quantity into a large number of small parts, approximate each small part, add the results, take the limit, and then evaluate the resulting integral.
8.3 Applications to Physics and Engineering
Erickson

3. Hydrostatic Force and Pressure
Water pressure increases the father down your go because the weight of the water above increases.
In general, we will submerge a thin horizontal plate with area of A m2 in a fluid of density  kg/m3 at a depth d m below the surface of the fluid. The fluid above the plate has a volume V = Ad so its mass is m = V = Ad.
The force exerted by the fluid
on the plate is:
F = mg = gAd
8.3 Applications to Physics and Engineering
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4. Hydrostatic Force and Pressure
The force exerted by the fluid on the plate is:
F = mg = gAd = Pd
Where A is the area of a thin plate
 is the fluid density in kg/m3
d is the depth in meters below the surface of the fluid.
The Pressure P on the plate is defined to be the force per unit area:
P = F/A = gd
The SI units for measuring pressure is newtons per square meter which is called pascal. (1 N/m2 = 1 Pa)
Water’s weight density is
62.5 lb/ft2 or 1000kg/m3
8.3 Applications to Physics and Engineering
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5. Force Exerted by a Fluid
The force F exerted by a fluid of a constant weight- density w against a submerged vertical plane region from y = c to y = d is
Where w=  g, h(y) is the depth of the fluid and L(y) is the horizontal length of the region at y.
Note: This integral comes from identifying the vertical axis as the y-axis.
8.3 Applications to Physics and Engineering
Erickson

6. Force Exerted by a Fluid
You can create a similar integral by choosing the vertical axis to be x.
The force F exerted by a fluid of a constant weight-density w against a submerged vertical plane region from x = a to x = b is
Where w=  g, h(x) is the depth of the fluid and L(x) is the horizontal length of the region at x.
8.3 Applications to Physics and Engineering
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7. Example 1 – pg. 560 #2
A tank is 8m long, 4m wide, 2m high, and contains kerosene with density 820 kg/m3 to a depth of 1.5m. Find:
The hydrostatic pressure on the bottom of the tank.
The hydrostatic force on the bottom.
The hydrostatic force on one end of the tank.
8.3 Applications to Physics and Engineering
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8. Example 2 – pg. 560
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.
8.3 Applications to Physics and Engineering
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9. Example 3 – pg. 560 #14
A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.
8.3 Applications to Physics and Engineering
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10. Moments and Centers of Mass
We can find the point P on which a thin plate of any given shape balances horizontally. This point is called the center of mass or center of gravity of the plate.
8.3 Applications to Physics and Engineering
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11. Moments and Centers of Mass
The rod below will balance if m1d1=m2d2.
The numbers m1d1 and m2d2 are called the moments of the masses.
8.3 Applications to Physics and Engineering
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12. Moments and Centers of Mass
If we put the rod along the x-axis, we will be able to solve for point P,
The numbers m1d1 and m2d2 are called the moments of the masses
8.3 Applications to Physics and Engineering
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13. Moments and Centers of Mass
Moment of the system about the origin
Moment of the system about the y-axis
Moment of the system about the x-axis
In one dimensions, the coordinates of the center of mass are given by
8.3 Applications to Physics and Engineering
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14. Moments and Centers of Mass
Now we will consider a flat plate (lamina) with uniform density  that occupies a region R of the plane. The center of mass of the plate is called the centroid of R.
They symmetry principle says that if R is symmetric about a line l, then
The centroid of R lies on l.
Moments should be defined so that if the entire mass of a region is concentrated at the center of mass, then its moments remain unchanged.
The moment of the union of two non overlapping regions should be the sum of the moments if the individual regions.
8.3 Applications to Physics and Engineering
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15. Moments and Centers of Mass
So we have the moment of R about the y-axis:
The moment of R about the x-axis:
8.3 Applications to Physics and Engineering
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16. Moments and Centers of Mass
The center of mass of the plate (the centroid of R) is located at the point:
8.3 Applications to Physics and Engineering
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17. Moments and Centers of Mass
If the region R is between two curves y = f (x) and y=g(x), where f (x) ≥ g(x), as shown below, the then we can say that the centroid of R is the point:
8.3 Applications to Physics and Engineering
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18. Example 4 – pg. 561 #22
Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass .
8.3 Applications to Physics and Engineering
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19. Example 5 – pg. 561 #24
The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system.
8.3 Applications to Physics and Engineering
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20. Example 6 – pg. 561 #32
Find the centroid of the region bounded by the given curves.
8.3 Applications to Physics and Engineering
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21. Theorem of Pappus
Let R be a plane region that lies entirely on one side of a line l in the plane. If R is rotated about l, then the volume of the resulting solid is the product of the area A of R and the distance d traveled by the centroid of R.
8.3 Applications to Physics and Engineering
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22. Example 7 – pg. 562
Use the Theorem of Pappus to find the volume of the given solid.
45. A cone with the height h and base radius r
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23. Book Resources Video Examples More Videos Wolfram Demonstrations
Example 1 – pg. 553
Example 3 – pg. 556
Example 7 – pg. 560
More Videos
Problem on hydrostatic force and pressure – A
Problem on hydrostatic force and pressure – B
Moments and center of mass of a variable density planer lamina
Finding the center of mass
Wolfram Demonstrations
Center of Mass of n points
Theorem of Pappus on Surfaces of Revolution
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24. Web Resources http://youtu.be/H5RcfMIZ_yw
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e_locator?course=General+Calculus +II&lesson=62&topic=1&width=80 0&height=684&topicTitle=Center+o f+mass+&+density&skinPath=http %3A%2F%2Fclassic.hippocampus. org%2Fhippocampus.skins%2Fdefa ult
8.3 Applications to Physics and Engineering
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