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Chapter 8 – Further Applications of Integration

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Chapter 8 – Further Applications of Integration

8.3 Applications to Physics and Engineering

8.3 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

- 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

- The force exerted by the fluid on the plate is:
F = mg = gAd

- 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

- The force F exerted by a fluid of a constant weight-density wagainst a submerged vertical plane region from y = cto y = dis
- Where w=rg, h(y) is the depth of the fluid and L(y) is the horizontal length of the region at y.

8.3 Applications to Physics and Engineering

- 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

- 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

- 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

- 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

- 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

- If we put the rod along the x-axis, we will be able to solve for point P,
- The numbers m1d1and m2d2 are called the moments of the masses

8.3 Applications to Physics and Engineering

- 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

- Now we will consider a flat plate (lamina) with uniform density that occupies a region = of the plane. The center of mass of the plate is called the centroid of =.
- They symmetry principle says that if = is symmetric about a line l, then
- The centroid of = 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

- So we have the moment of = about the y-axis:
- The moment of = about the x-axis:

8.3 Applications to Physics and Engineering

- The center of mass of the plate (the centroid of =) is located at the point:

8.3 Applications to Physics and Engineering

- If the region = 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 = is the point:

8.3 Applications to Physics and Engineering

- 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

- The masses mi are located at the points Pi. Find the moments Mxand My and the center of mass of the system.

8.3 Applications to Physics and Engineering

- Find the centroid of the region bounded by the given curves.

8.3 Applications to Physics and Engineering

- Let = be a plane region that lies entirely on one side of a line l in the plane. If = is rotated about l, then the volume of the resulting solid is the product of the area A of = and the distance d traveled by the centroid of =.

8.3 Applications to Physics and Engineering

- Use the Theorem of Pappus to find the volume of the given solid.
45. A cone with the height h and base radius r

8.3 Applications to Physics and Engineering

- Video Examples
- 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

8.3 Applications to Physics and Engineering

- http://youtu.be/H5RcfMIZ_yw
- http://youtu.be/12MhraQo0TY
- http://youtu.be/cXNmCaTod58
- http://youtu.be/h8kMaW2q9EM
- http://youtu.be/F2poHPZZBhE
- http://youtu.be/NYUyHj3c1Xg
- http://youtu.be/fJtxJv5sdqo
- http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=57&topic=2&width=800&height=684&topicTitle=Work+done+on+a+fluid&skinPath=http%3A%2F%2Fclassic.hippocampus.org%2Fhippocampus.skins%2Fdefault
- http://classic.hippocampus.org/course_locator?course=General+Calculus+II&lesson=62&topic=1&width=800&height=684&topicTitle=Center+of+mass+&+density&skinPath=http%3A%2F%2Fclassic.hippocampus.org%2Fhippocampus.skins%2Fdefault

8.3 Applications to Physics and Engineering