Create Presentation
Download Presentation

Download Presentation
## Chapter 8 – Further Applications of Integration

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Chapter 8 – Further Applications of Integration**8.3 Applications to Physics and Engineering 8.3 Applications to Physics and Engineering**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**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**Hydrostatic Force and Pressure**• 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**Force Exerted by a Fluid**• 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**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**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**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**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**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**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 m1d1and m2d2 are called the moments of the masses 8.3 Applications to Physics and Engineering**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**Moments and Centers of Mass**• 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**Moments and Centers of Mass**• So we have the moment of = about the y-axis: • The moment of = about the x-axis: 8.3 Applications to Physics and Engineering**Moments and Centers of Mass**• The center of mass of the plate (the centroid of =) is located at the point: 8.3 Applications to Physics and Engineering**Moments and Centers of Mass**• 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**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**Example 5 – pg. 561 #24**• 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**Example 6 – pg. 561 #32**• Find the centroid of the region bounded by the given curves. 8.3 Applications to Physics and Engineering**Theorem of Pappus**• 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**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 8.3 Applications to Physics and Engineering**Book Resources**• 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**Web Resources**• 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