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Chapter 7 Extra Topics Crater Lake, Oregon Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington Fg Centers of Mass: d Torque is a function of force and distance. Lake Superior, Washburn, WI Photo by Vickie Kelly, 2004 (Torque is the tendency of a system to rotate about a point.) If the forces are all gravitational, then torque m1 g mgx m2 g If the net torque is zero, then the system will balance. Since gravity is the same throughout the system, we could factor g out of the equation. M O mk xk This is called the moment about the origin. If we divide Mo by the total mass, we can find the center of mass (balance point.) M O mk xk MO x M x m k k m k x MO M x m k For a thin rod or strip: k d = density per unit length m (d is the Greek letter delta.) k moment about origin: M O x d x dx b a M d x dx b mass: center of mass: a MO x M For a rod of uniform density and thickness, the center of mass is in the middle. For a two dimensional shape, we need two distances to locate the center of mass. y strip of mass dm x distance from the y axis to the center of the strip x y distance from the x axis to y x the center of the strip y(pronounced dm Moment about x-axis: M xx tilde Center mass: ecksoftilda) Moment about y-axis: M y x dm Mass: M dm x My M Mx y M For a two dimensional shape, we need two distances to locate the center of mass. y For a plate of uniform thickness and density, the density drops out of the equation when finding the center of mass. x y x Vocabulary: center of mass = center of gravity = centroid constant density d = homogeneous = uniform 9 8 yx 2 Mx 3 Mx 3 0 7 6 0 1 2 2 x x dx 2 1 4 x dx 2 3 M y x x 2 dx 0 3 M y x3dx 0 5 4 3 2 x 1 .5x 0 1 2 2 3 xx coordinate of 1 2 centroid y =x (2.25, 2.7) 2 1 53 Mx x 10 0 1 43 My x 4 0 243 Mx 10 81 My 4 1 33 M x dx x 9 0 3 0 243 81 Mx 27 My 10 9 4 y x M 9 10 M 9 4 3 2 Note: The centroid does not have to be on the object. If the center of mass is obvious, use a shortcut: square rectangle circle h 3 b 3 right triangle Theorems of Pappus: When a two dimensional shape is rotated about an axis: Volume = area . distance traveled by the centroid. Surface Area = perimeter . distance traveled by the centroid of the arc. Consider an 8 cm diameter donut with a 3 cm diameter cross section: 1.5 V 2 r area V 2 2.5 1.5 V 11.25 2 2.5 2 V 111cm 3 We can find the centroid of a semi-circular surface by using the Theorems of Pappus and working back to get the centroid. 1 2 A r 2 y 4 3 V= r 3 1 2 4 3 2 y r r 2 3 4r y 3