W8.2 - February 26, 2008 Physics for...

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Unformatted text preview: February 26, 2008 Physics for Scientists&amp;Engineers 1 1 Physics for Scientists &amp; Physics for Scientists &amp; Engineers 1 Engineers 1 Lecture 28 February 26, 2008 Physics for Scientists&amp;Engineers 1 2 Example - Center of Mass Example - Center of Mass Assume that we have a uniform, square metal plate with side L = 3.40 cm and mass 0.205 kg. The plate is located with its lower left corner at ( x , y ) = (0,0) as shown in the figure. We remove a square from of the plate with side L /4 with lower left edge located at ( x , y ) =(0,0). What is the distance of the center of mass of the remaining plate from the origin? m = mass of plate m missing = m /16 = mass of missing area x plate = L / 2 x missing = L / 8 X = mx plate ! m mising x missing x plate ! x missing = m L / 2 ( ) ! m /16 ( ) L / 8 ( ) m ! m /16 X = L / 2 ( ) ! L /128 ( ) 15 /16 February 26, 2008 Physics for Scientists&amp;Engineers 1 3 Example - Center of Mass (2) Example - Center of Mass (2) X = L / 2 ( ) ! L /128 ( ) 15 /16 X = 3.40 cm / 2 ( ) ! 3.40 cm /128 ( ) 15 /16 X = 1.785 cm Y = X = 1.785 cm R = X 2 + Y 2 = 2 X 2 R = 2 1.785 cm ( ) 2 = 2.52 cm February 26, 2008 Physics for Scientists&amp;Engineers 1 4 Calculating the Center of Gravity Calculating the Center of Gravity How do we calculate the center of gravity for an arbitrarily shaped object? We represent the object by small identical cubes Each cube has its own center of gravity (in its geometric center) and we can calculate the center of gravity of the system of cubes, just like we did for a collection of particles February 26, 2008 Physics for Scientists&amp;Engineers 1 5 Density Distributions and Center of Gravity Density Distributions and Center of Gravity Clearly not all the cubes in the hammer have the same mass Density is defined as If the density is uniform then we can write If we assume that the density of each cube is different but each cube is uniform then we can write Reducing the size of the cubes gives us ( ) i i dm r dV ! = ! (for constant ) M V ! ! = 1 ( ) V R r r dV M ! = &quot; ! ! ! 1 1 1 1 ( ) n n i i i i i i R rdm r r dV M M ! = = = = &quot; &quot; ! ! ! ! i dm February 26, 2008 Physics for Scientists&amp;Engineers 1 6 Position of the Center of Gravity Position of the Center of Gravity We can then write the position of the Cartesian components of center of gravity of an object as If the density is constant for the whole object we can write Or 1 1 1 ( ) ; ( ) ; ( ) V V V X x r dV Y y r dV Z z r dV M M M ! ! ! = = = &quot; &quot; &quot; ! ! ! 1 (for constant ) V V R rdV rdV M V ! ! = = &quot; &quot; ! ! ! X = 1 V xdV ; V !...
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This note was uploaded on 04/06/2008 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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W8.2 - February 26, 2008 Physics for...

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