![]() Step #5: plug “dm” into Step #6: use the correct “bounds” of integration. Step #3: find a relationship between the variables that change with position (in this case x & y) Step #4: find “dm” in terms of the variable needed in your integral for step #5. Step #2: write your 1D, 2D, or 3D density function. ![]() If the object is non-uniform (its density is not the same everywhere) use to solve for M.ġ5 Recap (for 2D objects) Step #1: draw your “tiny mass slice” Step #6: use the density function to plug back in for “M” to finish the problem. Step #3: find “dm” Step #4: plug “dm” into Step #5: use the correct “bounds” of integration. Step #1: draw your “tiny mass slice” Step #2: write your 1D, 2D, or 3D density function. ![]() Mass varies according to the function Y L x Mass varies according to the function Y L x #2ġ2 Find the center of mass of a thin rod of length L whose Step #6: use the density function to plug back in for “M” to finish the problem.ġ1 #2 Find the center of mass of a thin rod of length L whose Need to use geometry to get a relationship between x & y dx y z xĩ Find the center of mass of a uniform, thin rod of length Lġ0 Recap Step #1: draw your “tiny mass slice” x 1D thin rod x 2D triangle x 3D beamħ Finding “dm” (part 4) x 1D thin rod dx dx 2D triangle y x dx y z x This is equivalent toĤ Finding “dm” (part 1) For 1D objects, use the 1D density function, lįor 2D objects, use the 2D density function, s For 3D objects, use the 3D density function, rĥ Finding “dm” (part 2) For 1D objects, use the 1D density function, lĦ Finding “dm” (part 3) Whether the object is 1-Dimensional, 2D, or 3D, you need to figure out what “tiny slices” you are dealing with. ![]() To find the center of mass, we need to find the product of “xdm” for each different tiny piece, add them all together, and then divide by the total mass of the object. Tiny mass, “dm” Notice that each “dm” is at a different location in the solid rod. For dealing with solid objects that are made up of infinitely many tiny masses, each of which is called “dm”. Presentation on theme: "Calculating the Center of Mass of Solid Objects (using Calculus)"- Presentation transcript:ġ Calculating the Center of Mass of Solid Objects (using Calculus)Ĭompact notation when dealing with a lot of masses at once. ![]()
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