Calculating the Center of Mass of Solid Objects (using Calculus) Lyzinski, CRHS-South

Center of Mass For two or more masses Compact notation when dealing with a lot of masses at once. For dealing with solid objects that are made up of infinitely many tiny masses, each of which is called “dm”.

Example of a solid object (solid rod) Tiny mass, “dm” Notice that each “dm” is at a different location in the solid rod. 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. This is equivalent to

Finding “dm” (part 1) For 1D objects, use the 1D density function, l For 2D objects, use the 2D density function, s For 3D objects, use the 3D density function, r

Finding “dm” (part 2) For 1D objects, use the 1D density function, l For 2D objects, use the 2D density function, s For 3D objects, use the 3D density function, r

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. Draw them on the object in question. x 1D thin rod x 2D triangle x 3D beam

Finding “dm” (part 4) x 1D thin rod dx dx 2D triangle y x dx y z x Need to use geometry to get a relationship between x & y dx y z x

Some Examples

Find the center of mass of a uniform, thin rod of length L Y L x #1

Recap Step #1: draw your “tiny mass slice” Step #2: write your 1D, 2D, or 3D density function. Step #3: find “dm” Step #4: plug “dm” into Step #5: use the correct “bounds” of integration. Step #6: use the density function to plug back in for “M” to finish the problem.

#2 Find the center of mass of a thin rod of length L whose Mass varies according to the function . Y L x #2

Find the center of mass of a thin rod of length L whose Mass varies according to the function . Y L x

Recap (for non-uniform objects) Step #1: draw your “tiny mass slice” Step #2: write your 1D, 2D, or 3D density function. Step #3: find “dm” Step #4: plug “dm” into Step #5: use the correct “bounds” of integration. Step #6: use the density function to plug back in for “M” to finish the problem. If the object is non-uniform (its density is not the same everywhere) use to solve for M.

a b c y dm x #3

Recap (for 2D objects) Step #1: draw your “tiny mass slice” Step #2: write your 1D, 2D, or 3D density function. 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 #5: plug “dm” into Step #6: use the correct “bounds” of integration. Step #7: use the density function to plug back in for “M” to finish the problem.

Find the distance from the center of a semi-circle that the CM lies. #4 r

Find the distance from the center of a semi-circle that the CM lies. The integral in this example is difficult, and would either require “u-substitution” or use of an integral table to solve it.

dm #5

dm #5

dm #5

dm #5

A difficult 3D Center of Mass Problem #6

#6