Center of Mass Lyzinski, CRHS-South

What is the “Center of Mass” of a system of masses What is the “Center of Mass” of a system of masses? It is the location that all the mass would be located at if the system were shrunken down to an infinitely small point. It is the location at which you could “balance” the object. It is the location that the masses would “rotate around” if the object were rotated.

Center of Mass For two or more masses Compact notation when dealing with a lot of masses at once.

Center of mass “analogy” It’s like a weighted average. You do a weighted average every time you calculate your grades. For example … Grades: 5/5 on a HW 10/10 on small quiz 50/50 on big quiz 40/100 on test 5 5 10 100 50 0 20 40 60 80 100%

Center of mass “analogy” (continued) So what if the grades were instead….masses, all on massless bar. Where would you place a pivot to balance the bar? 5 kg 5 kg 10 kg 100 kg 50 kg 0 20 40 60 80 100%

When masses in a system are rearranged, the center of mass of the system changes if external forces are applied to the system. HOWEVER, if no external forces are applied to a system, and if the system rearranges itself only due to internal forces, the center of mass of the system remains in the same location. WHY?

If no external forces act on a system, then momentum is conserved… Multiply all terms by Dt since v = x/Dt and thus x = vDt Divide both sides by m1 + m2 … and thus, if no external forces act on a system, the center of mass does not move.

The man on a log problem If the man starts at one end of the log, and walks to the other end, how far will the log move? Assume the water that the log is floating on is calm. 80 kg 200 kg 4 meters

The man on a log problem (continued) Assuming that no external forces are present, momentum is conserved and thus the center of mass does not move. x + 2 80 200 x 2 4

The man on a log problem (continued) Assuming that no external forces are present, momentum is conserved and thus the center of mass does not move.

Does it matter where you measure “x” from Does it matter where you measure “x” from? In other words, can you pick any “arbitrary” location to measure from? NO 4 80 200 2 2 - x 4-x

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