Center of Mass Lab

Workshop Remark: We didn’t bring the can of sand, but the rest of the lab materials are available.

Goal: Demonstrate your knowledge of centers of mass and verify the calculus via simple geometric techniques

Lab Description:

You will choose a planar lamina from the stock. You will be given coordinate paper as well.

First, suspend a plumb line and the lamina to mark a center line. Then suspend the lamina from a different spot and mark a second center line. The intersection is of course the center of mass, determined geometrically. Mark this on your lamina with a dot and “G”.

Using the coordinate paper, place your lamina on it in an appropriate way so that you can determine the equation of its boundaries. You will do this by finding points on the edge of the lamina. The more points you have, the more accurate your work will be. Your lamina will consist of a 3 straight edges and a curved one. Simply generate one curve - do not use a piecewise function.

Determine the equations(s) of all boundaries. Use regression in your calculator or FIT on DERIVE to find these equations (you pick the type of function). Now use calculus to locate the centroid of the region, showing all work. You must display the integrals that are being evaluated. You may use your calculator or DERIVE to evaluate the integrals. Use the coordinate paper to locate this computed centroid on your lamina, mark it with a dot and “C” for calculus.

You will then balance the lamina on a sharp—pointed pencil placed into a can of sand (the Centroid Tester) below. If it balances at either mark, you will submit all drawings and equations with your lamina. If it doesn’t balance, you will redo your work, and make new marks, etc.

Grading - the assignment is worth 15 points. If your work is acceptable and understandable, you will receive full credit if the lamina balances at the calculus point. If the lamina balances at the geometric point instead, you lose two points. You lose 4 points if the lamina does not balance. The setups and computations should be clear as usual, but with less writing (more of a worksheet than a report, but assessed for completeness as usual). You may try to balance the lamina at any time without making your mark. But once you mark it, that is your answer.

Nb: Thanks to Stu Schwartz from the AP Calc Listserve and the GASOU/NSF demos (now on for suggesting the two components of this lab.