# Centre of Mass of a Uniform Cuboid -- Show that it is at the Center

patrykh18
Homework Statement:
Starting from a definition of Centre of Mass, show explicitly that for a uniform cuboid the centre of mass is at the centre.
Relevant Equations:
Provided below.
So, I volunteered to run a seminar to first year students in my college. They got a question like this for homework recently and a lot of them made a mistake in the calculation. I am not asking for help with the question itself because I know how to do it. However, a lot of students made a mistake that I shown in the image below. I'm curious about what would be the best way to explain to them why that is a wrong approach (without going into too much details about mathematics).

Woahhhhhh, why you putting volumes in the limits?

patrykh18
Woahhhhhh, why you putting volumes in the limits?
Well I am going from x1 to x2 but it's all multiplied by yz

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Well I am going from x1 to x2 but it's all multiplied by yz
It should be multiplied by the constant cross sectional area, surely?

etotheipi
patrykh18
It should be multiplied by the constant cross sectional area, surely?

Yeah yz is the constant cross area

I was trying to figure out how to answer this but there's too much wrong at the moment to make a start. You have ##x##'s in the limits for your integration with respect to ##m##, you use ##V## both as the volume of the cuboid and as an integration variable, you set ##V=xyz## at one point, when this is clearly incorrect [##x##, ##y## and ##z## are coordinates..., and I don't even know which ##V## you're trying to refer to], you have some weird limits. Too much to untangle for me, sorry.

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Yeah yz is the constant cross area
The standard approach should integrate with respect to ##x##, and not ##V##. In any case, you cannot have ##V## as the fixed volume of the cube and an integration variable. That's a problem that encourages the invalid cancellation.

patrykh18
I was trying to figure out how to answer this but there's too much wrong at the moment to make a start. You have ##x##'s in the limits for your integration with respect to ##m##, you use ##V## both as the volume of the cuboid and as an integration variable, you set ##V=xyz## at one point, when this is clearly incorrect [##x##, ##y## and ##z## are coordinates..., and I don't even know which ##V## you're trying to refer to], you have some weird limits. Too much to untangle for me, sorry.

Yeah. I solved a lot of integrals. I know I need to distinguish between ##V## the volume and ##V## the integration parameter. I just never asked myself why you fundamentally have to do it.

patrykh18
The standard approach should integrate with respect to ##x##, and not ##V##. In any case, you cannot have ##V## as the fixed volume of the cube and an integration variable. That's a problem that encourages the invalid cancellation.

Yeah, if I did this question I would naturally distinguish between those two but I never asked myself why that is fundamentally the case.