Center of mass of two connected different density blocks

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Homework Help Overview

The problem involves determining the center of mass of a composite slab made of two materials with different densities: aluminum and iron. The dimensions of the slab and the densities are provided, and the task is to find the coordinates of the center of mass in three dimensions.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Some participants suggest calculating the mass and center of each material separately before finding the overall center of mass. Others question the vertical coordinate calculation, specifically whether it should be 6.7 cm based on symmetry.

Discussion Status

The discussion is ongoing, with participants exploring different methods to approach the problem. There is a focus on the implications of density ratios and the potential for misunderstanding the coordinate calculations. Some guidance has been offered regarding the calculation of the center of mass, but no consensus has been reached on the specific values.

Contextual Notes

Participants are grappling with the interpretation of the center of mass coordinates, particularly in relation to the negative side of the axis and the effects of density on the overall calculation.

J-dizzal
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Homework Statement


The figure shows a composite slab with dimensions d1 = 11.6 cm, d2 = 2.85 cm, and d3 = 13.4 cm. Half the slab consists of aluminum (density = 2.70 g/cm3) and half consists of iron (density = 7.85 g/cm3). What are (a) the x coordinate, (b) the y coordinate, and(c) the z coordinate of the slab's center of mass?

http://edugen.wileyplus.com/edugen/courses/crs7165/art/qb/qu/c09/fig09_40.gif

Homework Equations


20150707_161138_zpsvokykdw2.jpg


The Attempt at a Solution


I thought i applied the formulas correctly
20150707_161132_zpsa7yycxn5.jpg
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Its easier without calculus. Compute the mass and center of the two separate pieces, then compute the net CM.
 
Dr. Courtney said:
Its easier without calculus. Compute the mass and center of the two separate pieces, then compute the net CM.
I don't see why the com in the vertical (x-axis) is not 13.4/2 = 6.7cm.
 
I would approach this by getting the COM for each of the metal blocks. You then look at the different mass density ratio's of the two blocks. In the case above iron to aluminium is pretty much a 3:1 ratio. You then take the block as a whole and get the COM if both sides were the same density. Then use the COM for both blocks and adjust it with respect to the COM of the separate Fe and Al blocks in the 3:1 ratio.

For example the x direction, the COM of Fe is halfway through the block at 5.8cm, and the same for Al. Since Fe is 3 times denser than Al move the COM for both blocks closer to the COM of the Fe in a 3:1 ratio. So the overall COM in the x direction is 2.9cm from the COM of the Fe block.

Repeat that train of thought for y and z and then get the point that is closest to all three.

Sorry if that answer seems convoluted.
 
J-dizzal said:
I don't see why the com in the vertical (x-axis) is not 13.4/2 = 6.7cm.
It should be. The answer says it's not?
 
Nathanael said:
It should be.
yea, I've done it by symmetry and using the xcom formula and each time 6.7cm
 
Nathanael said:
It should be. The answer says it's not?
20150707_172625_zpsy7gbuml4.jpg
 
Perhaps it's because it lies on the negative side of the axis. Try -6.7 cm
 
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Nathanael said:
Perhaps it's because it lies on the negative side of the axis. Try -6.7 cm
yep that's why thanks sir.
 

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