Calculating Center of Mass with Added Particles

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http://examsolutions.co.uk/a-level-maths-papers/Edexcel/Mechanics/Mechanics-M2/2010-June/paper.php

Question 3)

part a) I got the answer to be 2.5 which is correct

part b) I don't really get how to do this, I understand as a new particle is being added which is the same mass of the total frame that the center of mass will change, but how do I figure out how much it will change by?

 

[Doc Al]

part b) I don't really get how to do this, I understand as a new particle is being added which is the same mass of the total frame that the center of mass will change, but how do I figure out how much it will change by?

Just like you found the center of mass of the frame by finding the combined center of mass of its pieces, treat the added mass and the frame as two pieces. If you have two equal masses, where will the combined center of mass be?

 

[HallsofIvy]

The center of mass of two objects of different masses can be calculated as the weighted average of the two centers of mass, "weighted" by their masses.

 

[phospho]

Just like you found the center of mass of the frame by finding the combined center of mass of its pieces, treat the added mass and the frame as two pieces. If you have two equal masses, where will the combined center of mass be?

hm well the total mass is 32 as I figured out from part a), and if I do the same thing as I did in part a):

center of BC is (6,0), center of BA is (3,4), center of AC(9,4)

mass ratio for BC will be M, BA, 10, AC, 10 as M is the total mass which I found to be 32 in part a) then taking moments about B to find y bar:

32*0 + 10*4 + 10 * 4 = 32y
80/32 = y = 2.5 which is the same as part a :s

 

[Doc Al]

You already found the center of mass of the frame. (No need to start from scratch.) Treat that as mass 1. (What are its coordinates?) Treat the added mass as mass 2. What are its coordinates? Now find the center of mass of those two together.

 

[HallsofIvy]

To find the centre of mass of two bodies (or groups of bodies) of different mass, find the "weighted" average of the centres of mass, weighted by the mass of each part. That is, if the tow bodies have masses m1 and m2 and the x coordinates of centres of mass are x1 and x2 respectively, then the x coordinate of the centre of mass of both is [m1/(m1+m2)]x1+ [m2/(m1+m2)]x2= (m1x1+ m2x2)/(m1+m2).

 

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got it, thank you both