# Coordinates of centre of mass of lamina

1. May 15, 2013

### diorific

An industrial tool is made from alaminar material occupyingthe region between thestraight lines y =2, y=1 2x and y =3−x. The density of thematerial varies and is given by thefunction σ(1 + x), where x is the horizontal distancefrom the y-axis and σ is aconstant. The mass of the tool andthe position of its centre of mass are sought.

(a) Sketch the region, and ﬁnd the coordinates of the vertices.

(b) Calculate the mass of the tool in terms of σ.

(c)Find the coordinates of the centre of mass of the tool.

a)

The vertices are the points (1,2), (2,1) and (4,2)

b)
The mass M is the sum of these double integrals

When you do all the calculations M=4δ

c)
Now I'm confused on how I find the coordinates of the centre of mass.
Since the coordinates are

But I have two double integrals, then do I need to sum both of the X double integrals to find the X-coordinate?

2. May 15, 2013

### SteamKing

Staff Emeritus
Yes, and there will be two integrals involved in finding the y-coordinate of the C.O.M.as well.

3. May 16, 2013

### diorific

ok, so this is what I did

But the result is 303/16≈18.94 and this clearly is not the X-coordinate of the centre of mass.

Can anyone help??

4. May 16, 2013

### SteamKing

Staff Emeritus
You are not showing the details of your calculations. If you post these, you might get some suggestions.

5. May 16, 2013

### diorific

Ok, I think I did a mistake when I did the calculations. I've done it again and the X-coordinate is 89/16≈5.56

6. May 16, 2013

### SteamKing

Staff Emeritus
C.O.M. coordinates generally lie within the boundaries of the body. Your calculation shows the C.O.M well outside.

7. May 16, 2013

### haruspex

You don't show your working for M. I get 5, not 4.
In the calculation starting at (2), the coefficient of x3 is wrong in the second line.

8. May 17, 2013

### diorific

Yes, you are right.
I've done the calculations again and the M=5
Also I've done the calculations again for the coordinates and everything looks correct since the coordinates lie within the shape.

9. May 18, 2013

### fatima123

hi
i also stuck on this problem can you show me how do you get 5 sigma coz i try it many time but i got 4
thanks

10. May 18, 2013

### haruspex

11. May 19, 2013

### fatima123

Hi I'm stuck on the integral if sigma(1+x) it self as I every time I got different value can you please show me how to integrate this
Thank you

12. May 19, 2013

### arildno

That is why you should post your working, so that we can point out to you where your mistake is. We are not mind readers.

13. May 19, 2013

### fatima123

14. May 19, 2013

### haruspex

The range y= 3-x to y=2 gives ∫(1+x) (2 - (3-x))dx. If you don't understand POST ALL YOUR WORKING!