# Rectangular Plate with Varying Density Across the Width

1. Jul 31, 2014

### taits2204

So the question is

A rectangular plate has length l, width w and thickness t. Its density is constant across the width, but varies with distance from one end as ρ=ρo [1 + (x/l)^2] Find the plate’s mass and the coordinates of it’s centre of mass.

I Have had a bash at this question, thinking that you would take the equation given in the question and then just double integrate to find for a small area and then work from there ? , but i don't even know if i'm in the right ball park?

2. Jul 31, 2014

### Orodruin

Staff Emeritus
Why don't you try doing just that and show us what you get?

3. Jul 31, 2014

### taits2204

Ive attached a copy of my working to the post, i have a feeling that its my maths thats causing the problems...

#### Attached Files:

• ###### IMG_3222.jpg
File size:
30.1 KB
Views:
68
4. Jul 31, 2014

### Orodruin

Staff Emeritus
What is the primitive function to 1 when integrated wrt x?

You also need to integrate over y, even if the density is constant in y. Otherwise your units will not make sense in the end as you should end up with a result in units of mass. Note that density has units mass/length^3. The integral over z should be trivial (see the first comment in this message).

5. Jul 31, 2014

### HallsofIvy

Staff Emeritus
You have only a double integral when you should have either a triple integral or, since the density is constant of the width, that double integral multiplied by the width, w. You say the density is constant over the width but say nothing about how it varies with depth. Are we to assume that it is constant over the thickness? If so then the density is1- the integral with respect to x multiplied by w and t.

By symmetry, the x and z coordinates (length and depth) of the center of mass are (1/2)l and (1/2)t. The y coordinate is $$wt\int_0^l x(\rho_0)(1- (x/l)^2)dx$$.

6. Jul 31, 2014

### taits2204

oops .. didnt quote right, sorry :P

7. Jul 31, 2014

### taits2204

8. Jul 31, 2014

### Orodruin

Staff Emeritus
You mixed x and y coordinates. You also need to divide by the total mass, the given expression does not have dimension of length.

9. Jul 31, 2014

### taits2204

so where do i go from here then ?
i solved the integral HallsofIvy gave me, giving me 1/4 ρ0l2ωt woudlnt that just mean that the coordinates for the COM are (1/4 ρ0l2ωt , ω/2 , t/2)

10. Jul 31, 2014

### Orodruin

Staff Emeritus
No, as I said he is expression for the x-coordinate must be divided by the total mass, which you still have to solve for correctly.

11. Aug 4, 2014

### taits2204

so would it be right to say

dm = ρowt[1+(x/l)^2) dx

wo xdm = ∫woρowt[1+(x/l)^2) x dx
and then solve from there ?