Rectangular Plate with Varying Density Across the Width

[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=ρ_o [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?

 

[Orodruin]

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

 

[taits2204]

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

 

[Orodruin]

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).

 

[HallsofIvy]

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 [tex]wt\int_0^l x(\rho_0)(1- (x/l)^2)dx[/tex].

 

[taits2204]

oops .. didnt quote right, sorry :P

 

[taits2204]

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.

I've just copied the question, so yeah i would assume that the density is also constant over the thickness of the plate. I'll take a crack at that and see what i get :) thanks

 

[Orodruin]

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 [tex]wt\int_0^l x(\rho_0)(1- (x/l)^2)dx[/tex].

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

 

[taits2204]

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

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)

 

[Orodruin]

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.

 

[taits2204]

so would it be right to say

dm = ρ_owt[1+(x/l)^2) dx

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