# How do you find linear density given mass per kg?

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1. Nov 22, 2015

### andkand97

1. The problem statement, all variables and given/known data
I need to find the final impact speed of the flame of a non-uniform candle. I am given that the mass per unit length increases linearly from top to bottom starting with 1kg/m at the bottom and 10kg/m at the top. I need to find how fast the candle flame is moving once the candle hits the table. I also know the length of the candle, 2m

2. Relevant equations
Xcm=1/M∫xdm
m=Lλ
dm=dxλ
I=∫x2dm

3. The attempt at a solution
If dm=dLλ then I have to find lamba ( which I'm unsure of how to do) then integrate from 0 to L to find the total mass. Then from there I can find the center of mass then rotational inertia and use energy to find the final velocity. Can someone tell me if I am thinking correctly and also help me find the total mass? Thanks in advance.

2. Nov 22, 2015

### SteamKing

Staff Emeritus
Well, a sketch of the value of density per unit length versus position would be helpful here.

Since you are given no other information besides the values of density at the top and bottom of the candle, along with the length of the candle, assume that there is a linear variation in the density of the candle per unit length.

3. Nov 22, 2015

### Staff: Mentor

Sorry, but your question makes no sense (at least not to me). Is there a figure or something that goes with this problem?

4. Nov 22, 2015

### andkand97

Oh sorry. I am told that the candle will fall and hit the table.I'm given a drawing of a candle but nothing else. I know that the mass per unit length increases from 1kg/m at the bottom to 10kg/m at the top. I need to find the impact speed of the candle's flame once the candle hits the table.Maybe this helps you understand the question better. Sorry about that.

5. Nov 22, 2015

### andkand97

Could you explain a little more, please?

6. Nov 22, 2015

### SteamKing

Staff Emeritus
Did you draw the picture yet?

7. Nov 22, 2015

### andkand97

Do you mean something like this? (But lighter at the bottom than the top)

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8. Nov 22, 2015

### andkand97

9. Nov 22, 2015

### SteamKing

Staff Emeritus
No, I mean draw a graph which plots the value of the density of the candle along its length.

You know the length of the candle (2 meters; rather large, actually), and you know the density in mass per unit length at the top and bottom of the candle (10 kg/m and 1 kg/m, respectively).

In other words, make a graph of this information.

10. Nov 22, 2015

### CWatters

If you consider thin slices at the top and bottom I think you can work out the ratio of the cross sectional area/radius. I don't know for sure but I suspect things like the height of the centre of mass or moment of inertia will turn out to be proportional to the same ratio.

11. Nov 22, 2015

### vela

Staff Emeritus
SteamKing wants you to draw a graph of $\lambda$ vs. $x$ so you can figure out an expression for $\lambda(x)$.

12. Nov 22, 2015

### andkand97

Oh I understand now. I got a total mass of 22 kg. Is this correct?

13. Nov 22, 2015

### SteamKing

Staff Emeritus
Why don't you show your calculations?