# linear density

• Oct 2nd 2008, 09:00 PM
thecount
linear density
The mass of a part of a metal rod that lies between its left and a point x meters to the right is 3x^2 kg. Find the linear density when x is:

(a) 1 m

(b) 2 m

(c) 3 m

Also, where is the density the highest? The lowest?

If anyone could help me out with this one that would be great! maybe show me how to do a,b or c and then I can try the rest? thanks!
• Oct 2nd 2008, 11:08 PM
CaptainBlack
Quote:

Originally Posted by thecount
The mass of a part of a metal rod that lies between its left and a point x meters to the right is 3x^2 kg. Find the linear density when x is:

(a) 1 m

(b) 2 m

(c) 3 m

Also, where is the density the highest? The lowest?

If anyone could help me out with this one that would be great! maybe show me how to do a,b or c and then I can try the rest? thanks!

Let the mass to the right of a point $\displaystyle x$ be $\displaystyle F(x)$

The linear density at $\displaystyle x$ is the limit of the mass in a small element of the rod near $\displaystyle x$ divided by the length of the element, so:

$\displaystyle f(x)=\lim_{\delta \to 0} \frac{F(x+\delta/2)-F(x-\delta/2)}{\delta}=F'(x)$

In this case $\displaystyle F(x)=3x^2$

RonL