Question

A rod extending between x=0 and x= 14.0cm has a uniform cross-
sectional area A= 9.00cm^2. It is made from a continuously changing
alloy of metals so that along it's length it's density changes
steadily from 2.70g/cm^3 to 19.3g/cm^3.

<b>a)</b> Identify the constants B and C required in the expression p= B +
Cx to describe the variable density.

<b>b)</b> The mass of the rod is given by
m= integral(pdV)= integral(pAdx)= integral 14cm/0 (14cm above integral sign and 0 below integral)(B+Cx)*(9.00cm^2)dx

(below the firs integral it says <b>all material</b> and below the
second integral it says <b>all x</b>

~this part a has gotten me confused as to what to do at all and the second part has gotten me even more confused!!


HELP!!!

Thank YOU!

Answer 1

I forgot to include what it says under the integration:

<b>Carry out the integration to find the mass of the rod</b>

Now about that..what in the equation for the integral is the mass???
it's not like they have a <b>m</b> anywhere in there..unless it's p but I started thinking it was to represent density..

Answer 2

Yes, p represents density in this case. The mass of the rod is
A*(Integral of) p (x) dx
where A is the cross sectional area of the rod.
For p(x), use the function you derive in part (a).
p(x) = A + bx
p(0) = A = 2.70 g/cm^3 (That already tells you what A is)
p(14) = A + B*14 = 19.3 g/cm^3
Solve for B.

Answer 3

first of all I don't know how to use the #s given to put into the equation or do something else with them.

I know...

x=0
x=14.0cm
A=9.00cm^2

density changes from: 2.70g/cm^3 to 19.3g/cm^3

<b>How do I put it into p= B+Cx??</b>

would x= 14-0??
C and B I'm not sure what they represent except that you said the mass is equal to B right?

Answer 4

and since p=m/V does that play a role in this as well?

Again I have no idea what C represents.

Answer 5

I forgot t use the expression B + Cx for p, and used A + Bx instead. The method of solution is the same. In your case, B = 2.70, and C = 1.186
p(x) = 2.70 + 1.186 x
Note that p = 2.70 g/cm^3 when x = 0 and 9.30 g/cm^3 when p = 14 cm, as required.

Now integrate A p(x) dx for the total mass

Answer 6

for this I got

m= A integral 14\0 (B+Cx)dx
(took out the A contstant A=9.00cm^2)

m= 9.00 integral 14\0 (2.70+1.186x)dx =

9.00(2.70x + 0.593x^2)|14 0 =
(the 14 is above and 0 below the | sign)

9.00[ 2.70(14) + 0.593(14)^2 ]=

1,386.252g


<b>~I'm not sure if I did the integration right and also I'm confused as to why you said
"note that p= 2.70g/cm^3 when x=0 and 9.30g/cm^3 when p= 14cm, as required" does this apply to what I plug into the integration ?? I would think it wouldn't since I have to put in 14 and 0 in anyways and the 0 cancels out and all is left is the 14 right?

Is this alright what I got??</b>