- #1
Craptola
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Been doing some physics problems from my mechanics class. This is the first time I've attempted a problem like this and am not so confident with solving it, I could be correct and just being paranoid but I just have a feeling I've made a mistake somewhere, would appreciate if someone who is comfortable with this kind of thing would tell me if/where I'm going wrong.
Consider a thin beam of length L=3 m, and height and width equal to w= 2 cm and h=2 cm respectively. Assume that the composure of the beam is from a mixture of materials, that have given it a non-uniform mass density, which varies continuously along x [where x represents the coordinate for the direction along the length of the beam, measured from the left edge of the beam]. Assume that such mass density as a function of x is given by:
[tex]\rho = \rho _{0}e^{\alpha x}[/tex]
where ρo= 9.10^3 kg/m3 and α=1/L
Find the total mass for the beam.
[tex]M=\int_{0}^{3}dm[/tex]
[tex]dm=\rho dv[/tex]
[tex]dv=0.0004dx[/tex]
[tex]\therefore dm=0.0004\rho dx = 3.6e^{-\frac{1}{3}x}dx[/tex]
[tex]M=\int_{0}^{3}dm = 3.6\int_{0}^{3}e^{-\frac{1}{3}x}dx[/tex]
[tex]= -10.8\left ( e^{-1} - 1 \right ) = 6.83kg[/tex]
Homework Statement
Consider a thin beam of length L=3 m, and height and width equal to w= 2 cm and h=2 cm respectively. Assume that the composure of the beam is from a mixture of materials, that have given it a non-uniform mass density, which varies continuously along x [where x represents the coordinate for the direction along the length of the beam, measured from the left edge of the beam]. Assume that such mass density as a function of x is given by:
[tex]\rho = \rho _{0}e^{\alpha x}[/tex]
where ρo= 9.10^3 kg/m3 and α=1/L
Find the total mass for the beam.
Homework Equations
[tex]M=\int_{0}^{3}dm[/tex]
[tex]dm=\rho dv[/tex]
The Attempt at a Solution
[tex]dv=0.0004dx[/tex]
[tex]\therefore dm=0.0004\rho dx = 3.6e^{-\frac{1}{3}x}dx[/tex]
[tex]M=\int_{0}^{3}dm = 3.6\int_{0}^{3}e^{-\frac{1}{3}x}dx[/tex]
[tex]= -10.8\left ( e^{-1} - 1 \right ) = 6.83kg[/tex]