A long thin rod lies along the x-axis from the origin to x=L, with L= 0.830 m. The mass per unit length, $\lambda$ (in kg/m) varies according to the equation $ \lambda = \lambda_0 (1+1.110x^3). The value of $\lambda_0$ is 0.200 kg/m and x is in meters. Calculate the total mass of the rod.

Question

A long thin rod lies along the x-axis from the origin to x=L, with L= 0.830 m. The mass per unit length, $\lambda$ (in kg/m) varies according to the equation  $ \lambda = \lambda_0 (1+1.110x^3). The value of $\lambda_0$ is 0.200 kg/m and x is in meters. Calculate the total mass of the rod.

- So I tried:

lambda = m/x
m = (lambda)(x)
  = (0.200kg/m)(0.830m)
  = 0.166kg

...which was apparently wrong so then I retried this time using the given equation:

$ \lambda = \lambda_0 (1+1.110x^3).
          = (0.200kg/m)(1+(1.110)(0.830m)^3)
          = 0.271kg

But that is also incorrect and I don't know what else to do.

Scott · Answer

this is integral calculus

the mass of a segment of the rod is
... segment length * lambda

lambda = .200 (1 + 1.110 x^3)
... = .200 + .222 x^3

segment length is dx

integrate from x = 0 to .83

.200 dx + .222 x^3 dx ... sum the two integrals

.166 + .026

Jason · Answer

I have no idea because I'm not in your grade. SORRY!

Saugat · Answer

dM=dx(0.200*(1+1.110x^3)) taking integral on both sides, M=0.200(L+1.110*x^4/4) put L=0.830 M=0.19234