Non-uniform inertia

  • Thread starter LASmith
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  • #1
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Homework Statement



A cylinder with radius R and mass M has density that increases linearly with radial distance r from the cylinder axis, ie. [itex]\rho[/itex]=[itex]\rho[/itex][itex]_{0}[/itex](r/R), where [itex]\rho[/itex][itex]_{0}[/itex] is a positive constant. Show that the moment of inertia of this cylinder about a longitudinal axis through the centre is given by I=(3MR[itex]^{3}[/itex])/5



Homework Equations



I=[itex]\int[/itex]r[itex]^{2}[/itex].dm
volume = 2[itex]\pi[/itex]rL.dr



The Attempt at a Solution



I=[itex]\int[/itex]r[itex]^{2}[/itex][itex]\rho[/itex].dv
=[itex]\int[/itex](r[itex]^{3}[/itex][itex]\rho[/itex][itex]_{0}[/itex]/R.)dv
=[itex]\int[/itex](r[itex]^{3}[/itex][itex]\rho[/itex][itex]_{0}[/itex]/R.)(2[itex]\pi[/itex]rL).dr

integrate between 0 and R to obtain
2[itex]\rho_{0}[/itex][itex]\pi[/itex]R[itex]^{4}[/itex]L/5

However, I do not understand how to express this without using the term [itex]\rho_{0}[/itex]
 

Answers and Replies

  • #2
Doc Al
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However, I do not understand how to express this without using the term [itex]\rho_{0}[/itex]
Find an expression for M in terms of ρ0.
 
  • #3
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Find an expression for M in terms of ρ0.

I realise this, however as the density is not constant, I am unsure of how to do this.
 
  • #4
Doc Al
Mentor
45,093
1,398
I realise this, however as the density is not constant, I am unsure of how to do this.
Set up an integral to solve for the total mass, just like you set one up for the rotational inertia.

Once you get M in terms of ρ0, you can rewrite your answer in terms of M instead of ρ0.
 

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