moment of inertia of cylendrical body
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The moment of inertia of a cylinder of mass M and radius R is
12MR2
The moment of inertia about a given axis is defined by the integral:
I=∫r2dm
where r is the distance from the axis of rotation and m the mass distibution. If we assume that the mass is distributed uniformly then we can define the costant mass density ρ as follows:
ρ=dmdV
where dV is the volume element.
We can therefore rewrite the integral as:
I=∫r2ρdV=ρ∫r2dV
It's convenient to write dV in cylindrical coordinates (r,θ,z):
dV=rdθdrdz
The integral thus becomes:
I=ρ∫h0dz∫2π0dθ∫R0r3
where h is the height of the cylinder.
The result of the integral is:
I=2πρhR44=12πρhR4=12MR2
where we used M=ρV=ρ⋅(πR2h)
12MR2
The moment of inertia about a given axis is defined by the integral:
I=∫r2dm
where r is the distance from the axis of rotation and m the mass distibution. If we assume that the mass is distributed uniformly then we can define the costant mass density ρ as follows:
ρ=dmdV
where dV is the volume element.
We can therefore rewrite the integral as:
I=∫r2ρdV=ρ∫r2dV
It's convenient to write dV in cylindrical coordinates (r,θ,z):
dV=rdθdrdz
The integral thus becomes:
I=ρ∫h0dz∫2π0dθ∫R0r3
where h is the height of the cylinder.
The result of the integral is:
I=2πρhR44=12πρhR4=12MR2
where we used M=ρV=ρ⋅(πR2h)
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