Moment of inertia for a non-uniform rod with varying density

[dinospamoni]

Homework Statement

The thin rod shown has a length L = 100 cm, and a density that varies from 26 g/cm at the origin to 3 g/cm at the far end. Determine a) the moment of inertia about axis 1 (passing through the center of mass of the rod), and b) the moment of inertia about axis 2 (passing through the heavy end of the rod).


picture attached

Homework Equations

I already found
Mass of rod = 1.45*10^3 g
Center of mass of rod = 36.8 cm from the more dense end

The density as a function of position I found to be: ρ(x)=.23x + 3

The Attempt at a Solution

For the first part:

I used I = ∫ r^2 * m/L dr
where -36.8 < r < 63.2

and got 1.46*10^6 g*cm^2

For the second part I know it has to do with the parallel axis theorem:

I = I_cm+mh^2

where h = 36.78 cm

and got 3.4*10^6 but that's not right.

Any ideas?

 

[voko]

For the first part:

I used I = ∫ r^2 * m/L dr
where -36.8 < r < 63.2

and got 1.46*10^6 g*cm^2

Since the density is not uniform, you can't just assume it is m/L. Use the density function you found above. Pay attention to the relation between r in the integral and x in the density function.

 

[SammyS]

Homework Statement

The thin rod shown has a length L = 100 cm, and a density that varies from 26 g/cm at the origin to 3 g/cm at the far end. Determine a) the moment of inertia about axis 1 (passing through the center of mass of the rod), and b) the moment of inertia about axis 2 (passing through the heavy end of the rod).

Homework Equations

I already found
Mass of rod = 1.45*10^3 g
Center of mass of rod = 36.8 cm from the more dense end

The density as a function of position I found to be: ρ(x)=.23x + 3

The Attempt at a Solution

For the first part:

I used I = ∫ r^2 * m/L dr
where -36.8 < r < 63.2

and got 1.46*10^6 g*cm^2

For the second part I know it has to do with the parallel axis theorem:

I = I_cm+mh^2

where h = 36.78 cm

and got 3.4*10^6 but that's not right.

Any ideas?

Your density function gives a linear density of 3 g/cm at the origin rather than at x = 100 cm ,

and a linear density of 26 g/cm at x = 100 cm rather than at the origin.