Why Is My Calculation of the Wheel's Moment of Inertia Incorrect?

[dannee]

Homework Statement

i've attched a picture of the wheel, which need to calculate moment of intertia with respect to 8 rods.

i've tried to use stiener rule but it's not getting the answer,

moment of wheel is (m1*r^2)/2
moment of rod is (m2*r^2/3)

i use the rule i total = icm + mr^2

so it's 8*( (m1*r^2)/2 + (m2*r^2)/3 )

what am i missing?

Homework Equations

m1 = mass of wheel
m2 = mass of rod
r = length of rod

The Attempt at a Solution

 

[Doc Al]

moment of wheel is (m1*r^2)/2

Revisit this formula.

so it's 8*( (m1*r^2)/2 + (m2*r^2)/3 )

There are 8 spokes but only 1 wheel.

 

[dannee]

that's right, 8 * 3(m2*r^2), i refer the moment of wheel is (m1*r^2)/2 and sum them,

but the correct answer is (3m2/8 + m1) * r^2

 

[Doc Al]

that's right, 8 * 3(m2*r^2), i refer the moment of wheel is (m1*r^2)/2 and sum them,

(1) Your formula for the moment of inertia of the wheel portion is incorrect.
(2) In your final result in post #1 you multiplied all terms by 8, thus counting the wheel 8 times.

but the correct answer is (3m2/8 + m1) * r^2

Double check that answer; looks like a typo.

 

[rwisz]

Thank you for sharing your work and question. It seems like you are on the right track with using the parallel axis theorem (also known as the Steiner rule) to calculate the moment of inertia of the wheel with respect to the 8 rods. However, it looks like you may have made a mistake in your calculations. The moment of inertia of the wheel should be (m1*r^2)/2, but the moment of inertia of the rod should be (m2*r^2)/12 (not 3). This is because the rod is rotating around its end, not its center of mass, which changes the moment of inertia calculation.

So, the correct equation for the total moment of inertia would be:

I_total = I_cm + m1*r^2/2 + 8*m2*r^2/12

= I_cm + (m1 + 2*m2)*r^2/2

I hope this helps and good luck with your calculations!