# Moment of inertia of thin rod with non-uniform mass (1 Viewer)

### Users Who Are Viewing This Thread (Users: 0, Guests: 1)

#### shotgunshogun

1. The problem statement, all variables and given/known data
A thin rod is exactly 1.9 meters long. The density in this rod varies in a peculiar manner. If we call the left-hand end of the rod x=0 and the right-hand end of the rod x=L , then the linear density can be expressed in units of kilograms per meter as http://coswebhost.rit.edu/webwork2_files/tmp/equations/6b/d06d55bce8c4f91789b960b87d21021.png [Broken]

You grab the RIGHT-hand end of the rod and prepares to swing the rod this end. What is the moment of inertia of the rod around this end?

3. The attempt at a solution

I tried to intergrate from -L to 0 since your changing the axis of rotation and i got 1.16 kgm^2. You integrate the linear density times x^2 dx if you revolve from the left end , the bounds would be from 0 to L (1.85 kgm^2). The right side though... it doesn't work just the same, what needs to be changed to fufill the question, change of bounds, change of the density equation?

Last edited by a moderator:

#### LowlyPion

Homework Helper
1. The problem statement, all variables and given/known data
A thin rod is exactly 1.9 meters long. The density in this rod varies in a peculiar manner. If we call the left-hand end of the rod x=0 and the right-hand end of the rod x=L , then the linear density can be expressed in units of kilograms per meter as http://coswebhost.rit.edu/webwork2_files/tmp/equations/6b/d06d55bce8c4f91789b960b87d21021.png [Broken]

You grab the RIGHT-hand end of the rod and prepares to swing the rod this end. What is the moment of inertia of the rod around this end?

3. The attempt at a solution

I tried to intergrate from -L to 0 since your changing the axis of rotation and i got 1.16 kgm^2. You integrate the linear density times x^2 dx if you revolve from the left end , the bounds would be from 0 to L (1.85 kgm^2). The right side though... it doesn't work just the same, what needs to be changed to fufill the question, change of bounds, change of the density equation?
What did you try to integrate? Can you show your equation?

Last edited by a moderator:

#### shotgunshogun

I tried to integrate $$\lambda(x)x^2$$ from -L to 0

#### LowlyPion

Homework Helper
I tried to integrate $$\lambda(x)x^2$$ from -L to 0
And not 0 to L ?

Edit: Oops I see they are going to hold it by the other end away from 0

#### LowlyPion

Homework Helper
OK. You need to revise your treatment of the distance of the moment arm then don't you?

Instead of multiplying by x2 and then integrating 0 to L if you were holding it at the origin, your moment arms now are (L - x)2 aren't they?

Then you can integrate from 0 to L as before?

#### shotgunshogun

i worked it out and i don't believe it works because the inertia is greater than the first situation. The value should be lower because the mass increases as it approaches 1.9. So when you rotate it from the right side, it should be alot easier to rotate the object and thus, a lesser value moment of inertia

#### LowlyPion

Homework Helper
Choosing to go along the negative x axis still puts your hands on the wrong end of the bat. Besides I'm not certain that λ is valid for -x.

Consider my suggestion a little more carefully.

For any mass element along the X axis you have a mass element of λ (x) a distance of L - x away. Is that not true?

#### Dr.D

The easy way to work this problem is to figure the MMOI with respect to the left end and then use the parallel axis theorem to move it to the right end.

### The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving