Inertia of a uniform rod

In summary, the Moment of Inertia for a uniform rod held at one end is (1/3)ML^2. This is due to the parallel axis theorem, which states that the Moment of Inertia is equal to the sum of the Moment of Inertia about the center of mass and the product of the mass and the square of the distance from the center of mass to the axis of rotation. For a rod pivoted at the center, the Moment of Inertia is (1/12)ML^2, as the center of mass is at the center and there is no distance from the center of mass to the axis. To find the Moment of Inertia, we must integrate the contributions of each mass element, which is
  • #1
cair0
15
0
i've been wondering why the I for a uniform rod held at one end is [tex] \frac{1}{3}ML^2 [/tex]

i know that it has something to do with the parallel axis theorm, but after finiding the first I, being: [tex] M(\frac{L}{2})^2 [/tex]
I am confused as how to find the I of the meter stick rotating about its parallel axis in the center. Since the center of mass is at the center, and it is being pivioted about the center, it seems there would be no I at all.

can anyone help me out?
 
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  • #2
Well one thing you have incorrect is that the Moment of Inertia of a Long Rod pivoted at the center is (1/12) ML^2. The reason why its (1/3) ML^2 is because of the parallel axis theorem. You have:

I = (1/12) ML^2 + MD^2.
D is this case is (L/2). Therefore, if you add (1/12) + (1/4), you should get (1/3).
 
  • #3
cair0 said:
I am confused as how to find the I of the meter stick rotating about its parallel axis in the center. Since the center of mass is at the center, and it is being pivioted about the center, it seems there would be no I at all.
Rotational inertia depends on the distribution of all the mass, not just the location of the center of mass. To find the rotational inertia of an object about an axis, you must integrate the contributions of each mass element. The rotational inertia of a small piece of mass Δm is ΔmR^2, where R is the distance from the axis. Doing the integration for a stick rotating about its center will give you (1/12)ML^2, as harsh noted.
 
  • #4
im not entirely sure how to integrate R^2 theough dm

i mean in a uniform rod, M = Lp, so dm = p dL, but then integrating that, i get pR^3 /3 where am i going wrong here?
 
  • #5
cair0 said:
im not entirely sure how to integrate R^2 theough dm

i mean in a uniform rod, M = Lp, so dm = p dL, but then integrating that, i get pR^3 /3 where am i going wrong here?

dm = p dL:
Wrong!
dm = p dR, where 0<=R<=L


Hence, we have:
[tex] I=\int_{0}^{L}pdR=\frac{p}{3}L^{3}=\frac{M}{3}L^{2}[/tex]

When doing integrations, you should ALWAYS INTRODUCE A DUMMY VARIABLE OF INTEGRATION!
This example is mathematically meaningless, but, unfortunately, it's common in physics texts:
[tex] F(t)=\int_{0}^{t}f(t)dt[/tex]
Never use this "convention"!

When integrating a function f, depending on time (for example) from 0 to an arbitrary time value t, write instead:
[tex] F(t)=\int_{0}^{t}f(\tau)d\tau[/tex]

Here, [tex]\tau[/tex] is called a dummy variable of integration.
 
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1. What is the definition of inertia of a uniform rod?

Inertia of a uniform rod is a measure of the resistance of the rod to changes in its state of motion. It is a property of matter that causes an object to resist changes in velocity.

2. How is inertia of a uniform rod calculated?

Inertia of a uniform rod can be calculated using the formula I = (1/12) * M * L^2, where I is the moment of inertia, M is the mass of the rod, and L is the length of the rod.

3. What factors affect the inertia of a uniform rod?

The inertia of a uniform rod is affected by its mass and length. The larger the mass and length of the rod, the greater its inertia will be.

4. How does inertia of a uniform rod differ from that of a point mass?

The inertia of a uniform rod is different from that of a point mass because the rod has a physical size and shape, whereas a point mass is considered to be infinitely small and have no physical dimensions.

5. What is the role of inertia in the motion of a uniform rod?

Inertia plays a crucial role in the motion of a uniform rod. It is responsible for keeping the rod in a state of motion or rest, and for resisting any changes in its motion caused by external forces.

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