Applying the parallel axis theorem to find inertia

In summary, the question asks for the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end. The parallel axis theorem is used to calculate the moment of inertia, which states that I = Icm + MD2, where I is the moment of inertia about the given axis, Icm is the moment of inertia about the object's mass centre, M is the mass of the object, and D is the distance between the given axis and the mass centre. Using the known equation for a long thin rod with rotation axis through end (I = 1/3 ML2), the moment of inertia is calculated to be 1/3 ML2, which is different from
  • #1
vetgirl1990
85
3

Homework Statement


Calculate the moment of inertia of a uniform rigid rod of length L and mass M, about an axis perpendicular to the rod through one end.

Homework Equations


Parallel axis theorem: I = Icm + MD2
Long thin rod with rotation axis through centre: Icm = 1/12 ML2
Long thin rod with rotation axis through end: I = 1/3 ML2

The Attempt at a Solution



I know this is a straightforward substitution problem into the parallel axis theorem EQN.
However, for this question, I'm not sure why the answer key uses Icm of the rotation axis through the CENTRE. The question specifically states that the rotation axis is at the end...

The answer given is I = 1/3ML2;
I have calculated I = 7/12ML2, using the Icm of a long thin rod with the rotation axis through the end.
Calculations:
D = 1/2L, since the centre of mass of a rod is right down the middle
I = Icm + MD2
I = 1/3 ML2 + M(1/2L)2
I = 7/12ML2

To reiterate, I'm just confused about why Icm was used for am axis through the centre, rather than through the end.
 
Last edited:
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  • #2
Can you show your calculation?
 
  • #3
azizlwl said:
Can you show your calculation?
Just edited my post with the full calculations.
 
  • #4
Why Icm[/SUB=1/3ML2. Different from you stated in relevant equations.
 
  • #5
azizlwl said:
Why Icm[/SUB=1/3ML2. Different from you stated in relevant equations.
I'm not sure which part of the solution you're referring to.
I = Icm + MD2 is the parallel axis theory, and I plugged in I = 1/3 ML2 the Icm component of that equation.
 
  • #6
vetgirl1990 said:
I'm not sure which part of the solution you're referring to.
I = Icm + MD2 is the parallel axis theory, and I plugged in I = 1/3 ML2 the Icm component of that equation.
You seem to be applying the parallel axis theorem twice over.
The MoI about the rod's centre, axis perpendicular to the rod, is ML2/12. By the parallel axis theorem (or otherwise) the MoI about one end is ML2/12+M(L/2)2=ML2/3. You quoted this as one of your known equations, and it is directly the answer to the question. There is no need to go adding another ML2/4.
Note that the Icm in the parallel axis theorem is the MoI for an axis through the object's mass centre. It is not valid to substitute in there the MoI about any other axis.
 

FAQ: Applying the parallel axis theorem to find inertia

What is the parallel axis theorem?

The parallel axis theorem is a formula used in physics to calculate the moment of inertia of a rigid body about an axis that is parallel to and at a distance from the body's center of mass.

How do you apply the parallel axis theorem?

To apply the parallel axis theorem, you need to know the moment of inertia of the object about its center of mass and the distance between the center of mass and the new axis. Then, you can use the formula I = Icm + md2, where I is the moment of inertia about the new axis, Icm is the moment of inertia about the center of mass, m is the mass of the object, and d is the distance between the center of mass and the new axis.

Why is the parallel axis theorem important?

The parallel axis theorem is important because it allows us to calculate the moment of inertia of an object about any axis, not just the center of mass. This is helpful in many real-world situations where the object may not rotate about its center of mass.

Can the parallel axis theorem be used for any shape?

Yes, the parallel axis theorem can be used for any shape as long as the shape is rigid and the mass is uniformly distributed. It is a general formula that applies to all objects, regardless of their shape.

Are there any limitations to the parallel axis theorem?

The parallel axis theorem assumes that the object is rigid and that the mass is evenly distributed. It also only applies to rotational motion, not translational motion. Additionally, it is only accurate for objects with a constant moment of inertia, meaning that the mass and shape cannot change during rotation.

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