Moment of Inertia Using Parallel Axis Theorem

Q^2/2.In summary, the task is to use the parallel axis theorem to find the moment of inertia of a cube with edge length Q if the axis is along one of its faces. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2. To find Pface, we use the equation Pface = PCM + md^2, where d is the distance from the axis to the center of mass and m is the mass of the cube. The correct value for Pface is (5/12)mQ^2.
  • #1
woaini
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Homework Statement



VlBTZHQ.png


Q is the edge length of the cube. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2 (PCM in the diagram). Use the parallel axis theorem to find the moment of inertia if the axis is along one face of the cube (Pface).

PCM=(1/6)mQ^2
Pface=?
d=?

Homework Equations



Pface = PCM+md^2

The Attempt at a Solution



d=(1/2)Q^2

Pface=[itex]\frac{mQ^2}{6}[/itex]+[itex]\frac{mQ^2}{4}[/itex]

Pface=[itex]\frac{5mQ^2}{12}[/itex]
 
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  • #2
woaini said:

Homework Statement



VlBTZHQ.png


Q is the edge length of the cube. The moment of inertia of the cube about an axis passing through its center and the center of two opposing faces is (1/6)mQ^2 (PCM in the diagram). Use the parallel axis theorem to find the moment of inertia if the axis is along one face of the cube (Pface).

PCM=(1/6)mQ^2
Pface=?
d=?

Homework Equations



Pface = PCM+md^2

The Attempt at a Solution



d=(1/2)Q^2

Pface=[itex]\frac{mQ^2}{6}[/itex]+[itex]\frac{mQ^2}{4}[/itex]

Pface=[itex]\frac{5mQ^2}{12}[/itex]

Correct...except d = Q/2
 
Last edited:

What is the parallel axis theorem?

The parallel axis theorem is a mathematical formula used to calculate the moment of inertia of a rigid body, by relating it to the moment of inertia of the same body about a parallel axis through its center of mass.

How is the moment of inertia calculated using the parallel axis theorem?

The moment of inertia using the parallel axis theorem is calculated by adding the moment of inertia of the body about its center of mass to the product of the mass of the body and the square of the distance between the parallel axis and the center of mass.

Why is the parallel axis theorem important in physics?

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 axis through its center of mass. This is useful in many physics and engineering applications, such as calculating the rotational energy and stability of objects.

How does the parallel axis theorem relate to the concept of rotational inertia?

The parallel axis theorem is a mathematical representation of the physical concept of rotational inertia, which is the resistance of an object to changes in its rotational motion. It helps us understand how the distribution of mass in an object affects its rotational behavior.

Can the parallel axis theorem be applied to all rigid bodies?

Yes, the parallel axis theorem can be applied to all rigid bodies, as long as the body is in a state of pure rotation and the axis of rotation is parallel to the axis through the center of mass. It is a fundamental principle in rotational mechanics and can be used to solve various problems in physics and engineering.

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