Deriving Moment of Inertia for a Uniform Rod Rotating about a Non-Center Axis

In summary, the period of oscillation for small angular displacements of a uniform rod of mass M and length L rotating about an axis through one end is 2pi*root(2L/3G). When the axis is a distance x from the center of mass, the period is 2pi*root[(1/12L^2 + x^2)/(gx)]. The moment of inertia for this axis can be derived using the parallel axis theorem.
  • #1
turdferguson
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0

Homework Statement


A uniform rod of mass M and length L is free to rotate about a horizontal axis perpendicular to the rod and through one end. A) Find the period of oscillation for small angular displacements. B) Find the period if the axis is a distance x from the center of mass

Homework Equations


I = summation(Mx^2)
T restoring = k*theta = mgtheta*x
period = 2pi*root(I/k)

The Attempt at a Solution


The first part is no problem. I = 1/3ML^2 and the restoring constant is LMg/2. T = 2pi*root(2L/3G)

For the second part, I know that mgtheta acts at the distance x, but how do I derive moment of inertia for an axis x distance from the com? I know it changes with the axis and it has to fall inbetween 1/3ML^2 and 1/12ML^2. But I am not familliar with the integration that goes into deriving I.

Should I leave it as 2pi*root(newI/xMg) ?
 
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  • #2
turdferguson said:
For the second part, I know that mgtheta acts at the distance x, but how do I derive moment of inertia for an axis x distance from the com?

This should help: http://en.wikipedia.org/wiki/Parallel_axis_theorem" .
 
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  • #3
Thanks a lot. So the period is 2pi*root[(1/12L^2 + x^2)/(gx)]
 

Related to Deriving Moment of Inertia for a Uniform Rod Rotating about a Non-Center Axis

What is moment of inertia?

Moment of inertia is a physical property of a rigid body that determines its resistance to rotational acceleration. It is defined as the sum of the products of the mass of each particle in the body and the square of its distance from the axis of rotation.

How is moment of inertia related to mass and distribution of mass?

The moment of inertia is directly proportional to the mass of the body and the square of its distance from the axis of rotation. This means that a body with a larger mass or a greater distribution of mass away from the axis of rotation will have a greater moment of inertia.

What is the formula for calculating moment of inertia?

The formula for calculating moment of inertia depends on the shape and distribution of mass of the rigid body. The most commonly used formula is I = mr², where I is the moment of inertia, m is the mass of the body, and r is the distance from the axis of rotation.

How does changing the axis of rotation affect the moment of inertia?

Changing the axis of rotation can greatly affect the moment of inertia of a body. If the axis of rotation is moved closer to the center of mass of the body, the moment of inertia will decrease. On the other hand, if the axis of rotation is moved further away from the center of mass, the moment of inertia will increase.

What is the practical application of knowing moment of inertia?

Knowing the moment of inertia is important in many applications, such as designing structures and machines that need to rotate, understanding the stability of objects in motion, and predicting the behavior of rotating bodies in different scenarios. It is also crucial in fields such as physics, engineering, and robotics.

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