Specifically they want to know ab out its radius of gyration. Unfortunately, he forgot what material he used. Assume the plates are rigidly attached to one another. Method of Composite Parts for Moments of Inertia and the Parallel Axis Theorem As an alternative to integration, both area and mass moments of inertia can be calculated via the method of composite parts, similar to what we did with centroids.
The Parallel Axis Theorem When we calculated the area and mass moments of inertia via integration, one of the first things we had to do was to select a point or axis we were going to take the moment of inertia about. The distances used in our moment integrals depends on the point or axis chosen. These distances will be at a minimum at the centroid and will get larger as we start further from the centroid.
The distance r in the Parallel Axis Theorem represents the distance we are moving the axis we are taking the moment or intent about. Using the Method of Composite Parts to Find the Moment of Inertia To find the moment of inertia of a body using the method of composite parts, you need to start by breaking your area or volume down into simple shapes.
Start by breaking down your area or volume into simple parts and number those shapes. Holes or cutouts will count as negative areas or masses. Most work in the method of composite parts will revolve around filling out a table such as this one. This table contains the rows and columns necessary to find the rectangular area moments of inertia I xx and I yy for this composite body.
The area or mass for each piece area for area moments of inertia or mass for mass moments of inertia. Remember cutouts should be listed as negative areas or masses. The centroid or center of mass locations X, Y and possibly Z coordinates. Most of the time, we will be finding the moment of inertia about centroid of the composite shape, and if that is not explicitly given to you, you will need to find that before going further.
The moment of inertia values about each shape's centroid. To find these values you will plug numbers for height, radius, mass, etc. Do not use these formulas blindly though as you may need to mentally rotate the body, and thus switch equations, if the orientation of the shape in the table does not match the orientation of the shape in your diagram.
The adjustment distances r for each shape. For this value you will want to determine how far the x-axis, y-axis, or z-axis moves to go from the centroid of the piece to the overall centroid, or point you are taking the moment of inertia about.
To calculate these values generally, you will be finding the horizontal, vertical, or diagonal distances between piece centroids and the overall centroids that you have listed earlier in the table. See the parallel axis theorem section of this page earlier for more details. Finally, you will have a column of the adjusted moments of inertia. Ben Davis December 27, Why do we use parallel axis theorem? Which of the following is correct for parallel axis theorem?
What is the formula of perpendicular axis theorem? What is the formula of thermal of parallel axis? What is the difference between parallel axis theorem and perpendicular axis theorem? What is unit of radius of gyration?
Add a comment. Active Oldest Votes. Improve this answer. Sahil Sahil 2 2 silver badges 8 8 bronze badges. Then it's alright. The axes you choose are usually the "natural ones", about which the body is symmetrical.
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