Moment of inertia of a rectangle about centroid
Understand the Problem
The question is asking for the calculation of the moment of inertia for a rectangle about its centroid. This involves using the formula for the moment of inertia based on the dimensions of the rectangle (width and height).
Answer
$$ I = \frac{1}{12} b h^3 $$
Answer for screen readers
Let $b$ be the width and $h$ be the height; then the moment of inertia about the centroid is given by: $$ I = \frac{1}{12} b h^3 $$
Steps to Solve
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Identify the formula for the moment of inertia To calculate the moment of inertia ($I$) of a rectangle about its centroid, we use the formula: $$ I = \frac{1}{12} b h^3 $$ where $b$ is the width and $h$ is the height of the rectangle.
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Substituting the values into the formula First, identify the values of $b$ (width) and $h$ (height) of the rectangle. Substitute these values into the formula.
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Calculate the moment of inertia After substitution, perform the arithmetic calculations to find the moment of inertia. Remember to follow the order of operations when calculating powers and products.
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Final result Write out the final value of the moment of inertia, ensuring it is in the correct units (typically in $kg \cdot m^2$ if using SI units).
Let $b$ be the width and $h$ be the height; then the moment of inertia about the centroid is given by: $$ I = \frac{1}{12} b h^3 $$
More Information
The moment of inertia is a measure of an object's resistance to changes in its rotation about a certain axis. For rectangles, this formula is critical in various applications in engineering and physics, particularly in structural analysis and dynamics.
Tips
- Forgetting to cube the height $h$ in the formula which can lead to incorrect calculations.
- Using the wrong dimensions for width and height, which must be consistent.
- Not using the correct units for $b$ and $h$, leading to erroneous unit conversions.
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