Breaking up the semi-circle into infinitesimally small rings

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The discussion focuses on breaking a semi-circle into infinitesimally small rings to calculate the moment of inertia. The user derived the mass element dm as pi*r*dr and integrated to find the moment of inertia, resulting in .25*pi*R^4, which differs from the expected .5*pi*R^4 for a full disk. A key point raised is the omission of a constant factor for density in the calculations. Including this factor and expressing it in terms of mass and area is crucial for accurate results. The conversation emphasizes the importance of correctly accounting for density in such integrative problems.
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I attempted to solve this problem by breaking up the semi-circle into infinitesimally small rings of mass dm and width dr a distance r away from the center [0<r<R]. I then wrote dm in terms of the area: dm=pi*r*dr. Then, I plugged into the formula di=integral(dm*r^2) and integrated from 0 to R giving me .25*pi*R^4. This is not what I would have gotten for a complete disk though I would have gotten .5*pi*R^4 using the same process. Can someone please help I don't even know if I started correctly.
 
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spencerw105 said:
I plugged into the formula di=integral(dm*r^2) and integrated from 0 to R giving me .25*pi*R^4. This is not what I would have gotten for a complete disk though I would have gotten .5*pi*R^4 using the same process. Can someone please help I don't even know if I started correctly.
You have left out a constant factor for density. What happens when you include that and express it in terms of the mass and area of the object?
 

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