The moment of inertia of a thin-walled hollow sphere can be derived using the equation I = ∫ r² dm, where dm represents a differential mass element. To compute dm, one can express it in terms of the area of a ring and the sphere's density. The integration should be performed from y = -R to y = +R, utilizing the equation y² + x² = R² to define the geometry of the sphere. This approach avoids multivariable integrals while allowing for trigonometric substitution as needed.
PREREQUISITESStudents studying physics or engineering, particularly those focusing on mechanics and rotational dynamics, as well as educators seeking to explain the moment of inertia of hollow objects.
You are missing a z^2 there for the third dimension.parsa418 said:Could anyone prove the moment of inertia of a thin walled hollow sphere using the y^2 + x^2 = r^2. I have only studied up to single variable calculus. I can take regular integrals but not multivariable integrals. I don't want to use the angle method or any polar coordinate systems except in the part where I might use it for trigonometric substitution to evaluate the integral.
Thanks