henry3369 said:Homework Statement
Find the moment of inertia of a solid sphere of uniform mass density (like a billiard ball) about an axis through its center
Homework Equations
I = ∫rρdV
The Attempt at a Solution
I =ρ ∫r4πr2dr = ρ4π∫r4
Then I integrate this from 0 (the center) to R, so I = (ρ4π)*(R5/5)
And ρ = mv so ρ = M/(4/3)πR3 = 3M/4πR3. Put ρ into the equation for moment of inertia to get I = 3MR2/5.
My book tells me the answer is (2/5)MR^2. Where did I go wrong?
The surface area of a spherical shell from the center of the sphere to the outer shell of radius R.PeroK said:Can you explain how you are setting up the integral? What shape are you integrating from 0 to R?
Exactly!henry3369 said:I figured it out. If I use spherical shells, the points from the shell aren't all the same distance form the axis of rotation. I was thinking about the distance from the center.
The moment of inertia of a solid sphere is a physical property that measures the resistance of the object to changes in its rotational motion. It is the sum of the products of each individual mass element in the object with the square of its distance from the axis of rotation.
The moment of inertia of a solid sphere can be calculated using the formula I = (2/5) * mr^2, where m is the mass of the sphere and r is the radius. This formula assumes that the object has a uniform mass distribution.
The units of moment of inertia depend on the units of mass and distance used in the calculation. In the International System of Units (SI), the units of moment of inertia are kg*m^2.
The moment of inertia of a solid sphere determines how much torque is required to accelerate the object in rotational motion. A larger moment of inertia means that more torque is needed to produce the same angular acceleration compared to an object with a smaller moment of inertia.
Yes, the moment of inertia of a solid sphere can change depending on the distribution of mass within the object. For example, if the mass is concentrated towards the outer edge of the sphere, the moment of inertia will be larger compared to a sphere with a more uniform mass distribution.