When treating the rigid motion of a solid body involving both rotation and translation the analysis is often simplest and cleanest when one makes use of orthogonal degrees of freedom which have cancelling interfering cross terms. This means it is preferred to analyze the composite motion by simply breaking it down as a translation of the body's center of mass and a rotation about an axis through the body's center of mass because such a breakdown has the infinitesimal translations orthogonal to infinitesimal rotations in their corresponding subspaces of the body's overall configuration space. (This is especially so when treating total kinetic energy of motion and not analyzing local torques about eccentric pivot points.) In such a situation the relevant moment of inertia is the one for an axis *through* the center of mass. So for the problem BW considers things work out simplest (for the energy conservation analysis) if one uses the moment of inertia about an axis through the sphere's center, and *not* the moment of inertia about an axis through the sphere's contact point with the incline.
BTW, the moment of inertia for an axis through the center of a uniform infinitely thin spherical shell of radius R is (2/3)*M*R^2. If the moment of inertia of a hollow sphere of outer radius R is (3/5)*M*R^2 then the sphere has a shell thickness of 0.10773447...*R, assuming the shell has a constant mass density.
David Bowman
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From: Phys-l [mailto:phys-l-bounces@mail.phys-l.org] On Behalf Of brian whatcott
Sent: Wednesday, May 20, 2020 3:18 PM
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Subject: [Phys-L] Rotation of a Rolling Ball.
I was considering a Galilean problem on Quora of a ball moving at a given velocity climbing a ramp of given height without slipping.
The question asked about its final velocity after the ascent.
This depends upon its kinetic energy, and how much is converted to potential energy.
I considered two possibilities: solid sphere and hollow sphere.
I used two moments of inertia for these cases:2/5*m*r^2 & 3/5*m*r^2
These happen to be the values used by Dan MacIsaac in his 1996 video tutorial at Buffalo, and by John Yelton at Oxford U., when upvoting a similar calculation recently.
Then I happened on a list of Moments:
1) a sphere spinning on a central axis 2/5*m*r^2
2) a sphere rolling on a surface 7/5*m*r*2
And I was taken aback.
The academic sources seem to be juggling the measure for rotation to make the lower value of Moment work, ignoring the axis of rotation offset correction.
What am I missing?