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How does one know that a uniform solid sphere rolling purely on a smooth surface does so about an axis passing through its CENTRE OF MASS and not any other axis? Is there a mathematical or intuitional proof to show that it rotates about CENTRE OF MASS ?

GodP7
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  • If you cover the surface in numbered dots, then it will be obvious that most of the dots move in large circles and the two dots that just translate horizontally are on the axis and becomes equivalent to analysing a wheel, or are you talking about a sphere that is not allowed to have any identifiable markings? – KDP Feb 12 '24 at 04:33
  • The point on the solid sphere in contact with the ground can be taken as instantaneous axis of rotation too. – Pumpkin_Star Feb 12 '24 at 06:12
  • When you say "rolling purely", do you mean rolling without slipping? When you say a "smooth surface", do you mean flat or frictionless? – David Bailey Feb 12 '24 at 06:48

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If a sphere rolls in a straight line across the surface then the sequence of points making contact with the surface trace out a circle intersecting the center of the sphere. Thus, it is rotating about a horizontal axis through its center.

You can make a sphere not travel in a straight line: put your palm over an orange and rotate your hand in small circles parallel to the surface. That motion of the sphere is not a rotation about its center.

Ben H
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For the sake of simplicity, I will not generalize this problem to all rotation, but simply speak of the problem at hand.

Consider the frame of reference in which the sphere experiences no translational movement.

Euler's rotation theorem states that any displacement of a rigid body such that a point on the rigid body remains fixed is equivalent to a single rotation about some axis that runs through the fixed point. The proof of this theorem is well-established.

There is a torque exerted on the sphere from friction. The torque causes rotation about the center of mass (linked by brainfreeze). In the frame of reference we have chosen, we find that the only point that is translationally invariant is the center of mass; this is the fixed point through which an axis of rotation must pass. Intuitively, the geometrical center is invariant; by symmetry, the center of mass of the uniform sphere lies at the same point, which you can verify with calculus.

If you're wondering whether this result holds if the frame is not inertial (i.e. an inclined plane), the fictitious force experienced in this frame of reference will act about the center of mass and thus have no effect on rotation.

Raphael Esquivel
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