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Physics LibreTexts

30.6: Ball Rolling on Inclined Rotating Plane

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Plane tilted at an angle of alpha to the z axis. i unit vector is perpendicular to the plane and j unit vector is along the plane
Figure 30.6.1

We’ll take unit vectors ˆz pointing vertically up, ˆi perpendicularly up from the plane, the angle between these two unit vectors being α. (We will need a set of orthogonal unit vectors ˆi,ˆj,ˆk, not fixed in the plane, but appropriately oriented, with ˆk horizontal.) The vector to the center of the sphere (radius a, mass m ) from an origin on the axis of rotation, at a point a above the plane, is r. The contact reaction force of the plane on the sphere is R.

The equations of motion are:

m¨r=Rmgˆz,I˙Ω=aˆi×R

and the equation of rolling contact is ˙raΩ׈i=^ωi×r.

First, we eliminate R from the equations of motion to give

˙Ω=(am/I)(¨r+gˆz)׈i

Note that ˙Ωˆi=0, so the spin in the direction normal to the plane is constant, Ωˆi=n say. (Both forces on the sphere have zero torque about this axis.)

Integrating,

Ω+ const. =(ma/I)(˙r+gtˆz)׈i

Now eliminate Ω by multiplying both sides by ×i and using the equation of rolling contact

˙raΩ׈i=ωˆi×r

to find:

(ma2/I)[(˙r+gtˆz)׈i]׈i=aΩ׈i+ const. =˙rωˆi×r+ const. 

then using (˙r׈i)׈i=˙r,(ˆz׈i)׈i=ˆj, we find

˙r(1+ma2/I)+(ma2/I)gtˆjsinα+ const. =ω^veci×r

The constant is fixed by the initial position r0, giving finally

˙r=ω1+ma2/Iˆi×[(rr0)+ma2/Iωgtˆksinα]

The first term in the square brackets would give the same circular motion we found for the horizontal rotating plane, the second term adds a steady motion of the center of this circle, in a horizontal direction (not down the plane!) at constant speed (ma2/Iω)gsinα.

(This is identical to the motion of a charged particle in crossed electric and magnetic fields.)

Bottom line: the intuitive notion that a ball rolling on a rotating inclined turntable would tend to roll downhill is wrong! Recall that for a particle circling in a magnetic field, if an electric field is added perpendicular to the magnetic field, the particle moves in a cycloid at the same average electrical potential—it has no net movement in the direction of the electric field , only perpendicular to it. Our rolling ball follows an identical cycloidal path—keeping the same average gravitational potential.


This page titled 30.6: Ball Rolling on Inclined Rotating Plane is shared under a not declared license and was authored, remixed, and/or curated by Michael Fowler.

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