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Unformatted text preview: r arm attained by choosing the dish’s center
of curvature as the pivot point.
The ball also undergoes rotational motion about its own center of mass due to the presence of static
92 friction between the ball and the dish. We must re-examine what it means to “roll without slipping”
in the case of motion on a curved surface. Rolling without slipping means that, as the ball rolls,
the contact point between the ball and dish travels the same arc length along the ball’s surface as it
does along the dish’s surface (see Figure 7). Using the formula for arc length, we get ∆β r = ∆θ R.
In a small time interval ∆t we have
R −→ ωβ r = ωθ R ,
∆t (18) which relates ωβ , the angular velocity of the rotation of the ball, to ωθ , the angular velocity of the
oscillation. Furthermore, since there is a net torque on the ball, the angular velocities will change
by a small increment ∆ω over a small increment of time ∆t , so
R −→ α r = αθ R ,
∆t (19) and we’ve found the appropriate expression relating the angular acceleration of rotations for the
ball, α , to the angular acceleration of oscillatory motion, αθ .
pivot (R-r) r R rg
Ff contact pt Fg Force Diagram Angle Relations Figure 7: Left: The force diagram for the ball in the dish. The force of gravity originates from the
ball’s center of mass while friction acts at the contact point between the dish and ball. The lever
arms for these two forces are rg = R − r and r f = R, respcectively. Right : A sphere rolling in
a spherical dish, demonstrating the relation between the two different angular displacements, ∆θ
and ∆β . The rolling without slipping condition says that the two red arcs in the diagram are of
equal lengths, which means that β and θ change at different rates as the ball rolls without slipping
along the dish’s surface. The mathematical relation between these two angles is given in Equation
Finally, combining Equations 17, 19,...
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This note was uploaded on 09/13/2011 for the course ECONOMICS 101 taught by Professor Gerson during the Spring '11 term at University of Michigan.
- Spring '11