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When a wheel is rolling without
slipping, as we saw before,

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that the contact point
here, the contact point

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is instantaneously at rest.

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Now, if the wheel is rolling
on a surface with friction,

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then it's possible that we may
have a static friction force.

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So there could be
static friction may act.

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However, static
fiction is always

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depends on the circumstances.

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Now let's just
consider two cases.

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Suppose here's a wheel, Vcm.

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This is a horizontal surface.

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And in that case,
the static friction,

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if the wheel is rolling
with velocity V,

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in this case f static is 0.

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And because of
that, the wheel will

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roll without any friction,
which is an idealization.

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There's other types of friction
called rolling resistance, air

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resistance, et cetera, which
will slow the wheel down.

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But in a perfect hard wheel,
idealized wheel in a vacuum,

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it will keep on rolling.

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However, if a wheel is
going down an incline plane

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and it wants to maintain
the rolling without slipping

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condition--

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so here we have Vcm.

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And it has some angular speed--

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we have to be careful
because in this case

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if we differentiate
Acm is our alpha.

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So what is making the
wheel spin faster?

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It has a non-zero,
angular acceleration.

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And if we looked at the
forces acting on the wheel,

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then we have a normal force.

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We have gravity acting
at the center of mass.

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And we also have
static friction.

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And it's precisely
the static friction

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that is producing a torque
about the center of mass.

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And that torque about
the center of mass

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will produce an
angular acceleration.

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So in order for the
wheel to continue

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rolling without
slipping, it must

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have both a linear
acceleration, which

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comes from gravitational
force component going down

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an inclined plane minus
the friction plus the alpha

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is coming from the torque.

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And so this side, if
we wrote it as a vector

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equation, the torque
about the center of mass

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would be the vector from--

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let's right this
is r cm to where

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the static friction is acting.

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So that's the vector r cm
F static cross f static.

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So in this case, f
static is non-zero.

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So there are many circumstances
in which static friction can

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vary between a zero value,
some possible value,

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and some maximum value.

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And here we see
one more example.

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In order for the wheel to
continue to accelerate down

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the inclined plane and
roll without slipping

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that the static friction
must produce a torque

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that causes the
angular acceleration.

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And so we see that
static friction

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depends on the other
constraints in the problem.