When a wheel is rolling without
slipping, as we saw before,
that the contact point
here, the contact point
is instantaneously at rest.
Now, if the wheel is rolling
on a surface with friction,
then it's possible that we may
have a static friction force.
So there could be
static friction may act.
However, static
fiction is always
depends on the circumstances.
Now let's just
consider two cases.
Suppose here's a wheel, Vcm.
This is a horizontal surface.
And in that case,
the static friction,
if the wheel is rolling
with velocity V,
in this case f static is 0.
And because of
that, the wheel will
roll without any friction,
which is an idealization.
There's other types of friction
called rolling resistance, air
resistance, et cetera, which
will slow the wheel down.
But in a perfect hard wheel,
idealized wheel in a vacuum,
it will keep on rolling.
However, if a wheel is
going down an incline plane
and it wants to maintain
the rolling without slipping
condition--
so here we have Vcm.
And it has some angular speed--
we have to be careful
because in this case
if we differentiate
Acm is our alpha.
So what is making the
wheel spin faster?
It has a non-zero,
angular acceleration.
And if we looked at the
forces acting on the wheel,
then we have a normal force.
We have gravity acting
at the center of mass.
And we also have
static friction.
And it's precisely
the static friction
that is producing a torque
about the center of mass.
And that torque about
the center of mass
will produce an
angular acceleration.
So in order for the
wheel to continue
rolling without
slipping, it must
have both a linear
acceleration, which
comes from gravitational
force component going down
an inclined plane minus
the friction plus the alpha
is coming from the torque.
And so this side, if
we wrote it as a vector
equation, the torque
about the center of mass
would be the vector from--
let's right this
is r cm to where
the static friction is acting.
So that's the vector r cm
F static cross f static.
So in this case, f
static is non-zero.
So there are many circumstances
in which static friction can
vary between a zero value,
some possible value,
and some maximum value.
And here we see
one more example.
In order for the wheel to
continue to accelerate down
the inclined plane and
roll without slipping
that the static friction
must produce a torque
that causes the
angular acceleration.
And so we see that
static friction
depends on the other
constraints in the problem.
