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.