Mr.p: Good morning. Let's learn about rolling without slipping.
"Flipping Physics" Intro
This is an object, which is rolling without slipping.
Now it might seem obvious, however, realize there is no slipping which occurs at the contact point
between the object which is rolling without slipping and the surface it is rolling on.
Billy: Yeah, that's what it is. Bo: It's rolling without slipping, that's what it means.
Mr.p: Right. So again rolling without slipping.
I'm going to isolate two points on the object which is rolling without slipping.
The center of mass of the object is a dot in orange and a point on the edge of the object is a dot in blue.
I have also added dots which track the locations of those points.
You should be able to see that the orange dot at the center of mass of the object rolling without slipping
is moving at roughly a constant velocity.
The dot in blue, which is at the edge of the object rolling without slipping, traces a more complicated pattern.
Billy: Oh, that's a cycloid. A cycloid is a curve traced by a point on a circle, which is rolling along a straight line.
Mr.p: Absolutely Billy, the blue dot is tracing out a cycloid.
Bobby: I actually think that looks pretty cool. Billy: Yeah, totally!
Bo: Sure. Mr.p: Believe it or not, I agree.
Okay, rolling without slipping is actually a combination of two types of motion.
It combines pure translational motion with pure rotational motion.
As you can see, every part of the object in pure translational motion has the same velocity to the right.
Therefore every part of the pure translational motion object
has a velocity equal to the velocity of the center of mass of the object.
This is not true of a pure rotational motion object.
The center of mass of the pure rotational motion object is not moving and therefore has a velocity of zero.
However, the outside edge of the pure rotational motion object has a tangential velocity
equal to the radius of the object times the angular velocity of the object.
When you add those together on the object which is rolling without slipping,
you can see the top of the object
has a velocity equal to the velocity of the center of mass of the pure translational object
plus the velocity of the edge of the pure rotational object, which is radius times angular velocity.
Both of those velocities are to the right.
The velocity of the center of mass of the object which is rolling without slipping equals
just the velocity of the center of mass of the pure translational object
because the velocity of the center of mass of the pure rotational object equals zero.
The velocity of the bottom of the object which is rolling without slipping
is the velocity of the center of mass of the pure translational object
plus the velocity of the edge of the pure rotational object, which is radius times angular velocity.
Going back to the cycloid for a moment, do y'all see
how the very bottom of the object rolling without slipping actually does not move relative to the surface
and therefore has a velocity at that brief moment of contact, which is equal to zero.
Bobby: Okay. Bo: Sure.
Billy: Absolutely, it is not slipping, so it is static friction,
so the velocity of the bottom of the edge of the object rolling without slipping has to equal zero.
Bo: Oh right, that means
the velocity of the center of mass of pure translational motion minus radius times angular velocity equals zero.
Which means the velocity
of the center of mass of pure translational motion equals the radius of the object times angular velocity.
Bobby: Therefore the velocity of the center of mass of an object which is rolling without slipping
equals the radius of the object times the angular velocity of the object
Billy: Wow. That is easy to remember
because it is the same as what the tangential velocity of an object equals, radius times angular velocity.
Bo: Except in the equation for rolling without slipping it's always a capital R for the radius of the object.
Billy: Right. Mr.p: Very nice.
The velocity of the center of mass of an object which is rolling without slipping
equals the radius of the object times the angular velocity of the object.
Notice, this means the velocity at the top of the object rolling without slipping
equals two times the velocity of the center of mass of the object.
Let's take a moment to look at the velocity of a point on the edge of the object rolling without slipping.
Remember it is the addition of the velocity of the pure translational object
and the velocity of the pure rotational object.
I've represented that using a tip-to-tail vector diagram.
With the resultant vector as the velocity of the edge of the object rolling without slipping.
Okay, now let's talk about kinetic energy.
The total kinetic energy of an object rolling without slipping
equals the sum of the translational and rotational kinetic energies of the object.
or one half mass times velocity squared plus one half orotational inertia times angular velocity squared.
Also realize the equation for the acceleration of the center of mass of an object
which is rolling without slipping is similar to the equation for tangential acceleration.
It equals the object's radius times its angular acceleration.
And the equation for the distance travelled by the center of mass of an object rolling without slipping is
similar to the arc length equation. It equals the object's radius times its angular displacement.
Both of those equations can be derived using the same logic
we just used to derive the equation for the velocity of the center of mass of an object rolling without slipping.
Thank you very much for learning with me today. I enjoyed learning with you.