KAHIBARO
Table of Contents
Pure Rolling
Rolling without slipping is a special kind of motion in which an object both translates and rotates in a linked way. A wheel, ball, or cylinder moves forward while also spinning, but the part touching the ground does not slide across the surface.
This condition is very important because it connects linear motion and rotational motion with a simple rule.
For rolling without slipping,
$$v_{\text{cm}} = R\omega$$
and, if the motion is changing,
$$a_{\text{cm}} = R\alpha$$
where $v_{\text{cm}}$ is the speed of the center of mass, $a_{\text{cm}}$ is the linear acceleration of the center of mass, $R$ is the radius, $\omega$ is the angular speed, and $\alpha$ is the angular acceleration.
What "Without Slipping" Means
Imagine a wheel moving along a road. If the wheel slips, the contact point rubs across the ground. If it rolls without slipping, the contact point is instantaneously at rest relative to the ground.
This does not mean the whole wheel is at rest. It means only the point touching the ground has zero velocity at that instant.
A useful way to think about this is that the wheel is doing two motions at once. The center of mass moves forward, and the wheel rotates around its center. At the bottom point, these two effects cancel exactly when there is no slipping.
The Rolling Condition
Suppose a wheel of radius $R$ rotates through an angle $\theta$. If it rolls without slipping, the distance traveled by its center is
$$s = R\theta$$
Differentiating with respect to time gives the speed relation
$$v_{\text{cm}} = \frac{ds}{dt} = R\frac{d\theta}{dt} = R\omega$$
Differentiating again gives
$$a_{\text{cm}} = R\alpha$$
These relations are only valid for rolling without slipping.
The geometric rolling condition is
$$s = R\theta$$
This leads directly to
$$v_{\text{cm}} = R\omega$$
and
$$a_{\text{cm}} = R\alpha$$
Only use these formulas when there is no slipping.
Velocity of Different Points on a Rolling Wheel
Different points on a rolling object do not all move at the same speed.
The velocity of any point is the combination of the translational velocity of the center of mass and the rotational velocity about the center.
For a wheel rolling to the right,
| Point on wheel | Speed relative to ground |
|---|---|
| Center | $v_{\text{cm}}$ |
| Top point | $2v_{\text{cm}}$ |
| Bottom point | $0$ |
The top point moves fastest because the forward translational motion and the rotational motion point in the same direction there. The bottom point has zero speed because the two motions cancel.
Why Static Friction Matters
Rolling without slipping is usually made possible by static friction. Static friction acts at the contact point and prevents relative motion between the surface and the rolling object.
This is a subtle but important point. Even though friction is involved, the contact point is not sliding, so the friction is static, not kinetic.
Static friction can speed up or slow down the rotational motion, depending on the situation. For example, if a spinning wheel is placed on the ground, friction can reduce its spin and increase its forward motion until the rolling condition is satisfied.
For rolling without slipping on a surface, the friction involved is usually static friction, not kinetic friction.
Direction of Friction
The direction of static friction is not always opposite to the motion of the object. It opposes the tendency to slip at the contact point.
This means the friction direction depends on what the wheel would do if friction were absent.
If the bottom of the wheel tends to slide backward relative to the surface, friction acts forward. If it tends to slide forward, friction acts backward.
This is why friction on a rolling object can sometimes point in the same direction as the motion and sometimes in the opposite direction.
Rolling on a Horizontal Surface
On a level surface, an object rolling steadily without speeding up has constant $\omega$ and constant $v_{\text{cm}}$. In that case, the rolling condition remains
$$v_{\text{cm}} = R\omega$$
If there is no net force and no net torque, the rolling motion continues unchanged.
If a torque or horizontal force acts, then both the translational and rotational motion can change, but they must still satisfy
$$a_{\text{cm}} = R\alpha$$
as long as slipping does not begin.
Rolling Down an Incline
When an object rolls down a slope without slipping, gravity causes both translation and rotation. Part of the gravitational effect increases the forward speed of the center of mass, and part increases the rotational speed.
Because some energy goes into rotation, a rolling object usually accelerates down a slope more slowly than an object that simply slides without friction.
The exact acceleration depends on the moment of inertia, which is treated in other chapters, but the rolling condition still applies:
$$a_{\text{cm}} = R\alpha$$
Instantaneous Axis of Rotation
A helpful idea is that a rolling wheel can be viewed, at one instant, as rotating about the contact point with the ground. Since that point has zero velocity relative to the ground, it acts like an instantaneous axis of rotation.
This is a useful picture for understanding why the bottom point is momentarily at rest and why the top point moves faster than the center.
However, the wheel is not permanently rotating about that point, because the contact point changes continuously as the wheel moves.
Recognizing Slipping
A rolling object is slipping if the rolling condition fails. For example, if
$$v_{\text{cm}} \ne R\omega$$
then the surface speed of the rim does not match the forward motion correctly, and the contact point moves relative to the ground.
If $v_{\text{cm}} > R\omega$, the object is moving forward too fast for its rate of spin. If $v_{\text{cm}} < R\omega$, it is spinning too fast for its forward motion.
In both cases, slipping occurs.
A rolling object does not slip only if
$$v_{\text{cm}} = R\omega$$
If this equality is not true, the point of contact slides relative to the surface.
Summary of the Key Ideas
Rolling without slipping links translation and rotation in a very simple way. The distance traveled equals the radius times angular displacement, and this gives direct relations between linear and angular quantities. The contact point is instantaneously at rest relative to the ground, the top of the wheel moves faster than the center, and static friction is what usually enforces the no slipping condition.
| Quantity | Rolling without slipping relation |
|---|---|
| Linear distance | $s = R\theta$ |
| Linear speed | $v_{\text{cm}} = R\omega$ |
| Linear acceleration | $a_{\text{cm}} = R\alpha$ |
These relations are the core of rolling motion without slipping.