Rolling is a combination motion in which an object rotates about an axis that is moving along a straight-line trajectory. For example, FIGURE 12.37 is a time-exposure photo of a rolling wheel with one lightbulb on the axis and a second lightbulb at the edge. The axis light moves straight ahead, but the edge light moves along a curve. Let’s see if we can understand this interesting motion. We’ll consider only objects that roll without slipping.

FIGURE 12.37 The trajectories of the center of a wheel and of a point on the rim are seen in a time-exposure photograph.

FIGURE 12.38 shows a round object—a wheel or a sphere—that rolls forward exactly one revolution. The point that had been on the bottom follows the curve you saw in Figure 12.37 to the top and back to the bottom. Because the object doesn’t slip, the center of mass moves forward exactly one circumference: $\text{Δ}{x}_{\text{cm}}=2\pi R$.

FIGURE 12.38 An object rolling through one revolution.

We can also write the distance traveled in terms of the velocity of the center of mass: $\text{Δ}{x}_{\text{cm}}=v_{\text{cm}}\text{Δ}t$. But $\text{Δ}t$, the time it takes the object to make one complete revolution, is nothing other than the rotation period T. In other words, $\text{Δ}{x}_{\text{cm}}=v_{\text{cm}}T$.

These two expressions for $\text{Δ}{x}_{\text{cm}}$ come from two perspectives on the motion: one looking at the rotation and the other looking at the translation of the center of mass. But it’s the same distance no matter how you look at it, so these two expressions must be equal. Consequently,

If we divide by T, we can write the center-of-mass velocity as

But $2\pi \text{/}T$ is the angular velocity $\omega$, as you learned in Chapter 4, leading to

Equation 12.36 is the rolling constraint, the basic link between translation and rotation for objects that roll without slipping.

Let’s look carefully at a particle in the rolling object. As FIGURE 12.39a shows, the position vector $\vec{r}{}_i$ for particle i is the vector sum $\vec{r}{}_i=\vec{r}{}_{\text{cm}}+\vec{r}{}_{i\text{,}\text{rel}}$. Taking the time derivative of this equation, we can write the velocity of particle i as

In other words, the velocity of particle i can be divided into two parts: the velocity $\vec{v}{}_{\text{cm}}$ of the object as a whole plus the velocity $\vec{v}{}_{i\text{,}\text{rel}}$ of particle i relative to the center of mass (i.e., the velocity that particle i would have if the object were only rotating and had no translational motion).

FIGURE 12.39 The motion of a particle in the rolling object.

Full Alternative Text

FIGURE 12.39b applies this idea to point P at the very bottom of the rolling object, the point of contact between the object and the surface. This point is moving around the center of the object at angular velocity $\omega$, so $v_{i\text{,}\text{rel}}=-R\omega$. The negative sign indicates that the motion is cw. At the same time, the center-of-mass velocity, Equation 12.36, is $v_{\text{cm}}=R\omega$. Adding these, we find that the velocity of point P, the lowest point, is $v_i=0$. In other words, the point on the bottom of a rolling object is instantaneously at rest.

Although this seems surprising, it is really what we mean by “rolling without slipping.” If the bottom point had a velocity, it would be moving horizontally relative to the surface. In other words, it would be slipping or sliding across the surface. To roll without slipping, the bottom point, the point touching the surface, must be at rest.

FIGURE 12.40 shows how the velocity vectors at the top, center, and bottom of a rotating wheel are found by adding the rotational velocity vectors to the center-of-mass velocity. You can see that $v_{\text{bottom}}=0$ and that $v_{\text{top}}=2R\omega =2v_{\text{cm}}$.

FIGURE 12.40 Rolling without slipping is a combination of translation and rotation.