Pure rolling motion and different poles

My question concerns a highly rated topic in university physics courses: pure rolling motion.

In particular, I would like to better understand the logic with which the problem can be studied using both the center of mass reference system and an inertial reference system.

Let's take the case in which the wheel is subjected to a force $$F$$ at the center of mass, and we want to identify the angular acceleration of the wheel and the static friction force $$f$$ necessary for pure rolling.

In the first case (reference system of the center of mass) the problem is studied as a roto-translation and the system of equations that allow us to study the pure rolling motion will be:

$$F-f = m \cdot a_g$$

$$f \cdot r = I \cdot \frac{a_g}{r}$$

where $$r$$ is the radius of the wheel, $$I$$ its moment of inertia around the axis passing through the center of gravity, and $$a_g$$ is the acceleration of the center of gravity. The second cardinal equation was developed using the center of mass as the pole. Substituting the first equation in the second one results, with appropriate steps:

$$F \cdot r-m \cdot a_g \cdot r = I \cdot \frac{a_g}{r}$$

$$F \cdot r = I \cdot (a_g / r) + m \cdot a_g \cdot r$$

thus:

$$F \cdot r = (I + mr ^ 2) \cdot \frac{a_g}{r}$$

Let us now consider the second case (inertial reference system). In this case the motion of the wheel can be studied as a pure rotation around the point of contact $$C$$ with the ground. In this case the equation that allows to identify the acceleration of the wheel will be:

$$F \cdot r = (I + mr ^ 2) \frac{a_g}{r}$$

(Using point $$C$$ as the pole).

So in both cases we have obtained an identical equation. In the first case the term $$mr^2$$ which is added to the moment of inertia $$I$$ of the wheel derives from having replaced the equation $$F-f ​​= m\cdot a_g$$ in the second equation. In the second case, however, the term $$mr ^ 2$$ derives directly from the application of the Huygens-Steiner theorem.

What is the physical significance of this thing? Why in the reference system of the center of mass do I get a term similar to that of Huygens Steiner but without having used it? How it is possible? And why using the point $$C$$ or the center of mass as a pole I obtained the same results?

1 Answer

With a rolling object, the angular acceleration about the center of mass is the same as the instantaneous angular acceleration about the point of contact. The two equations which yield the two accelerations should be the same.

• True, but I don't understand how they lead to the same result and why in one case the term mr ^ 2 derives from the moment of inertia and in the other it derives from the substitution of the equation $F-f = ma$ in the equation relative to the rotation. Is there a physical justification? Commented Apr 21, 2021 at 14:39