Rolling resistance is often neglected when solving problems involving rolling. For example, when modeling a cylinder rolling on a flat surface, it is often assumed that the cylinder and surface are rigid. This means that the cylinder and surface are made up of very hard material. Under this assumption there is no deformation of the cylinder at the contact location, and no deformation of the surface underneath the cylinder. As a result, the cylinder will continue rolling indefinitely (further assuming that there is no frictional resistance with the air). This means there is no rolling resistance, which can slow down the cylinder.

It is informative to analyze the dynamics of a uniform cylinder rolling on a flat horizontal surface, using the rigidity assumption.

Consider the figure below illustrating the motion of a uniform cylinder on a flat horizontal surface, and with sign convention shown. The cylinder rolls without slipping. This is modeled as a two-dimensional problem.

Where:

Apply Newton’s Second Law in the

Where:

Apply Newton’s Second Law in the

where

Therefore,

Next, apply the moment equation for rotation of a rigid body about its center of mass

Where:

Σ

Now,

Substituting this into equation (2) we have

Since the cylinder is rolling with no slipping we can write:

This is in accordance with the sign convention used.

Combining this equation with equation (1) we have

Combining this equation with equation (3) we have

Since the angular acceleration

However, we know from real life that the cylinder will slow down and eventually stop. In other words, we know that rolling resistance exists.

So we conclude that there must be some deformation at the contact interface, which causes the cylinder to slow down and eventually stop. With this new understanding we can now come up with a new solution, which captures the physics of this.

Consider the new figure below illustrating the motion of a uniform cylinder on a flat horizontal surface. For simplicity, let’s assume that the cylinder itself is rigid and the surface underneath is deforming due to the weight and motion of the cylinder. As before, assume that the cylinder rolls without slipping.

As before, this is modeled as a two-dimensional problem. The set up of this problem will be a bit different from before.

A new variable

Apply Newton’s Second Law in the

Apply Newton’s Second Law in the

Since

Next, apply the moment equation for rotation of a rigid body about its center of mass G (which coincides with the geometric center

Now, since

Substituting this into equation (6) we have

Since the cylinder is rolling with no slipping we can write:

This is in accordance with the sign convention used.

Combining this equation with equation (4) we have

Combining this equation with equation (7) we have

Substitute equation (5) and we get

For a uniform cylinder,

Therefore equation (8) becomes

Solving for angular acceleration we get

This tells us that the angular acceleration of the cylinder is negative, which makes sense since we know that the cylinder experiences rolling resistance, and slows down as a result.

Note that the resultant forces

The fact that the cylinder tends to slow down means that it takes a constant force to overcome rolling resistance and keep the cylinder rolling at a constant speed. We can find an expression for this force. Let’s call this force

Let’s assume that the applied force

Since the cylinder is rolling at constant velocity with no slipping, its angular acceleration is zero and its center of mass (coinciding with point

Apply Newton’s Second Law in the x-direction:

Apply Newton’s Second Law in the y-direction:

Since

Next, apply the moment equation for rotation of a rigid body about its center of mass

Now, since

Substituting this into equation (11) we have

Substituting equations (9) and (10) into the above equation we get

Therefore the force

So, to minimize the rolling resistance (and minimize

As mentioned, rolling resistance is neglected in many problems that involve rolling. This means that, by assumption, there is negligible deformation at the contact interface, and we set

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