# Smooth rolling motion - conservation of energy?

• stfz
In summary, the conversation is discussing the conservation of mechanical energy in a system with a rolling object and the presence of friction. The main question is whether mechanical energy is generally conserved in all paths as long as the rolling is smooth. The conclusion is that in a closed system without dissipative forces, mechanical energy remains constant, and in the case of perfect rolling without slipping, the energy will be conserved. However, the presence of static friction may complicate the situation, but it does not do any work and therefore does not affect the conservation of mechanical energy. To fully understand the role of friction in mechanical energy, one must consider the relative motion between the rolling object and the surface it is rolling on, as well as the energy dissipated

#### stfz

Member advised to use the homework template for posts in the homework sections of PF.
This isn't about a specific physics problem, but rather a question:
Given I have a ball or cylinder rolling smoothly along some path, is it generally true that mechanical energy is conserved?
I.e. if ##E_mech = K+U = K_{trans} + K_{rot} + U##, then ##\Delta E_mech = 0##?

I have been able to formulate a proof for a cylinder rolling down an inclined plane, with a change in height ##\Delta h##. I've been able to show that, at the bottom, ##K_{rot}+K_{trans} = mgh##.

But I just wanted to check that this is generally true along any path (e.g. curved paths), given that the rolling is always smooth? And also, are there any caveats here where this assertion doesn't work?

Thanks!

stfz said:
But I just wanted to check that this is generally true along any path (e.g. curved paths), given that the rolling is always smooth? And also, are there any caveats here where this assertion doesn't work?

The total amount of mechanical energy, in a closed system in the absence of dissipative forces (e.g. friction, air resistance), remains constant.
so, if you have conditons of perfect rolling (without slipping) the energy should be conserved.

Hmm. I was under the impression that the static friction present was a friction force and hence the there are non-conservative forces at work.
However, now that you mention it, I realize that static friction, by definition, can do no work. Hence there are no non-conservative forces doing work per se (although there are non-conservative forces present!)

Is that why mechanical energy is conserved?

What is the relative motion between the smoothly rolling object and the surface along which it is rolling at the point / line of contact?

The answer to that plus an equation for energy dissipated by static friction between two surfaces should provide you the insight you are looking for.

## 1. What is smooth rolling motion?

Smooth rolling motion is the movement of an object, such as a ball or wheel, in which the object rolls without slipping or sliding. This type of motion is characterized by a constant velocity and is often seen in everyday objects, such as cars, bikes, and balls.

## 2. What is conservation of energy?

Conservation of energy is a fundamental principle in physics that states that energy cannot be created or destroyed, but can only be transformed from one form to another. This means that the total amount of energy in a closed system remains constant over time.

## 3. How does smooth rolling motion demonstrate conservation of energy?

In smooth rolling motion, the object's kinetic energy is converted into both translational and rotational kinetic energy. As the object moves, its kinetic energy remains constant, demonstrating the conservation of energy. Additionally, any potential energy the object may have is also conserved during smooth rolling motion.

## 4. What factors affect an object's smooth rolling motion?

The main factors that affect an object's smooth rolling motion are its mass, radius, and the surface it is rolling on. A heavier object will have more inertia and require more force to move, while a larger radius will require less force. The surface also plays a role, as a smoother surface will result in less friction and allow for a smoother and longer rolling motion.

## 5. How is smooth rolling motion used in real-world applications?

Smooth rolling motion is used in many real-world applications, such as in transportation, sports, and machinery. Cars, bikes, and other vehicles use smooth rolling motion to move efficiently and with less energy. In sports, balls and other equipment use smooth rolling motion to perform optimally. Machinery, such as conveyor belts and gears, also utilize smooth rolling motion for efficient movement and energy conservation.