# Coefficient of rolling friction for a lab cart

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• Einstein44
In summary, the conversation discusses the calculation of friction arising from the axle and wheel of a lab cart, referencing a research paper that derived an equation for the coefficient of rolling friction. The first part of the equation is referred to as rolling resistance, while the second part is the friction force from the fixed axle-wheel contact patch. The conversation also addresses the use of the equation to find the frictional force and the potential impact of non-elastic effects on the force. The conversation concludes by discussing the applicability of the equation to different types of wheels and surfaces.
Einstein44
I was looking for a way to calculate the friction arising from the axle and wheel of a standard lab cart. I came across this research paper: https://www.usna.edu/Users/physics/mungan/_files/documents/Publications/PhysEd4.pdf
That derived the following equation for the coefficient of rolling friction:
$$\mu _{r}\approx \frac{D+kr}{R}$$
for what the variables stand can be seen on the diagram shown in the article (it would be too difficult to explain this with words).
Now my problem is that I had never heard of a such thing and want able to find much on the internet that explained it in detail. I wasn't able to follow the derivation exactly, but I think some people here will find it easier to follow.
My question is: is this really a correct indicator of the friction for a cart moving down a ramp, and how can this be used to find the frictional Force? Can you just simply use the same approach as for dynamic friction for instance? (just multiplying it by the Normal Force?)

The first part ##D /R## is usually referred to as rolling resistance. By definition (sum of moments):
$$f R = ND$$
or:
$$\frac{D}{R} = \frac{f}{N}= C_{rr}$$
Where ##C_{rr}## is the coefficient of rolling resistance. So there is no friction, but since the resistance force has a definition similar to a friction force, they are used in a similar way.

The second part ##k r/R## is the friction force from the fixed axle-wheel contact patch ##f_a = kN## as felt at wheel-road contact patch, ##f_w##. Again sum of moments:
$$f_a r = f_w R$$
or:
$$f_w = \frac{f_a r}{R}= \frac{k r}{R}N$$
$$\mu_r N = f + f_w = C_{rr}N + \frac{k r}{R}N = \frac{D}{R}N + \frac{k r}{R}N = \frac{D+k r}{R}N$$
$$\mu_r = \frac{D+k r}{R}$$

Einstein44 and Lnewqban
@jack action, why is N off center and what distance D depend on?

@Einstein44, consider that the values of N will be different for front and rear wheels, due to the slope.

jack action said:
The first part ##D /R## is usually referred to as rolling resistance. By definition (sum of moments):
$$f R = ND$$
or:
$$\frac{D}{R} = \frac{f}{N}= C_{rr}$$
Where ##C_{rr}## is the coefficient of rolling resistance. So there is no friction, but since the resistance force has a definition similar to a friction force, they are used in a similar way.

The second part ##k r/R## is the friction force from the fixed axle-wheel contact patch ##f_a = kN## as felt at wheel-road contact patch, ##f_w##. Again sum of moments:
$$f_a r = f_w R$$
or:
$$f_w = \frac{f_a r}{R}= \frac{k r}{R}N$$
$$\mu_r N = f + f_w = C_{rr}N + \frac{k r}{R}N = \frac{D}{R}N + \frac{k r}{R}N = \frac{D+k r}{R}N$$
$$\mu_r = \frac{D+k r}{R}$$
Thank you! On the website you have linked in your comment above I have found the following equation:
$$F=\frac{Nb}{r}$$
Would this then give me the force due to the rolling resistance directly? That means I wouldn't need to use the equation above? Because essentially I am trying to find out the Frictional Force.

Lnewqban said:
@jack action, why is N off center and what distance D depend on?
https://en.wikipedia.org/wiki/Rolling_resistance said:
It is mainly caused by non-elastic effects; that is, not all the energy needed for deformation (or movement) of the wheel, roadbed, etc., is recovered when the pressure is removed. Two forms of this are hysteresis losses (see below), and permanent (plastic) deformation of the object or the surface (e.g. soil).

Einstein44 said:
Would this then give me the force due to the rolling resistance directly?
Normally, you used ##F = C_{rr} N## as the values of rolling resistance coefficients are usually easy to find. I never saw a source for ##b## except for the one I cited (and there are not a lot except for steel-on-steel).

The equation from the OP is usually used by more advanced designs like bearing manufacturing for research purposes.

Einstein44 and Lnewqban
jack action said:
Normally, you used ##F = C_{rr} N## as the values of rolling resistance coefficients are usually easy to find. I never saw a source for ##b## except for the one I cited (and there are not a lot except for steel-on-steel).

The equation from the OP is usually used by more advanced designs like bearing manufacturing for research purposes.
Yes, I understand what you wrote. However, I feel like this mainly applies to wheels that deform on the surface, such as a deflated tire for example (correct me if I'm wrong), so how can you apply any equation to find the Frictional Force for basically a hard wheel on another surface? (besides approximating D from the first equation to be zero, due to no deformation of the wheel)

Every material deforms, none are purely inelastic. For example, a railroad steel wheel on a steel rail has ##C_{rr}= 0.0003##. This means that for a wheel with a 12" radius, the distance ##D## is 0.0036" or equivalent to about the diameter of a hair!

Einstein44 and Lnewqban
Einstein44 said:
Yes, I understand what you wrote. However, I feel like this mainly applies to wheels that deform on the surface, such as a deflated tire for example (correct me if I'm wrong), so how can you apply any equation to find the Frictional Force for basically a hard wheel on another surface? (besides approximating D from the first equation to be zero, due to no deformation of the wheel)
If the surface is soft you have a dent, and the normal forces on this slopes have a net backwards component.

Lnewqban and Einstein44

## What is the coefficient of rolling friction?

The coefficient of rolling friction is a measure of the resistance to motion between a rolling object and the surface it is rolling on. It is typically represented by the symbol μr.

## How is the coefficient of rolling friction determined?

The coefficient of rolling friction can be determined experimentally by measuring the force required to keep a rolling object moving at a constant speed on a flat surface. This force is then divided by the weight of the object to calculate the coefficient of rolling friction.

## What factors affect the coefficient of rolling friction?

The coefficient of rolling friction can be affected by several factors, including the type of surface the object is rolling on, the weight and shape of the object, and the speed at which it is rolling. Additionally, the presence of any lubricants or contaminants on the surface can also impact the coefficient of rolling friction.

## How does the coefficient of rolling friction differ from the coefficient of static friction?

The coefficient of rolling friction is a measure of the resistance to motion for an object that is already in motion, while the coefficient of static friction is a measure of the resistance to motion for an object that is at rest. This means that the coefficient of rolling friction is typically lower than the coefficient of static friction for the same object and surface.

## Why is understanding the coefficient of rolling friction important?

Understanding the coefficient of rolling friction is important for engineers and scientists in various fields, such as transportation and manufacturing, as it can impact the design and efficiency of machines and vehicles. It is also important in everyday life, as it can affect the ease of movement for objects such as shopping carts and suitcases.

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