# Calculating Friction between Tapered Cylinders: A Math & Engineering Challenge

• rob barbowski
In summary, Friction torque can be minimized by maintaining a clearance between inner and outer buckets.
rob barbowski
Hi everyone: I need the brain of an engineer if anyone out there cares to help. I have a masters in Math from UofT but could use some knowledge from the smartest people- Engineers, I am now an experienced builder and yes everyone, we need smart people doing construction too- my math degree has facilitated my work as most people in construction lack brains...

I would like to know how engineers calculate the force of friction between surfaces but in a not so trivial context: I understand the logic of friction and calculating it when given a mass and a coefficient of static and kinetic friction, but my question is not so simple like OAC Physics I believe.

Q: how do we calculate friction considering surface area with two identical substances, in this case, slightly tapered CYLINDERS. consider this: I have one tapered cylinder inside another that is identical- like stacking buckets or plastic cups. i understand that if mass inside the top bucket increases by filling it with a liquid, it exerts lateral pressure on the sides of the elastic container and the container deforms slightly- this increases the horizontal force exerted on the container it is within/contained and therefore the frictional force between the two objects increases.

How is this force calculated? If i add water to a bucket and its within another bucket, how do we determine the increased frictional force between the two buckets? I think its a function of contact surface area of the objects with the mass of the substance added (uniformly distributed in the bucket) but i am not an engineer... If anyone can tell me how to calculate this, even roughly, it would be greatly appreciated- using math would be helpful as i am quite good at it.

I have a feeling this is a quite complicated to calculate accurately and many factors modify the result but if someone can educate me with a basic way to approximate such a force, it would be great as I am just trying to understand how frictional forces are effected when changing the external surface area contact? would two very tall and small diameter buckets result in less frictional force than two very short and wide buckets stacked?

The volume is constant. I am trying to minimize the friction here and just as we can maximize the volume of a constructed shape given a fixed surface area of material to make the shape, I believe we can also maximize or minimize friction as well but i don't know for certain. To me the horizontal cross section of the cylinder at any given height specifically, and the corresponding lateral force exerted at that height, will be a function of the pressure exerted with gravity on it from the mass above, thereby making the relative lateral force given a height in the container, differential- I am not sure...? I just need know how to change the shape of my object so that when filled and stacked in another identical (but empty) object, the friction between the objects is minimized.If anyone can answer or direct me to finding one of utility, its sincerely appreciated! and If the answer directs me to my ultimate result which i believe is possible, they will receive a lot more and this is seriously my intent- If someone helps me here and I use that help to bring my idea to reality, they will be very happy.Thanks.Rob B.

email: [email address removed by mentor]

Last edited by a moderator:
Small nominally square elements / Interface stresses / Normal forces / Tangential friction forces / Sum of tangential friction forces x radius of action -> Friction Torque .

Problem can actually be solved in practice by several different means and to several different levels of accuracy .

nb : There will be no single value answer . Friction torque can only be predicted to fall within a band of values .

Last edited:
I don't entirely understand the later part of the problem statement . You can reduce friction torque to zero by maintaining a clearance between inner and outer buckets .

If there must be some contact then you would need to look at several different configurations and find an optimal one .

Nidum said:
I don't entirely understand the later part of the problem statement . You can reduce friction torque to zero by maintaining a clearance between inner and outer buckets .

If there must be some contact then you would need to look at several different configurations and find an optimal one .
Agree, and I see this done (and not done) in practice with ~ 5 Gallon plastic buckets. In some cases, there is an outer lip near the top of the bucket that limits how far the top bucket will drop into the lower one. If this provides enough clearance to allow for the top bucket to be filled, but not deform enough to touch the sides of the lower bucket, there is no friction from the sides.

When not done, it can be a bear to separate the two buckets.

So a combination of defining where the lip needs to be for a given clearance, and designing the bucket for a given deformation when filled that is less than that clearance should get you near zero friction.

Alternately, it might help to provide one or more inlet(s) on the bottom bucket/sloped-cylinder, and pump air or liquid into the space. That pressure should help separate the two.

There are plenty of similar friction problems featuring masses on slopes or wedges and the solution to this question will be similar. Just calculate the normal force between the two tapered cylinders. Its probably just something like

Normal force = axial force * cos(the taper angle)

The axial force being mg where m is the mass of the top "bucket".

In the most common model of friction (F=μN) the friction force is independent of contact area (If the area is increased the pressure reduces and these effects are usually considered to cancel out). However this is only a model of how friction works, it's not derived from first principles. Other models may be more accurate in some cases - for example it's widely believed that reducing the pressure in car tyres gives you more grip on snow because it increases the contact area. If you ever need reliable results from a friction problem you have to do experiments. Airports usually measure runway friction, I don't think they try and calculate it.

## 1. What is the purpose of calculating friction between tapered cylinders?

The purpose of calculating friction between tapered cylinders is to determine the amount of resistance or force that is generated when two cylinders with different diameters come into contact with each other. This information is important in engineering and design as it can affect the performance and durability of mechanical systems.

## 2. How is friction between tapered cylinders calculated?

Friction between tapered cylinders can be calculated using the formula: F = μ*N, where F is the force of friction, μ is the coefficient of friction, and N is the normal force between the two cylinders. The coefficient of friction is dependent on the materials and surface properties of the cylinders.

## 3. What are some factors that can affect the friction between tapered cylinders?

Some factors that can affect the friction between tapered cylinders include the material and surface properties of the cylinders, the amount of force or pressure applied, the angle at which the cylinders come into contact, and the speed of the cylinders.

## 4. How can the calculated friction between tapered cylinders be used in engineering and design?

The calculated friction between tapered cylinders can be used in engineering and design to determine the appropriate materials and surface coatings to reduce friction and improve the performance of mechanical systems. It can also help in selecting the right amount of force and pressure to achieve optimal performance and prevent excessive wear and tear on the cylinders.

## 5. Are there any real-world applications for calculating friction between tapered cylinders?

Yes, there are many real-world applications for calculating friction between tapered cylinders. For example, it can be used in the design of gears and bearings in machinery, as well as in the automotive industry for determining the amount of friction between engine components. It can also be applied in the design of prosthetics and medical devices to ensure smooth and efficient movement.

• Classical Physics
Replies
37
Views
2K
• Mechanics
Replies
4
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
575
• Classical Physics
Replies
2
Views
1K
• Mechanical Engineering
Replies
9
Views
1K
• Classical Physics
Replies
5
Views
1K
• General Engineering
Replies
12
Views
3K
• General Engineering
Replies
25
Views
9K
• Mechanics
Replies
32
Views
1K
• Introductory Physics Homework Help
Replies
18
Views
2K