# Friction between roller coaster and track

• LaszloNagy
In summary: It's definitely too hard to learn in one day. See my suggestion above. I think the circular track is supposed to start at the bottom and curve behind the initial slope.
LaszloNagy

## Homework Statement

A roller coaster is going down on a track (see image below, the track itself is highlighted in red) without initial velocity. What is the coefficient of friction between the roller coaster and the track?

height at which the cart starts (h) = 12m
weight of the car (m) = 620kg
angle of the track (α) = 40°
radius of the circle (r) = 8m
normal force acting on the cart in the moment it reaches the lowest point (N) = 22,900N

## Homework Equations

I managed to calculate the energy lost during the process, but I have no idea what to do next.

## The Attempt at a Solution

I calculated the energy of the cart at the highest and lowest points (hopefully my thought process is visible on the image) and the difference of the two is the energy lost due to friction.

Okay, so you've got the energy loss. Where did that energy go?

All of that is lost through friction. I just don't know how to figure out the coefficient, since N is changing throughout the movement

LaszloNagy said:
All of that is lost through friction. I just don't know how to figure out the coefficient, since N is changing throughout the movement

Okay, I was wondering what you thought about that! Do you think you are supposed to make some assumption? If this is "introductory" physics, it looks a little hard. If not, then you'll have to integrate round the curved bit of the track, which isn't so easy. Do you know the maths to do that?

My high school teacher gave this problem so I thought it's introductory. I don't know the maths for that, but I think it has something to do with finding the area below a curve. I can look it up if that's what it takes to solve this. Could you give me some hints on how to go on from here?

LaszloNagy said:
My high school teacher gave this problem so I thought it's introductory. I don't know the maths for that, but I think it has something to do with finding the area below a curve. I can look it up if that's what it takes to solve this. Could you give me some hints on how to go on from here?

First, you need to do a bit of geometry. You're diagram is not bad, but note that if ##r = 8m## and ##h = 12m## then the circle is much bigger if drawn to scale. You need to work out how far down the slope the point of intersection is. Hint: first find the angle subtended at the centre of the circle.

Then, you'll need an expression for the normal force at each point of the curve and you'll have to integrate that round the curve.

If there's a quick way, I don't see it.

Here's an idea: redraw your diagram so that the straight slope finishes just at the ground, where the circular track begins. Don't have the straight track meet the circular track above the ground. That makes it easier and I think that's the problem that was intended.

I could calculate that the circle and the straight part of the track meet at a height of 1.9m
The function for the normal force is:
f(x) = gravitational force*cos(x)+centripetal force or in terms of numbers f(x) = 6200*cos(x)+16740
x is between 0°(at the lowest point of the circle) and 40°(where the circle becomes part of the track). I tried to understand integrals, but it seems like a much harder topic than I could cover in one day so I'm stuck at this point

LaszloNagy said:
I could calculate that the circle and the straight part of the track meet at a height of 1.9m
The function for the normal force is:
f(x) = gravitational force*cos(x)+centripetal force or in terms of numbers f(x) = 6200*cos(x)+16740
x is between 0°(at the lowest point of the circle) and 40°(where the circle becomes part of the track). I tried to understand integrals, but it seems like a much harder topic than I could cover in one day so I'm stuck at this point

It's definitely too hard to learn in one day. See my suggestion above. I think the circular track is supposed to start at the bottom and curve behind the initial slope. Like a real rollercoaster!

Oh, I think there is a misunderstanding. The car doesn't travel all the way around the circle. The circle is just there to show the curve of the track after the straight part. After it reaches the lowest point it doesn't go up, it continues straight, but that's not part of the problem anyways. So the whole movement is just going down the slope and the small curve of the circle where the track becomes flat. Sorry if I phrased it wrong, English isn't my primary language

LaszloNagy said:
Oh, I think there is a misunderstanding. The car doesn't travel all the way around the circle. The circle is just there to show the curve of the track after the straight part. After it reaches the lowest point it doesn't go up, it continues straight, but that's not part of the problem anyways. So the whole movement is just going down the slope and the small curve of the circle where the track becomes flat. Sorry if I phrased it wrong, English isn't my primary language

If it doesn't go up and start a loop, then there would be no centripetal acceleration.

How should I approach the problem then?

LaszloNagy said:
How should I approach the problem then?

Just assume the slope goes (almost) all the way to the bottom. This minimises the length of the circular track before it gets to the bottom and means that the length of curved track with variable normal force can be neglected. Effectively assume the track is straight all the way to the ground, where the circular upward loop begins.

Seems to me your teacher has set a far more difficult problem than intended. For the curved section, I get an equation of the form ##yy'=A\sin(x)-\mu A \cos(x)-\mu y^2## where A=g/r. I very much doubt that can be solved.
PeroK's approximation in post #14 looks as good as anything.

haruspex said:
Seems to me your teacher has set a far more difficult problem than intended. For the curved section, I get an equation of the form ##yy'=A\sin(x)-\mu A \cos(x)-\mu y^2## where A=g/r. I very much doubt that can be solved.
PeroK's approximation in post #14 looks as good as anything.
Correction: the differential equation can be solved, but the resulting equation for mu involves a mixture of functions which can only be solved by numerical approximation.

## 1. What is friction between a roller coaster and its track?

Friction between a roller coaster and its track is the resistance or force that occurs when two surfaces come into contact with each other. In this case, it is the resistance between the wheels of the roller coaster and the track it runs on.

## 2. How does friction affect a roller coaster ride?

Friction can affect a roller coaster ride in several ways. It can slow down the speed of the roller coaster, causing it to lose momentum and potentially affect the thrill of the ride. It can also create heat and wear on the wheels and track, which can impact the overall safety and maintenance of the roller coaster.

## 3. What factors can affect the amount of friction between a roller coaster and its track?

The amount of friction between a roller coaster and its track can be affected by several factors, such as the material and texture of the track and wheels, the force of gravity, the weight and speed of the roller coaster, and any external factors such as weather conditions.

## 4. How is friction between a roller coaster and its track calculated?

The calculation of friction between a roller coaster and its track involves the coefficient of friction, which is a measure of the roughness or smoothness of the surfaces in contact. The force of friction can then be determined by multiplying the coefficient of friction by the weight of the roller coaster.

## 5. How do engineers reduce friction in roller coasters?

Engineers use several methods to reduce friction in roller coasters, such as using smooth track surfaces and wheels, applying lubricants, and designing curves and turns to minimize friction. Regular maintenance and inspections are also crucial in identifying and addressing any potential sources of friction in a roller coaster.

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