What makes amusement park rides so much fun is the forces your body experiences when you're on them. There are turns, twists, and rapid acceleration. It's quite different from what we experience on a daily basis. But it is precisely these unusual sensations of having your body pushed and pulled in different directions, that keeps people coming back for more.

However, designing these rides is much more than just putting in random loops on a track. A solid understanding of the physics is necessary for the designers in order to push the safety limits for humans as much as possible. So having an idea of how much force the body will experience on a ride is a key factor when deciding how fast, how high, or how big a radius is required.

Click on the links below to learn about the physics involved in these particular rides.

Ferris Wheel Physics

Roller Coaster Physics

Another popular amusement park ride is the Gravitron. In this ride people lean against the external wall and the force generated by centriptetal acceleration, during rotation, keeps the riders from sliding down the wall. The figure below shows a schematic of the ride.

Where:

The coordinate system of the ride is given as shown and it is assumed that the ride is horizontal.

We wish to determine the minimum rotational speed (

Let us first determine the minimum rotational speed such that the rider does not slide down the external wall.

Apply Newton's second law to the rider, in the x-direction:

where

Apply Newton's second law to the rider, in the y-direction:

The limiting case for slipping occurs when the friction force

Combine equations (1) - (3) and solve for

This equation is only valid for cos

Next, let's determine the maximum rotational speed such that the rider does not slide up the external wall. For this case we solve the same equations as given above but the force

This equation is only valid for sin

There are many other fun rides at amusement parks, dozens in fact, and many of them exhibit complex three-dimensional motion which is difficult to analyze from a physics perspective. Nevertheless I took it upon myself to create three Excel spreadsheets which can be used to analyze most of the three-dimensional rides found at amusement parks. The way to use these spreadsheets is to first match a ride to one of these spreadsheets, and then input the corresponding parameters of the ride into the spreadsheet. The spreadsheet will then calculate the maximum g-force experienced by a rider on the ride. The maximum g-force is an important value since it tells you what the maximum force experienced by a rider is on the ride. This can be very useful (and interesting) information when deciding which rides you want to avoid (or go on), depending on your personal preferences.

These three spreadsheets are categorized into three separate cases, which as mentioned, can be used to analyze most of the three-dimensional rides found at amusement parks. I describe these three cases below, and provide the corresponding spreadsheets to use when analyzing them.

Case 1

A general schematic of this ride is given in the figure below, with the

The parameters shown in this figure are defined as follows:

The angular velocity vector

The angular velocity vector

The angular velocity vector

The angular velocity vector

Note that the angular velocity vector

The rider is located at point P.

Case 2

A general schematic of this ride is given in the figure below, with the

The parameters shown in this figure are defined as follows:

The angular velocity vector

The angular velocity vector

The angular velocity vector

The angular velocity vector

The angular velocity vector

Note that the angular velocity vector

The rider is located at point P.

Case 3

A general schematic of this ride is given in the figure below, with the

The parameters shown in this figure are defined as follows:

The angular velocity vector

The angular velocity vector

Note that the angular velocity vector

The rider is located at point

The Excel spreadsheets used for analyzing all three cases can be downloaded here:

Case 1

Case 2

Case 3

The spreadsheets are in compressed "zip" format and you have to uncompress them before you can use them.

The following are example solutions for different amusement park rides. These are obtained using the spreadsheets for the three cases.

Case 3 can be used to analyze this ride.

From observation we have the following ride parameters (approximate):

Mass of

Rope length

Wheel radius

Wheel spin rate

Precession rate

Arm length

From the spreadsheet for Case 3, the calculated maximum g-force experienced by a rider at point

Note: We are assuming that the gondolas (containing the riders) can be modeled as a point mass (

Case 2 can be used to analyze this ride.

From observation we have the following ride parameters (approximate):

Precession rate

Precession rate

Wheel spin rate

Elbow CBO spin rate

Arm length

Arm length

Arm length

Wheel radius

Gravity is pointing in direction 6 (the -

From the spreadsheet for Case 2, the calculated maximum g-force experienced by a rider at point P is approximately 2.8

Case 3 can be used to analyze this ride.

From observation we have the following ride parameters (approximate):

Mass of

Rope length

Wheel radius

Wheel spin rate

Precession rate

Arm length

From the spreadsheet for Case 3, the calculated maximum g-force experienced by a rider at point

Case 1 can be used to analyze this ride.

From observation we have the following ride parameters (approximate):

Precession rate

Precession rate

Wheel spin rate

Arm length

Arm length

Wheel radius

Gravity is pointing in direction 6 (the -

From the spreadsheet for Case 1, the calculated maximum g-force experienced by a rider at point P is approximately 2.1

Case 3 can be used to analyze this ride.

From observation we have the following ride parameters (approximate):

Mass of

Rope length

Wheel radius

Wheel spin rate

Precession rate

Arm length

From the spreadsheet for Case 3, the calculated maximum g-force experienced by a rider at point

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