# Forces on the rider in Mechanica

## Acceleration and forces

### The main rotation

- The main rotation in T=8 seconds per turn of the 12m long arm.
- The centre of the star thus moves in a circle with radius R=12m, and a velocity V=2πR/T=9.4m/s = 34km/h.
- This acceleration leads to a centripetal acceleration of the center of the star: a
_{c}=V^{2}/R=0.75g. - Alternatively, the motion can be described in terms of an angular velocity Ω=2π/T≈0.79/s. The centripetal acceleration can then be written as a
_{c}=RΩ^{2}. - For a rider sitting close to the centre of the star - or on a gondola parallel to the axis of rotation - we expect an upward force from the ride of 1.75 mg in the lowest point and 0.25 mg in the highest point.
- For a rider at the end of a 4m long gondola arm perpendicular to the rotation axis, the radius becomes slightly longer, 12.6m. However, the centripetal acceleration in the "vertical" direction is unchanged. The effect on the "lateral" acceleration is discussed under "The star in a rotating coordinate system".

The graph below shows the "vertical" component of the accelerometer graph with slightly larger maximum values, to be discussed below. We also note the damped oscillations at the top (after 60s).

The coordinate axes follow the rider (see the rabbit in the right column).

### The rotation of the star

- The star has a radius d=4m. The star itself makes a full turn in 10s (measured most easily at the top, when the main rotation has stopped for some time), giving an angular velocity ω≈0.63/s.
- The centripetal force due to this rotation is 0.16g towards the center of the star. (The acceleration centripetal acceleration is indicated in turquoise in the photo with the star at the top). This acceleration requires a sideways, "lateral" force on the rider of 0.16mg.
- When the star is in a vertical plane, the lateral forces must also compensate for the force of gravity and we would expect a maximum lateral force of 1.16mg from the right (defined as positive) in the lowest point and a negative lateral force of 0.84mg for a the highest point of the star.
- The graph below of lateral forces shows slightly larger variations, that are related to the main rotation.

### The star in a rotating coordinate system

In the photos to the right we have marked a coordinate system for the star, with the x'-axis pointing in the same direction as the main rotation axis and the z' axis along the main arm, toward the center of the ride.

The main rotation also causes a rotation around the x'-axis diameter of the star, (in addition to the star rotation around the z' axis, perpendicular to the plane of star). This results in an additional centripetal acceleration, directed towards the x-axis, and proportional to the distance to the x-axis. For the highest and lowest part of the star in the photo, this centripetal acceleration of 0.25g is purely lateral. Taking this acceleration into account, we expect a lateral force on the rider can be up to 1.4g from the right (in the bottom of the star) and up to 0.6g from the left (at the top), i.e. -0.6g. These results are consistent with the data in the blue graph above.

### The Coriolis effect in the star

The star itself is also rotating within the rotating x'y'z' coordinate system. This velocity **v' **a of the rider leads to an additional acceleration - the "Coriolis" effect, and is largest, 2Ωωd≈0.40g, when the motion is orthogonal to the main rotation axis. When the star is at the bottom, this acceleration is in the vertical direction, 0.4g upwards on one side and downwards on the other. Combined with a centripetal acceleration of around 0.79g, this brings the maximum upward force on the rider to around 2.2mg (=1+0.75+0.4)mg, which is also observed in the "vertical component", shown in the red graph above.

(See more graphs on a separate page)

### The rotation of the gondolas

Due to the individual rotation of the gondolas the longitudinal and vertical components, are interchanged in the system of the rider. The oscillations may also give an additional centripetal acceleration. Read more.

### Modelling of the motion

On a separate page we present a mathematical model of the motion, assuming constant angular velocities for the main rotation and the rotation of the star. The motion is used to calculate the resulting G-forces, but without taking the gondola motion into account.

### Uneven main rotation

Comparison with the total G-force in the measured data indicate that the forces at the bottom are somewhat underestimated by the model. From the video of the main rotation, we conclude that with the bottom half of a turn takes about 3.6s compared to 4.4s for the top. This implies that the angular velocity at the bottom is at least 10% larger than assumed by a period of 8s for the rotation. The centripetal acceleration due to the main rotation thus becomes at least 20% larger at the bottom (and 20% larger at the top).