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The Starship 2000
J. Witthuhn, D. Tedeschi
University of South Carolina

Introduction:
            The Starship 2000 is a ride at the South Carolina State Fair’s Midway Physics Day. It is a ride where the rider is spun in a circular motion (s. Fig. 1). Throughout the ride, the rider is on a sled, which is against an angled wall as shown in Figure 2. Once the ride is spinning fast enough, the rider will slide up the ramp on the sled. Data was recorded using a three-axes accelerometer. Figure 2 also shows the orientation of the positive x, y, and z-axes as well as the angle of the wall. Does the angle of the wall effect the forces felt by the rider? This ride offers an excellent opportunity to study the nature of the forces as formulated in Newton’s Second Law.

 
Fig. 1:
Fig. 2:
  A diagram of the forces acting  on a person in circular motion
The orientation of  the accelerometer to the
wall of the Starship 2000.

                                              
                                                                                                                                 
                                                                         

Centripetal Force:
During this ride, the rider experiences only two forces: the gravitational force (Fg) and the centripetal force (Fc). It is the centripetal force, which is responsible for the enjoyment of this ride. However, what exactly is centripetal force? To better understand centripetal forces, imagine a ball attached to a string. When the ball is spun in circular motion, what keeps it traveling in a circle? Centripetal force, which is provided by the hand pulling on the string, keeps the ball in its circular path. All objects in circular motion experience this force. In addition, the centripetal force is directed towards the center of the circular path the object is in. In the ball analogy, the hand provides the centripetal force, but in the Starship 2000, the wall provides the centripetal force.
 
 
 
Ride Data:
The data used to study the forces on the Starship 2000 was collected using three-axes accelerometers. As Figure 2 shows, the wall of the ride is not perpendicular to the floor of the ride. The tilting causes a redistribution of forces between the axes of the accelerometer, which will be discussed in more detail in a later section, but this redistribution does not change the total force felt by the rider. To relate the forces a person undergoes, it is common to use the force of earth’s gravity as a reference. Newton’s Second Law states that the force on an object is the product of the mass and the acceleration of that object. Everyone is under the influence of earth’s gravity, which for our purpose has a constant acceleration (g) of 9.8 m/s2. By dividing the centripetal force (Fc) by the gravitational force (Fg), the centripetal force a person is exposed to can be measured in multiples of g. For the purpose of analyzing the data on the Starship 2000, the masses in the equations below are equivalent therefore they can be canceled out. This allows the forces in the Starship 2000 to be analyzed using only the accelerations recorded by the accelerometer and the acceleration due to gravity (g). Using the acceleration recorded in the z-axis in Figure 3 as the dominating part of centripetal acceleration that is felt by the rider and the formula below, the maximum number a g’s a rider experiences during this ride can be approximated, which once calculated is approximately 2 g’s.

Fig. 3:
The acceleration recorded in each axis plotted against the time of the ride.

Calculations:
           
How does the angle effect the analysis of the Starship 2000?
Figure 3 shows the acceleration recorded in each axis plotted against the time of the ride. Using this data, the angle of the accelerometer can be calculated. Had the wall been perpendicular to the floor, the x-axis would have recorded only the acceleration due to gravity throughout the ride, while the z-axis would have recorded only the centripetal acceleration. As Figure 4 shows, the x and z-axes are reoriented due to the tilting of the wall. The data recorded in these axes varies throughout the ride because the tilting of the wall causes the axis to be influenced by both the centripetal force and the gravitational force (s.Fig.5). After the ride stopped (),  the angle of the accelerometer can be determined using the diagram in Figure 4 and the formula below. Using this formula, the angle (ø) was found to be 87 degrees. Why does the rider feel more force against a perpendicular wall? People’s bodies become accustomed to gravity acting on their bodies. If a person experiences a new force on their body, it can be an exhilarating, as in the Starship 2000, or frightening, falling down. If a person were to lie down against the floor of the Starship during the ride, the only force the person would feel would be their head pressing into the wall while gravity kept them against the floor of the ride. This would not be a very fun situation, but if the rider were standing against the tilted wall, the ride becomes more interesting. The Starship 2000 allows the rider to slide up the wall on a sled. By allowing the rider to slide up the wall, he is no longer pressed against the ground under the influence of gravity. (Some versions of this ride use a perpendicular wall (s. Fig. 1) and remove the floor from the rider’s feet which makes friction responsible for keeping the rider from falling to the ground.) It is the action of foreign forces acting on the rider’s body combined with the “loss of gravity” which makes this ride so enjoyable and explains why a rider feels more force against a perpendicular wall.

Fig.4:
The forces on the rider as oriented with the accelerometer.
 

 
 

 
Resultant Acceleration
To better analyze all the forces on the rider, the resultant acceleration (aR= v2/r) needs to be determined. The equation below is used to determine the resultant acceleration, which is plotted against time in Figure 6. As stated previously, the Starship 2000 is under the influence of only two forces: gravitational and centripetal. Gravitational acceleration is constant, but centripetal acceleration is proportional to the square of the velocity (ac=v). The equation below also shows how the resultant acceleration is not influenced by the angle of the wall or the accelerometer; the gravitational and centripetal accelerations are redistributed only along the x and z axes, but by taking the sum of the square of each acceleration value, the total acceleration due to each force is accounted for. The figure below shows how much the acceleration changes during the ride, which implies a large contribution of centripetal acceleration to the resultant acceleration. The change in velocity can also be analyzed from Figure 6. To interpret the acceleration versus time graph in Figure 6, the positive slope shows an increase in velocity (speeding up), the negative slope shows a decrease in velocity (slowing down), and zero slope, shows no change in speed.
 

 

Fig.5:
The resultant acceleration plotted against time

 
 
 
What is the angle of the wall?
The angle of the wall can be calculated even though the angle of the accelerometer was positioned at a different angle than shown in Figure 2. Using the resultant acceleration and the relationship ac=gtanø (this relationship is proved in the next section), it is possible to calculate the angle of the wall. The angle at which the accelerometer was placed does not effect the centripetal acceleration on the rider as explained earlier.  The total centripetal acceleration must be calculated to determine this angle. Determining the centripetal acceleration is similar to the method used to find the resultant acceleration, however, the gravitational acceleration must be taken away from the resultant acceleration to determine the centripetal acceleration. Since there are only two forces contributing to resultant acceleration, the centripetal acceleration can be found by taking the difference of the square of the resultant acceleration and the gravitational acceleration as shown below. Using the formulas below and the data recorded in Figure 3, the angle of the wall is approximately 65º from the horizontal.
 

 

 
 
When will the ramp slide up the wall?
To slide up the wall, the force pulling the sled up the wall needs to be greater than the force pulling the sled down the wall. Since the wall is tilted (s. Fig. 2), only a potion of the gravitational force is pulling the sled down the wall. Also, the force pulling the sled up the wall is a portion of the centripetal force. Using vector composition, the force pulling the sled downward (s. Fig. 6) is given by Fx=mgsinø, while the force pulling the sled up the wall (s. Fig. 6) is given by F1=maccosø. Once the force pulling the sled up the wall (F1) is equal to the force pulling the sled down the wall (Fx), the sled is in equilibrium. The equilibrium point is important because when the force pulling the rider up overcomes the force pulling the rider down, the rider loses all contact with the ground. This is a foreign feeling to the rider, because the rider’s body has become accustomed to gravity keeping him down, which he no longer fells at this time, so that when the person feels this new force, which normally does not act on his body, it can be exciting or fun. To determine the point at which the forces are in equilibrium, the velocity (v) at the equilibrium point needs to be calculated.  The formulas below show how this velocity is derived. Using the formula for this velocity and the angle of the accelerometer, the critical velocity was calculated to be 27 m/s. Using this velocity and the radius of the ride to be 4 m with the angular frequency formula (v=wr), the frequency is calculated to be 1 revolution per second. This seems to be a high frequency for such a large object.  This implies that the angle of the accelerometer cannot be used to determine the critical velocity. The angle of the wall is needed to calculate the critical velocity. By using the angle of the wall, which was calculated in the previous section, the critical velocity is 9.24 m/s. This velocity gives an angular frequency of 0.37 revolutions per second. This is a much more reasonable result.
 
 
 
 

Fig.6:
The forces on the sled and their contributors.

 
 
 





 
 
 Future Analysis:
            Although all the physics, which we wanted to analyze, was analyzed, there is room for improvement in the accuracy of the data. In calculating the critical velocity, the radius of the ride is necessary. The radius used in this calculation is a poor estimate of the actual radius of the ride. To obtain a more precise measurement of the radius, a more accurate method is needed. The floor of the ride has approximately the same circumference as the skirt covering the base of the ride. Stringing a rope around this skirt and measuring the length of this rope, would allow one to calculate the radius of the ride using the formula for circumference (C=2pr). The placement of the accelerometer directly against the wall of the ride would also reduce those errors in the data analysis that depend on these angles.  The accelerometers had to be placed in a hip sack for safety reasons. This allowed the accelerometer to be moved around during the ride. If the accelerometer were allowed to be placed directly against the wall behind the rider, much of the error from the motion of the accelerometer would be reduced. This would also allow one to calculate the angle of the wall using the formula aztanø=ax, and would therefore, give two methods for determining the angle of the wall, which would allow a comparison between these two results.


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This page updated 24 Aug. 2001 by David J. Tedeschi
and copyright © 2001, the Board of Trustees of the University of South Carolina.