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Centripital Force
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The varying forces during a roller coaster ride help make it thrilling. These forces can cause you to feel weightless at times, and press you into your seat or push you laterally from side to side. However, as with most things in life... too much of a good thing can be bad.
In roller coaster jargon, forces are usually given in terms of a factor called "G's". This is just a way of measuring a force compared to the normal force of gravity. The "G Force" is calculated by simply dividing the force by the force due to gravity:
G Force = G = (F)(1/mg)
m = mass (Kg)
g = gravity constant (9.82 m/s²).
For example, if you were simply standing on the ground, the force being exerted on you from gravity is equal to your weight, and you would be experiencing 1 G. If the force were 2 G's, it would be twice as strong as gravity, and you would seem to weigh twice your normal weight.
If the forces during a roller coaster ride get excessive, they can cause discomfort -- and if large enough, can cause injury. They can also cause physical problems. For instance, when performing extreme flight manuevers, a typical "Top Gun" jet pilot may begin to lose consciousness if subjected to a sustained force of 5 G's or more.
When experiencing circular motion in a loop, the major force you feel is Centripital Force -- which is the force that presses you into your seat.
Note the spelling of the word for the force. As described by Newton's Third Law of Motion, for every force on an object, there is an equal and opposite force exerted by the object. For a roller coaster car in a circular loop, the force that pushes on you (from the track) is called "Centripital Force". The equal and opposite force that the car exerts on the track is called "Centrifugal Force". When analysing roller coaster forces, we are generally interested in the forces acting on us -- that is, the Centripital Force.
Consider the case of a roller coaster car going through a loop that is a simple circle of constant radius, as shown in the diagram.
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The Centripital Force is equal to:
Centripital Force = F = (mv²)(1/R)
m = mass (Kg)
g = gravity constant (9.82 m/s²)
v = velocity (m/s)
R = loop radius (m).
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The centripital force pushes you toward the center of the loop. It is the force that causes you to move in the circular path. To use the more precise physics terms: the centripital force causes the acceleration resulting in circular motion.
Note that centripital force is inversely proportional to the loop radius. Therefore, the smaller the loop, the larger the force, and vice versa.
If you look closely at any real roller coaster, you will notice that the loops are not simple circular arcs. For most real-life loops, the radius of the arc varies in a complex way, with the radius being smaller near the top of the loop. The name for this type of geometric shape is a "Klothoid" (or sometimes spelled "Clothoid").
With
Coaster Dynamics,
the loops can be modeled as either simple circular arcs, or as "Teardrop Loops".
In this case, we use a simple circular arc, since it makes the mathematical analysis of the loop forces easier.
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Apply Your Knowledge
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Your Task:
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Knowing the initial velocity of the roller coaster car as it enters the loop, what will be the largest loop radius for a maximum G Force of 4.5 G's?
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What We Know:
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For our simplified analysis, we will ignore the affect of friction.
The known parameters are:
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mass of the car = 453.59 kg
g = 9.82 m/s²
maximum G Force = 4.5 G.
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Use the "Coaster Lab" Option in
Coaster Dynamics,
to determine v, the initial velocity of the car as it enters the loop.
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What To Do:
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Using the equations below, solve for the loop radius, R, at which the G Force will be 4.5 G's:
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1) F = (mv²)(1/R)
2) G = (F)(1/mg).
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To test your work, change the height of the loop to your calculated value, and then re-run the roller coaster ride.
Even though you did not include the affect of friction in your calculations, remember that the
Coaster Dynamics
simulation does include friction. So, where do you expect the maximum G Force to occur?
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