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**Roller Coasters: The Clothoid Loop**

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Why the tears??? Have you ever examined a roller coaster’s loop? They will always look more like a tear drop than a circle. This tear drop shape is called a clothoid loop. A clothoid loop is essentially a loop where the radius of curvature changes. Why a clothoid loop? The answer lies in the engineers desire to make the ride thrilling, but SAFE. How does physics relate to the safety of a rider?? gees!!!! Over the next several slides, we will see why this shape has to be.

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**Circular Loop vs. Clothoid Loop Two different shaped loops – same total height**

RBOT = 8.0 m RTOP = 4.0 m h = 12.0 m R = 6.0 m h = 12.0 m Note: We will consider a simplified version of a clothoid loop – one in which the clothoid is made up of two different radii. In this case, both the entrance and exit have radii = 8.0 m. The top has a radius = 4.0 m.

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**Circular Loop – Calculation**

R = 6.0 m h = 12.0 m Circular Loop – Calculation Our ultimate goal is to figure out how many “gees” a rider would experience in a circular loop and compare this to a clothoid loop. Work through this series of questions without looking at the solution. If you get stuck, try clicking on the speaker graphic next to the question for a hint. (Try it on your own first.) When you are finished, you may look at the next few page for the solution. A rider will always feel the largest number of “gees” at the bottom of a loop. Why? Assume that the mass of the rider is 75 kg and use g = 10.0 m/s2. Find the minimum speed that the roller coaster can go at the top of the loop without relying on the seat belt to hold the person up. 3) Find the speed at the bottom of the loop (assuming no friction). 4) Find the normal force at the bottom. 5) Find the # of gees the person experiences at the bottom.

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**Circular Loop - Solution**

R = 6.0 m h = 12.0 m Circular Loop - Solution A rider will always feel the largest number of “gees” at the bottom of a loop. Why? The # of “gees” we feel is related to the normal force that is acting on us. At the bottom of a loop, the normal force must overcome gravity to create an upward acceleration. At the top of the loop, the normal force and gravity are both acting downward so the normal force will be smaller. Assume that the mass of the rider is 75 kg and use g = 10.0 m/s2. Find the minimum speed that the roller coaster can go at the top of the loop without relying on the seat belt to hold the person up.

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**Circular Loop – Solution Cont**

R = 6.0 m h = 12.0 m 3) Find the speed at the bottom of the loop (assuming no friction).

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**Circular Loop – Solution Cont**

R = 6.0 m h = 12.0 m Find the normal force at the bottom. 5) Find the # of gees the person experiences at the bottom.

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**Clothoid – Calculation**

RBOT = 8.0 m RTOP = 4.0 m h = 12.0 m Clothoid – Calculation Now, we would like you to answer the same set of questions except for the clothoid loop. You will compare your solutions at the start of class with your partner(s). Make sure you use the appropriate radius for each question. Again, assume that the mass of the rider is 75 kg and use g = 10.0 m/s2. Find the minimum speed that the roller coaster can go at the top of the loop without relying on the seat belt to hold the person up. 3) Find the speed at the bottom of the loop (assuming no friction). 4) Find the normal force at the bottom, just as the rider is entering the loop. 5) Find the # of gees the person experiences as they enter the loop.

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**A first approximation of friction**

Let’s return to the circular loop problem. Let’s assume that a car 550 kg car enters the circular loop going 17 m/s and exits the loop going 16.5 m/s. What is the average frictional force acting on the car? Listen to this speaker graphic. R = 6.0 m h = 12.0 m

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