Determining Minimum Height to Complete a Loop

In summary, the question asks for the minimum height of a slope needed for a cart to safely complete a loop with a given radius and combined mass. The solution involves using the Conservation of Energy formula and Newton's Second Law. The correct answer is 2.5 times the radius. The asker expresses surprise at the approach suggested in Hint #3 and doubts their own understanding of circular motion.
  • #1
crushedcorn
7
0
This question was posted on Khan Academy. I attempted it after watching all of their great Work and Energy videos, but the way they arrived at the answer was way above my skill level even after reviewing the videos a number of times. Here is the question:

What is the minimum height h of a slope that must be constructed in order for a rider in the cart with a combined mass m = 80.0 kg to complete the loop with a radius r = 10.0 meters safely, assuming a frictionless trip?

4e906cd94e8d150a1b45f273793579c77b38f5e6.png

How would you answer this question? The way that they arrived at their answer is below. Is there any alternative/more concise way to arrive at the same answer? I would have never guessed to proceed in the way noted in Hint #3.

We can use the Conservation of Energy formula to evaluate these two points: at the top of the hill and at the top of the loop.

caffd013cb948bf84803ad975be9a21722638d0a.png

Hint #2
At the top of the hill, the kinetic energy is zero, and the potential energy is mgh.

863284b2d565dbf72b029fb3e30cc3989ddc2627.png

Hint #3
At the top of the loop, we must derive another formula to connect ideas together by using Newton’s Second Law to describe the forces acting on the cart and then to solve for v:

d26b49c5ae2f46c757736dc22d56742fe0011f3c.png

Hint #4
This expression is for the velocity at the top of the loop since there is no normal force only at that point. Additionally, h = 2r such that the potential energy is mg(2r).

40419bee3dd2001dd8df7d462b1a36a7f664028c.png

Hint #5
The correct answer is 2.5. Since the radius of the loop is 10 meters, then 2.5r is 25 meters.
 
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  • #2
crushedcorn said:
This question was posted on Khan Academy. I attempted it after watching all of their great Work and Energy videos, but the way they arrived at the answer was way above my skill level even after reviewing the videos a number of times.

I believe this is the easiest and smallest way to answer this question - I would certainly have solved the question this way .

Does this answer your doubt ?

crushedcorn said:
I would have never guessed to proceed in the way noted in Hint #3.

Do you know your circular motion ?
 

1. How do you determine the minimum height needed to complete a loop on a roller coaster?

In order to determine the minimum height needed to complete a loop on a roller coaster, you need to consider several factors such as the speed of the roller coaster, the radius of the loop, and the force of gravity. Using these factors, you can calculate the minimum height needed to generate enough centripetal force to keep the riders safely on the track.

2. What is the formula for determining minimum height for a loop on a roller coaster?

The formula for determining the minimum height for a loop on a roller coaster is h = r + v2/2g, where h is the minimum height, r is the radius of the loop, v is the speed of the roller coaster at the bottom of the loop, and g is the force of gravity (9.8 m/s2).

3. Can the minimum height for a loop be different for different roller coasters?

Yes, the minimum height for a loop can vary for different roller coasters. It depends on the speed, radius, and design of the loop. Some roller coasters may have a higher minimum height requirement to ensure the safety of the riders, while others may have a lower minimum height due to their design and speed.

4. Are there any other factors that can affect the minimum height for a loop on a roller coaster?

Aside from speed, radius, and gravity, there are other factors that can affect the minimum height for a loop on a roller coaster. These include the weight and size of the riders, the type and condition of the track, and the amount of friction between the wheels and the track. These factors should also be taken into consideration when determining the minimum height for a loop.

5. How important is it to determine the minimum height for a loop on a roller coaster?

Determining the minimum height for a loop on a roller coaster is crucial for the safety of the riders. If the minimum height is not met, there may not be enough centripetal force to keep the riders safely on the track, resulting in a potential accident. It is also important for the overall design and performance of the roller coaster to ensure a thrilling and enjoyable experience for the riders.

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