Roller coaster loop searching for Height

In summary, the problem involves a 100kg mass on a frictionless track that extends into a loop with a radius of 20m. The task is to find the minimum height for the mass to make it around the loop without falling off or going backwards. By using conservation of energy and the equation for centripetal force, it is determined that the minimum height is 50m from the top of the loop.
  • #1
gggorillaz
6
0

Homework Statement


A mass of 100kg is h meters high on a track that extends into a loop that has a radius of 20m. Assume track is frictionless. I need to find minimum height for the mass to make it around the loop without falling off or going backwards.


Homework Equations


Conservation of Energy:
K_1+U_2=K_2+U_2
v=sqrt(2gh)

Force:
F=ma
F=mg

Centripetal Force:
(mv^2)/r=F


The Attempt at a Solution


This may be right, i only need someone to check my work in case I am incorrect. So, from K_1+U_1=K_2+U_2 i get K=U
From there i have 1/2mv^2=mgh
masses cancel so v^2=2gh
from there i find centripetal force = (mv^2)/r=F
F=mg=(mv^2)/r
masses again cancel so i find that g=(v^2)/r
I replace v^2 with 2gh so i find that g=2gh/r
g's cancel so i find that h=r/2 plug in for r: h = 20/2=10meters.
Can someone verify that I am correct? thanks in advance!
 
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  • #2
gggorillaz said:

Homework Statement


A mass of 100kg is h meters high on a track that extends into a loop that has a radius of 20m. Assume track is frictionless. I need to find minimum height for the mass to make it around the loop without falling off or going backwards.


Homework Equations


Conservation of Energy:
K_1+U_2=K_2+U_2
v=sqrt(2gh)

Force:
F=ma
F=mg

Centripetal Force:
(mv^2)/r=F


The Attempt at a Solution


This may be right, i only need someone to check my work in case I am incorrect. So, from K_1+U_1=K_2+U_2 i get K=U
From there i have 1/2mv^2=mgh
masses cancel so v^2=2gh
from there i find centripetal force = (mv^2)/r=F
F=mg=(mv^2)/r
masses again cancel so i find that g=(v^2)/r
I replace v^2 with 2gh so i find that g=2gh/r
g's cancel so i find that h=r/2 plug in for r: h = 20/2=10meters.
Can someone verify that I am correct? thanks in advance!
You are not correct, how can the track be only 10 m high when the loop itself is 40 m high?:wink: Can you spot your error in your equation in the use of the variable 'h' in relation to the height of the top of the loop where the speed is critically at its minimum?
 
  • #3
iIs it that i have a skewed understanding of what centripetal force equals? maybe that ma=mv^2/r but my a isn't = g in this case?

Should i be getting h=2diameter? if so, how do i get there?
 
Last edited:
  • #4
I am not sure of your understanding of the centripetal force, but when you solved for the acceleration of a =g, that is correct, at the top of the loop, since at that point, there is no normal force acting if the coaster is just able to round the top of the loop without leaving the track. You may have assumed that a=g at the bottom of the loop?? Anyway, since v2 = rg is correct at the point at the top of the loop, then when applying conservation of energy , you must apply it between the point h and the point at the top of the loop, to equate velocities and solve for the distance from the top of the track to the top of the loop.
 
  • #5
Ohhh right, I see now, it would now be mgh=1/2mv^2+mg(height of loop) so h=1/2(r)+h =1/2(20)+40 = 50m right?
 
  • #6
gggorillaz said:
Ohhh right, I see now, it would now be mgh=1/2mv^2+mg(height of loop) so htrack=1/2(r)+hloop =1/2(20)+40 = 50m right?
Right:approve:
 
  • #7
Yay! thank you so much for helping!
 

Related to Roller coaster loop searching for Height

1. How do you calculate the ideal height for a roller coaster loop?

The ideal height for a roller coaster loop can be calculated using a combination of physics equations and roller coaster design principles. It involves considering factors such as the speed of the train, the radius of the loop, and the force of gravity. This calculation is crucial to ensure the loop is safe and provides a thrilling experience for riders.

2. What is the minimum height requirement for a roller coaster loop?

The minimum height requirement for a roller coaster loop varies depending on the design and type of coaster. Generally, the minimum height would be around 50-60 feet, but it can be higher for more intense loops. This is to ensure that the train has enough speed and momentum to complete the loop safely without causing discomfort to riders.

3. Can a roller coaster loop be too tall?

Yes, a roller coaster loop can be too tall. If the loop is too tall, it can cause excessive forces on the riders, which can be dangerous and uncomfortable. Additionally, a taller loop would require a longer track, which can be costly and may not fit into the desired location or theme of the roller coaster.

4. How does the height of a roller coaster loop affect the overall ride experience?

The height of a roller coaster loop plays a significant role in the overall ride experience. A higher loop will provide a more intense and thrilling experience for riders, while a lower loop may be less intense but still provide a fun and enjoyable ride. The height also affects the smoothness of the ride, as a taller loop will require more track length and can result in more forces on the riders.

5. Are there any safety regulations for the height of a roller coaster loop?

Yes, there are safety regulations for the height of a roller coaster loop. These regulations vary by country and state, but they typically include minimum and maximum height requirements, as well as guidelines for factors such as speed, g-forces, and track design. These regulations are in place to ensure the safety of riders and prevent accidents or injuries.

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