Finding the minimum speed of a loop the loop and finding the height of the hill

AI Thread Summary
To determine the minimum speed for a roller coaster cart to stay on track at the top of a loop with a radius of 10m, the formula Vtop = sqrt(rg) yields a speed of approximately 9.89 m/s. This speed ensures the cart has enough kinetic energy to maintain its path through the loop. The discussion also highlights the need to calculate the kinetic energy at the bottom of the loop and the potential energy at the top to find the height of the hill. The forces acting on the cart at the top of the loop and the necessary net force for acceleration are critical for understanding the dynamics involved. Ultimately, the height of the hill can be derived from the energy considerations once the necessary kinetic energy is established.
SherBear
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Homework Statement



I have a hill with a roller coaster cart on it and it goes down and around a loop-the-loop

the radius of the loop is 10m

Diameter=20m

What is the minimum speed the cart needs to go to keep it on the track?

There is no friction, what is the height of the hill?

Homework Equations



Vtop=sqrt rg



The Attempt at a Solution


to get the minimum speed the cart needs to go to keep it on the track is
Vtop= sqrt rg
=sqrt 10m(9.8 m/s^2)
Vtop=9.89 m/s ---------is this correct?

There is no friction, what is the height of the hill?

I don't know what to do for this problem
 
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I forgot to add the third part, there is a rider on the bottom of the loop in a cart, what is N?

I have Ar=v^2 / r

9.89^2 m/s / 10m = 9.78 m/s

Then F=N-mg=ma

N=mg+ma

N= m (g+a)

N= 60 kg (9.8 m/s^2 + 9.78 m/s) =

N= 1,174.8 is that m/s or N as in Newtons?---------is this correct?
 
The velocity you calculated gives it enough energy to reach the height of the top of the loop, but that doesn't necessarily mean it stays on the track. If that were the speed at the bottom of the loop, how much KE would it have left at the top? What speed do you think it needs to have at the top to stay on the track?
 
No idea Haruspex?
 
If the cart has speed u at the bottom of the loop, how much KE does it have?
How much PE does it gain in reaching the top of the loop?
How much KE then does it have left (as a function of m, r, g and u)?
What forces act on it when it is at the top of the loop?
What is its acceleration if it is still staying on the track at speed v?
What net force is required to achieve that acceleration?
These are the steps you need to go through.
 
Ok, thank you. Any idea about the height of the hill ?
 
SherBear said:
Any idea about the height of the hill ?
Once you know the KE it needs at the bottom of the loop, it's easy to work out the PE it needs at the top of the hill.
 
How can I go through those steps if it only gives me the diameter?
 
Is it 2R - 1/2 R ?

20-5 = 15m ?
 
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Do you have any answers for the steps I listed? Answer what you can and we can take it from there.
 
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