Circular motion of a rollercoaster car on a loop-the-loop

[Jolene]

Homework Statement

You are designing a rollercoaster and want to add an upside loop to it (a vertical circle). If the top speed that the rollercoaster cars can generate is 175 km/h, what is the maximum radius that you can make the loop so that the roller coaster does not fall off the track during the loop?

Relevant Equations

Fy= N + Fg = mac

Can someone please check if I got the correct answer. Thank you!
I got:
Fy= N + Fg = mac
N + mg = mv^2/r
g = v^2/r
r = v^2/g
r = (48.61)^2/9.8
r = 241.1 m

 

[PeroK]

Homework Statement: You are designing a rollercoaster and want to add an upside loop to it (a vertical circle). If the top speed that the rollercoaster cars can generate is 175 km/h, what is the maximum radius that you can make the loop so that the roller coaster does not fall off the track during the loop?
Relevant Equations: Fy= N + Fg = mac

Can someone please check if I got the correct answer. Thank you!
I got:
Fy= N + Fg = mac
N + mg = mv^2/r
g = v^2/r
r = v^2/g
r = (48.61)^2/9.8
r = 241.1 m

Just typing in some formulas without saying what you are doing or what your assumptions are is not a good solution. Note that a radius of approx 240m is huge. The height of a tall skyscraper and 480m long. Does that answer sound realistic?

 

[Orodruin]

The height of a tall skyscraper and 480m long. Does that answer sound realistic?

Show me a rollercoaster that reaches 175 kph … without any friction losses that in itself would require a height drop of over 100 m … The unrealism is already present in the problem statement.

 

[PeroK]

Show me a rollercoaster that reaches 175 kph … without any friction losses that in itself would require a height drop of over 100 m … The unrealism is already present in the problem statement.

The fastest rollercoaster gets to 240 kph. And the highest loop is about 50m. The problem is not that unrealistic, despite not taking mechanical energy loss into account.

A 480m high loop, however, is unrealistic.

 

[PeroK]

@Jolene Can you figure out what you're missing here?

 

[Orodruin]

A 480m high loop, however, is unrealistic.

The point is that - given only the constraint of no negative g-force - that’s what you get if it travels at 175 kph at the top of the loop.

 

[nasu]

If it's 175 km/h at the top, it won't be the maximum speed it "can generate". Unless it is a powered car, with an engine.

 

[Orodruin]

If it's 175 km/h at the top, it won't be the maximum speed it "can generate". Unless it is a powered car, with an engine.

Well, you already stated the obvious exception. We do not know what is actually assumed by the problem.

 

[Jolene]

Just typing in some formulas without saying what you are doing or what your assumptions are is not a good solution. Note that a radius of approx 240m is huge. The height of a tall skyscraper and 480m long. Does that answer sound realistic?

Thank you for the quick respond and sorry for not being specific. Here's what I did:

I considered the total force of the rollercoaster at the top of the loop: Fy = N + Fg = mac

Then, I re-write the equation as: N + mg = mv^2/r

As N approaches 0 (at the top of the loop), v goes to v max (when n=0), , the equation becomes: mg = mv^2/rI Rearranged the equation to solve for r r = v^2/g

Lastly, I plugged in the numbers to find the radius when the speed is at its top speed. r=241.1m

 

[PeroK]

Thank you for the quick respond and sorry for not being specific. Here's what I did:

I considered the total force of the rollercoaster at the top of the loop: Fy = N + Fg = mac

Then, I re-write the equation as: N + mg = mv^2/r

As N approaches 0 (at the top of the loop), v goes to v max (when n=0), , the equation becomes: mg = mv^2/rI Rearranged the equation to solve for r r = v^2/g

Lastly, I plugged in the numbers to find the radius when the speed is at its top speed. r=241.1m

If you read the other posts, you might see why that's an unrealistic analysis. Do you think the rollercoaster might slow down as it ascends the loop?

 

[Jolene]

If you read the other posts, you might see why that's an unrealistic analysis. Do you think the rollercoaster might slow down as it ascends the loop?

I'm so confused, isn't that why I used the rollercoaster's top speed for my calculation? So, is my starting equation Fy=N+Fg=mac incorrect?

 

[kuruman]

Your equation is correct. However, "top speed" does not mean "speed at the top of the track." It means "the fastest the roller-coaster can ever move."

 

[kuruman]

As an aside and for those interested, I found here an interesting little article about "real" roller coaster design.

 

[PeroK]

I'm so confused, isn't that why I used the rollercoaster's top speed for my calculation? So, is my starting equation Fy=N+Fg=mac incorrect?

Your equations are not wrong in themselves. But, they do not realistically model the motion of a rollercoaster.

I looked up on the Internet the fastest rollercoaster (which is faster than your one) and the highest loop, which is about 50m. That tells you that something has gone wrong somewhere in your calculations. Your loop is higher than the Empire State Building!

If a rollercoaster is doing 175 kph at the highest point, then won't it be going even faster when it comes back down?

 

[Orodruin]

If a rollercoaster is doing 174 kph at the highest point, then won't it be going even faster when it comes back down?

That would depend on if it slams the breaks or not!