# Rollercoaster ride problem, involving centripetal accelaration

• Hoppa
In summary, the owners of a fun-park are designing a new ride that will have a section in the form of a helix with a horizontal central axis. The carriages will travel along the helical path at a fixed speed, u, and will be upside down at the highest point of the helix. The designers want to ensure that the patrons are only just on the point of falling from their seats, and they can do this by making the centripetal acceleration equal to the acceleration due to gravity, g. The parameters that can be changed are the radius of the helix, p, and its pitch, p. By using a suitable choice of axes, the carriage's trajectory can be modeled as r(t) = (
Hoppa
The owners of a fun-park are designing a new ride. This will have a section in the form of a helix whose central axis is directed horizontally. Patrons will travel along the helical path in carriages whose speed is a fixed value, u. The carriages will be upside down when they are at a high point on the helix and the designers want to ensure that patrons are only just on the point of falling from their seats at this point (even though they are strapped in!). They can do this by ensuring that the centripetal acceleration just equals the acceleration due to gravity, g. The parameters that the designers can change are the radius of the helix p and its pitch p, which is the horizontal distance between neighbouring helical sections of the same orientation.
Show that, with suitable choice of axes, a carriage’s trajectory can be modeled as:
r(t) = (p cos wt, p sin wt, qt)
and find a relationship for the pitch p in terms of the modelling parameters w and q. Find a relation for the carriage speed u and the centripetal acceleration in terms of the modelling parameters p, w and q. By requiring that the centripetal acceleration should equal the acceleration due to gravity, find a relationship between p and p (involving u) that will assist in design decisions.

thats the problem that i am stuck with. i haven't got far with it either. any help would be appreciated.
since the helix's are circular it would be that 2(pi)r = p x (number of sections)
centripetal accelaration equals the accelaration due to gravity, would that equal v^2/r = 9.8 ??

Please start by restating your problem so we know for sure what is what is what. You have two different things labelled as p. One of them is pitch, the other is a radius that was probably labeled as the greek letter rho. Spell out rho where p means rho.

This ride is not really a roller coaster. The cars do not "coast", speeding up and slowing down as they go down and up the way a roller coaster does. Their speed is controlled by some driving mechanism that keeps them going at constant speed.

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ok i will try:
carriages speed is fixed value, u.
centripetal acceleration needs to be equal the acceleration due to gravity g.
parameters that are varible are the radius of the helix (rho) and its pitch p.

show that with suitable choice of axes, a carriage’s trajectory can be modeled as:
r(t) = (rho cos wt, p sin wt, qt)

Hoppa said:
ok i will try:
carriages speed is fixed value, u.
centripetal acceleration needs to be equal the acceleration due to gravity g.
parameters that are varible are the radius of the helix (rho) and its pitch p.

show that with suitable choice of axes, a carriage’s trajectory can be modeled as:
r(t) = (rho cos wt, p sin wt, qt)

You missed one. It would have to be

r(t) = (rho cos wt, rho sin wt, qt)

I'm assuming this is a calculus based course. If not, let me know. Start by taking the derivative of r(t) with respect to time to find all the velocity components. When you get that, figure the component of velocity directed in the plane perpendicular to the axis of the helix. That will help you figure out the weightlesness condition.

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yeh it a calculus based course

figure the component of velocity directed toward the axis of helix.
is that v^2/r = 9.8?

When you get that, figure the component of velocity directed toward the axis of the helix. That will help you figure out the weightlesness condition.

how do i figure the component of velocity? i think i have the derivative correct. how will it help to figure out weightlessess condition?

Hoppa said:
figure the component of velocity directed toward the axis of helix.
is that v^2/r = 9.8?

That would be the acceleration, and it is correct. I should have said the component of velocity in the plane perpendicular to the axis of the helix. The velocity in that plane is always perpendicular to the axis. The acceleration is toward the axis.

Hoppa said:
When you get that, figure the component of velocity directed toward the axis of the helix. That will help you figure out the weightlesness condition.

how do i figure the component of velocity? i think i have the derivative correct. how will it help to figure out weightlessess condition?

If you took derivatives of each of the three components of r(t), the first two of those are components perpendicular to the axis of the helix. The last one is the component in the direction of the helix. The one in the direction of the helix is related to the pitch. The other two combined are always directed tangent to a circle surrounding the axis. You can think of the problem as circular motion around the axis to figure out centripetal force independently of the translation along the axis. If you did it right, you will find that the sum of squares of the first two velocity components is a constant equal to the v^2 in the centripetal force equation. If the centripetal force equals mg, the riders will be weightless at the top. Once that condition is satisfied, you can express the velocity in the z direction in terms of the total velocity u and the acceleration due to gravity. When you have those relationships figured out, you are close to finished.

I'll be back later on this evening. Show us what you have done. What are your velocity components?

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## 1. What is the definition of centripetal acceleration?

Centripetal acceleration is the acceleration of an object moving in a circular path that is directed towards the center of the circle.

## 2. How is centripetal acceleration calculated?

Centripetal acceleration can be calculated by taking the square of the velocity of the object and dividing it by the radius of the circle it is moving in.

## 3. Why is centripetal acceleration important in rollercoaster rides?

Centripetal acceleration is important in rollercoaster rides because it is responsible for the change in direction of the ride, keeping the passengers in their seats and preventing them from flying off the ride.

## 4. How does centripetal acceleration affect the speed of a rollercoaster?

As the radius of the rollercoaster decreases, the centripetal acceleration increases, causing the ride to move at a faster speed. Similarly, as the radius increases, the centripetal acceleration decreases and the ride slows down.

## 5. How can centripetal acceleration be increased in a rollercoaster?

Centripetal acceleration can be increased in a rollercoaster by decreasing the radius of the circular path, increasing the speed of the ride, or increasing the mass of the riders. However, it is important to note that there are limits to these factors to ensure the safety and comfort of the riders.

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