Equation to represent the GPE of a pirate ship ride?

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  minamikaze

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  Oct 22,

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  Gpe Ship

In summary: Your fellow student is correct that the given equation is only an approximation for the motion of a pendulum that's reasonable for small swings. However, the given equation is still SHM in the small angle approximation.

• Oct 22,

• #1

minamikaze

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Homework Statement

I came across the question above - and one of my peers told me that the equation E = 100(1-coskt) + a in the question is actually inaccurate in that it does not represent the motion of the pirate ship ride properly, because the equation isn't simple harmonic - the negative example to prove this was that when t = 3, kt = π radians (since k = π/3). I'm not very strong in this topic so I was unable to debate too much with him, so I wanted to get some insights from the experts here.

Homework Equations

He claims that the equation to represent the GPE of the pirate ship should be E = 100[1-cos (θm sin ωt)] + 30, so that the value of θ varies from 0 to 2. This would make the equation simple harmonic.

The Attempt at a Solution

I drew the graph above, and I still don't really see any problem with the GPE equation, actually. Could someone please enlighten me on this one?

Thank you so much in advance for any help!

I came across the question above - and one of my peers told me that the equation E = 100(1-coskt) + a in the question is actually inaccurate in that it does not represent the motion of the pirate ship ride properly, because the equation isn't simple harmonic - the negative example to prove this was that when t = 3, kt = π radians (since k = π/3). I'm not very strong in this topic so I was unable to debate too much with him, so I wanted to get some insights from the experts here.He claims that the equation to represent the GPE of the pirate ship should be E = 100[1-cos (θsin ωt)] + 30, so that the value of θ varies from 0 to 2. This would make the equation simple harmonic.I drew the graph above, and I still don't really see any problem with the GPE equation, actually. Could someone please enlighten me on this one?Thank you so much in advance for any help!

 

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• Oct 22,

• #2

Chandra Prayaga

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I don't see anything wrong with your equation for E. It satisfies every condition laid out in the problem. So, can you now answer the questions (iii), (iv) and (v)?

 

• Oct 22,

• #3

haruspex

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minamikaze said:

He claims that the equation to represent the GPE of the pirate ship should be E = 100[1-cos (θm sin ωt)] + 30, so that the value of θ varies from 0 to 2. This would make the equation simple harmonic.

Your fellow student is correct that the given equation is only an approximation for the motion of a pendulum that's reasonable for small swings.
In the more complicated alternative, I assume θm represents some unspecified function. That probably works, but does not make it SHM. (Did you mean that?) It is only SHM in the small angle approximation.

In the present problem the swing period is 12 seconds, implying a pendulum length of around 35m. For an initial height of 6m, that gives an amplitude of 34°, rather too large for the small angle approximation. So your friend's criticism is valid. Nonetheless, this is what the questioner has instructed you to use.

Your fellow student is correct that the given equation is only an approximation for the motion of a pendulum that's reasonable for small swings.In the more complicated alternative, I assume θrepresents some unspecified function. That probably works, but does not make it SHM. (Did you mean that?) It is only SHM in the small angle approximation.In the present problem the swing period is 12 seconds, implying a pendulum length of around 35m. For an initial height of 6m, that gives an amplitude of 34°, rather too large for the small angle approximation. So your friend's criticism is valid. Nonetheless, this is what the questioner has instructed you to use.

 

FAQ: Equation to represent the GPE of a pirate ship ride?

What is the equation to represent the GPE of a pirate ship ride?

The equation to represent the GPE (Gravitational Potential Energy) of a pirate ship ride is GPE = mgh, where m is the mass of the ride, g is the acceleration due to gravity, and h is the height of the ride.

How is the GPE of a pirate ship ride calculated?

The GPE of a pirate ship ride is calculated by multiplying the mass of the ride by the acceleration due to gravity (9.8 m/s^2) and the height of the ride in meters.

What factors affect the GPE of a pirate ship ride?

The factors that affect the GPE of a pirate ship ride include the mass of the ride, the height of the ride, and the acceleration due to gravity.

Why is the GPE of a pirate ship ride important?

The GPE of a pirate ship ride is important because it determines the potential energy of the ride, which is later converted into kinetic energy as the ride moves. This energy is what creates the thrilling experience for riders.

Is the GPE of a pirate ship ride constant?

No, the GPE of a pirate ship ride is not constant. It changes as the ride moves up and down, and also depends on external factors such as changes in the mass or height of the ride.

Pirate ships are found in many amusement parks. They are examples of pendulum rides, where riders sit on both sides of the middle, facing each other. The further away from the middle, the higher you ride on one side, but not quite so high on the other.

The upper graph shows how the G forces vary during the ride in a pirate ship and the lower shows the corresponding angular velocity.

A few questions to consider:

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1. What forces act on the rider in the turning points?
2. What forces act on the rider as the swing passes the lowest point?
3. The force on the rider in the turning points depends on the angle θ:  N = mg cos θ. Use the accelerometer graph for the following questions. (Consider the middle part of the ride, when the pendulum reaches the largest angles)
   1. What is the angle between the rider and the acceleration of gravity at the highest turning point? 
   2. What is the angle at the lower turning point?
   3. During what parts of the ride is the angular momentum largest?
   4. What is the acceleration at the bottom of the pendulum motion?
4. At highest point you face towards the middle of the ride. Which way are you rotating as the swing moves down again (positive or negative "pitch"). (Check also the detailed graph)
5. The force at the bottom is larger than mg. (Why?). Show that for a mathematical pendulum (point mass on massless string) the force in the lowest point deviates from mg by twice as much as the deviation in the highest point. The relation holds also if you are located in the "radius of gyration". Does this hold for the graphs in the pirate ship?