MHB Arc length & similar questions

AI Thread Summary
The discussion revolves around calculating the arc length and maximum height of a pirate ship ride modeled as a swinging pendulum. The arc length, calculated using the correct radius of 40 feet and a total angle of 130 degrees, is approximately 90.76 feet. For the maximum height of the center of the ship, the formula h = r(1 - cos(θ)) is applied, yielding a height of about 23.1 feet. Participants clarify the importance of using the correct radius in calculations and provide step-by-step solutions. The thread emphasizes understanding the concepts of arc length and height in relation to pendulum motion.
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Hello! I really don't understand this concept, and I have an example problem that I am working on that I just CANNOT figure out! Any help? Thanks so much in advance! A group of people get on a pirate ship ride at the fair. This ride is a swinging pendulum with a maximum swing angle of 65 degrees from the center of the ship in either direction. The arm of the pendulum holding the ship has a 40 ft radius and the ship is 22 feet long with the last seats positioned 1 foot from the end of the ship.
What is the arc length traveled by the center of the ship between the two maximum points?
ALSO>>>>
What is the maximum height reached by the center of the ship? I attached a picture for reference! Thanks so much! :)View attachment 6233
 

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fluffertoes said:
Hello! I really don't understand this concept, and I have an example problem that I am working on that I just CANNOT figure out! Any help? Thanks so much in advance! A group of people get on a pirate ship ride at the fair. This ride is a swinging pendulum with a maximum swing angle of 65 degrees from the center of the ship in either direction. The arm of the pendulum holding the ship has a 40 ft radius and the ship is 22 feet long with the last seats positioned 1 foot from the end of the ship.
I found that the arc length traveled by the center of the ship between the two maximum points is 40 feet. Is this correct?
ALSO>>>>
What is the maximum height reached by the center of the ship? I attached a picture for reference! Thanks so much! :)

Between the two maximum points, an angle of $\displaystyle \begin{align*} 130^{\circ} \end{align*}$ is swept out, so the arclength would be

$\displaystyle \begin{align*} \mathcal{l} &= \frac{130}{360} \cdot 2\,\pi \cdot 45\,\textrm{ft} \\ &= \frac{65\,\pi}{2} \,\textrm{ft} \\ &\approx 102.1\,\textrm{ft} \end{align*}$
 
Prove It said:
Between the two maximum points, an angle of $\displaystyle \begin{align*} 130^{\circ} \end{align*}$ is swept out, so the arclength would be

$\displaystyle \begin{align*} \mathcal{l} &= \frac{130}{360} \cdot 2\,\pi \cdot 45\,\textrm{ft} \\ &= \frac{65\,\pi}{2} \,\textrm{ft} \\ &\approx 102.1\,\textrm{ft} \end{align*}$

The radius is 40 feet, not 45 feet. So technically wouldn't the arc length end up being (in terms of pi (π)):

l=130/360⋅ (2π⋅40ft) = 260π/9 feet
 
Yes:

$$s=r\theta=\left(40\text{ ft}\right)\left(2\cdot65^{\circ}\frac{\pi}{180^{\circ}}\right)=\frac{260\pi}{9}\,\text{ft}\approx90.76\text{ ft}$$

To find the maximum height (above the lowest point) of the middle of the ship, we can observe that the height $h$ of the ship in terms of the angular displacement $\theta$ is given by:

$$h=r(1-\cos(\theta))$$

So, use $r=40\text{ ft}$ and $\theta=65^{\circ}$ in the above formula to get the maximum height of the center of the ship. :D
 
MarkFL said:
Yes:

$$s=r\theta=\left(40\text{ ft}\right)\left(2\cdot65^{\circ}\frac{\pi}{180^{\circ}}\right)=\frac{260\pi}{9}\,\text{ft}\approx90.76\text{ ft}$$

To find the maximum height (above the lowest point) of the middle of the ship, we can observe that the height $h$ of the ship in terms of the angular displacement $\theta$ is given by:

$$h=r(1-\cos(\theta))$$

So, use $r=40\text{ ft}$ and $\theta=65^{\circ}$ in the above formula to get the maximum height of the center of the ship. :D

So would that leave me with:

(40)Cos(65°) = 16.9
40 - 16.9 = 23.095 feet
 
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