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## Homework Statement

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The "Eureka Hovercraft Corporation" wanted to hold hovercraft races as an advertising stunt. The hovercraft supports itself by blowing air downward, and has a big fixed propeller on the top deck for forward propulsion. Unfortunately, it has no steering equipment, so that the pilots found that making high speed turns was very difficult. The company decided to overcome this problem by designing a bowl-shaped track in which the hovercraft, once up to speed, would coast along in a circular path with no need to steer. They hired an engineer to design and build the track, and when he finished, he hastily left the country. When the company held their first race, they found to their dismay that the craft took exactly the same time T to circle the track, no matter what its speed. Find the equation for the cross section of the bowl in terms of T.

## Homework Equations

Newton's law of motion

## The Attempt at a Solution

Hard to start!

I wanted to work in polar coordinates to deal with circular motion, and at the end, once I find ##\theta##, go to cartesian coordinates so that ## y(t) = x(t) \tan(\theta(t)) ##.

Since the instructions say that for any time ##t##: ## T = \frac{2\pi R}{v(t)} ##; and since in circular motion ##v(t) =R \dot\theta ##, I am tempted to say that ##\theta(t) = 2\pi \frac{t}{T}##, but it does not explain why the engineer left the country.

I tried to draw forces in order to find a relationship between them, ##T##, and ##\theta## (or ##\tan(\theta)##).

I think there should be a reaction force ##\vec N## from the track radially, a propulsion force ##\vec F## tangentially, and a projection of the weight ##\vec W## tangentially and radially. With Newton's law of motion, I get :

##ma_r = -N - W\sin(\theta)##

##ma_\theta = F - W\cos(\theta)##

After reducing, I get

## F = mg \cos(\theta) ##

## N = m (\frac{4R\pi^2}{T^2} - g\sin(\theta)) ##

and now I'm lost.