Substituting everything into our formula gives us: The derivative under the square root is straightforward: The starting value for θ is a = 0 and after 7.5 turns, the end point is b = 7.5 × 2π = 15π = 47.12389. We'll use the formula for the Arc Length of a Curve in Polar Coordinates to find the length. R = 5 + 0.22282 θ Length of the first spiral ( 2π b is the distance between each arm.)įor both spirals given above, a= 5, since the curve starts at 5.Ģπ b = 1.4, giving us b = 1.4/(2π) = 0.22282 Finding the Length of the Spiralīefore we can find the length of the spiral, we need to know its equation.Īn Archimedean Spiral has general equation in polar coordinates: Here's the spiral with inner radius 5, outer radius 15.5 after 7.5 turns:Īrchimedean spiral, inner radius 5, outer radius 15.5 On the other hand, if is it important to have the outer radius being 15.5 units, then the increase per turn would need to be: Here's what the spiral with 0.81 radius increase per turn looks like:Īrchimedean spiral, 0.81 units between each arm
So the outer radius cannot be 15.5 units if the increase per turn is 0.81. If the inner radius is 5 units and the increase in radius per turn is 0.81 units, then 7.5 turns will give us an outer radius of: In this case, it's not possible because it has a fundamental flaw. (Some of them are not possible to solve either because there is not enough information given, the algebra is unreadable, or some key vocabulary is not used correctly.) This was another example of an impossible-to-solve reader question.
We can see Archimedean Spirals in the spring mechanism of clocks and in vinyl records (used by the recording industry before CDs and MP3s) and in tightly coiled rope. This is an example of an Archimedean Spiral, otherwise known as an arithmetic spiral, where the arms get bigger by a constant amount for each turn. To find the total length of a flat spiral having outer end radius = 15.5 units, inner radius = 5 units & the increase in radius per turn = 0.81 unit, the total No. Reader Anantha recently wrote asking the following interesting question:
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