(II) Digital bits on a 12.0-cm diameter audio CD are encoded a...

Question

(II) Digital bits on a 12.0-cm diameter audio CD are encoded along an outward spiraling path that starts at radius R₁ = 2.5 cm and finishes at radius R₂ = 5.8 cm. The distance between the centers of neighboring spiral-windings is 1.6 μm ( = 1.6 x 10^-6 m).

(a) Determine the total length of the spiraling path. [Hint: Imagine 'unwinding' the spiral into a straight path of width 1.6 μm, and note that the original spiral and the straight path both occupy the same area.]

Accepted Answer

Hey, everyone in this problem, we're asked to consider a copper wire that is being rolled into a circular coil. The copper wire starts coiling at an inner radius of 4.8 decimeters and finishes at an outer radius of 9.3 decimeters. Each coil is separated by a distance of 2.2 m. And we are asked to find the total length of the copper wire. We're given four answer choices all in meters. Option A 3.6 multiplied by 10 to the exponent two. Option B 5.5 multiplied by 10 to the exponent one. Option C 6.8 multiplied by 10 to the exponent one. And option D 9.1 multiplied by 10 to the exponent two. So let's start by drawing out what we have going on. So we have this wire and it's coiling around and we know that the distance between these coils and the distance is going to be 2.2 millimeters and it's gonna coil some amount, but it doesn't coil all the way to the sector. And if we think about looking at the area that is covered by the rolled coil and we can think about a, a circle, it's kind of a hole in the middle and we know that the outer radius is 9.3 decimeters and the inner radius is 4.8 decimeters. And the area that's actually covered is all the area between those two circles. OK. So that's kind of what we have going on. And we can think about how we can calculate this area that's covered by the coil. And so we think about the area covered by the coil, right? It's just gonna be the area of the outer. So, oh, so Pie Ro, we'll say ro for the outer radius squared minus pi R I squared and the inner radius squared. So this is all of the area that's gonna be taken up by this coil that includes the copper wire and those 2.2 millimeter spaces between them. OK. So now the question is we have to find the length of the copper wire. So how do we relate this area that we have to the length? One thing we can do is think about unraveling this coin. OK. The area that it takes up when it's coiled up is gonna be the exact same as the area it takes up when it's uncoiled, it's the same amount of material, it's gonna cover the same area. So if we think about uncoiling what we have and we have our wire and we have this space as well and the the width of this kind of rectangular shape we get when we uncoil it, that's gonna be that distance between the cos see that's that 2.2 millimeter, that's it. Then we have our wire and then we have the length of our wire. Oh, ok. So Ella is what we're looking for. Now, this forms a rectangle. So we know that the area can be calculated by the length multiplied by the width. And we know that that has to be the same as the area. If we calculate it, when it's coiled up again, the same material covers the same area. And so this is gonna be equal to the length multiplied by the width when we uncoil this material. I know when we say width again, we're not saying the width of the coil. It has to be the width of those separation distances because it has to match that area when it's coiled up. And we have that same distance between those coils. All right. So let's go ahead substitute in what we know and solve for the length, right. So we have pi multiplied by the outer radius. OK. We're in decimeters, we wanna convert to our standard unit of meters. So we need to divide by 10. So we get 0.93 m minus pi multiplied by 0.48 m squared. That's gonna be equal to the length of the coil L that we're looking for multiplied by the width when we unroll it, which is that separation distance of 2.2 millimeters by converting to meters, we need to multiply by 10 to the exponent negative. So our length out, well, that's just gonna be equal to pi multiplied by. If we simplify the left hand side, we can simplify this to 0.6345 m squared divided by 2.2 multiplied by 10 to the exponent negative 3 m meters squared, divided by meters is gonna leave us with a unit of meters, which is what we want for length. And we get that the length is approximately equal to 906.064 m. All right. So taking a look at our answer choices, we're going round to two significant digits or write this in scientific notation. And when we do, we can see that our answer is going to match with option D. The length of this copper wire is about 9.1 multiplied by 10 to the exponent 2 m. Thanks everyone for watching. I hope this video helped you in the next one.

Key Concepts

Spiral Geometry

Area of a Spiral

Integration in Calculus

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