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Chapter 1: Problem 88

The filament in an incandescent light bulb is made from tungsten metal \(\left(d=19.3 \mathrm{g} / \mathrm{cm}^{3}\right)\) that has been drawn into a very thin wire. The diameter of the wire is difficult to measure directly, so it is sometimes estimated by measuring the mass of a fixed length of wire. If a \(0.200 \mathrm{m}\) length of tungsten wire weighs \(42.9 \mathrm{mg}\), then what is the diameter of the wire? Express your answer in millimeters.

Short Answer

Expert verified
The diameter of the tungsten wire is approximately 0.048 millimeters.

Step by step solution

01

Calculate the volume of the wire.

First determine the volume of the tungsten wire. As we know, density = mass/volume. Therefore, we can rearrange this formula to find the volume, which gives us volume = mass/density. Given mass = 42.9 mg and density = 19.3 g/cm³, it's important to convert milligrams to grams, so mass = 42.9*10^-3 g. Substitute these values into the volume formula to result in a volume of: volume = (42.9*10^-3) / 19.3 cm³.
02

Convert length of wire to cm and calculate radius.

The length given (0.200 m) needs to be converted to centimeters since our volume measurement is in cm³, which results in L = 0.200*10^2 cm = 20 cm. Now, since wire is assumed as a cylinder, the volume of the cylinder is given by \(V=\pi r^{2}L\), where r is the radius. Rearranging this equation to find r gives us \(r=\sqrt{V/\pi L}\). Substituting the values into the formula, we get the radius of the wire.
03

Calculate the diameter of the wire.

The diameter of a circle (or in this case, the wire) is simply twice the radius. So, by multiplying the calculated radius by 2, we obtain the diameter of the wire in centimeters.
04

Convert Diameter from cm to mm.

The last step is to convert the found diameter from centimeters to millimeters. To do this, we multiply our diameter by 10 (since 1cm = 10mm). That gives us the final diameter in millimeters.

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Most popular questions from this chapter

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