/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 A wire is of mass \((0.3 \pm 0.0... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 4

A wire is of mass \((0.3 \pm 0.003) \mathrm{gm}\). The radius is \((0.5 \pm 0.005) \mathrm{mm}\) and length is \((6.0 \pm 0.06) \mathrm{cm}\) then \(\%\) error in density is (A) 3 (B) 4 (C) 6 (D) \(-2\)

Short Answer

Expert verified
To find the percentage error in the density of the wire, first calculate the density using the given mass, radius, and length: \(\rho = \frac{m}{V}\), where \(V = \pi r^2h\). Then, calculate the percentage errors in mass, radius, and length: \(\frac{\Delta m}{m}\), \(\frac{\Delta r}{r}\), and \(\frac{\Delta h}{h}\) respectively. Finally, find the percentage error in density: \(\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2\frac{\Delta r}{r} + \frac{\Delta h}{h}\). Comparing the calculated percentage error with the given options, the correct answer is (C) 6.

Step by step solution

01

Calculate the volume of the wire

The wire is cylindrical, so its volume can be calculated using the formula: \(V = \pi r^2h\) where r is the radius and h is the length. Here, \(r = 0.5 \times 10^{-3} \mathrm{m}\) and \(h = 6.0 \times 10^{-2} \mathrm{m}\).
02

Calculate the density

The density can be calculated using the formula: \(\rho = \frac{m}{V}\) where m is the mass. Here, \(m = 0.3 \times 10^{-3} \mathrm{kg}\). #Step 2: Calculate the percentage error in density#
03

Calculate the percentage error in mass

The percentage error in mass, \(\Delta m\), can be calculated using the formula: \(\frac{\Delta m}{m} = \frac{0.003}{0.3}\)
04

Calculate the percentage error in radius

The percentage error in radius, \(\Delta r\), can be calculated using the formula: \(\frac{\Delta r}{r} = \frac{0.005}{0.5}\)
05

Calculate the percentage error in length

The percentage error in length, \(\Delta h\), can be calculated using the formula: \(\frac{\Delta h}{h} = \frac{0.06}{6.0}\)
06

Calculate the percentage error in density

The percentage error in density, \(\Delta \rho\), can be calculated using these errors and the formula: \(\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2\frac{\Delta r}{r} + \frac{\Delta h}{h}\) Using the calculated percentage errors in mass, radius, and length, we can find the percentage error in density. Finally, compare the calculated percentage error with the given options to find the correct answer.

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