Q3TY By what factor must the mass inc... [FREE SOLUTION]

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Chapter 28: Q3TY (page 775)

By what factor must the mass increase to reduce the sampling standard deviation by a factor of 2?

Short Answer

Expert verified

The mass needs to be increased four times to reduce the sampling standard deviation by a factor of 2.

Step by step solution

Assumption

Let,

K_s= Sampling constant

m = mass of particle

R= relative standard deviation

蟽 = Standard deviation

m₁ = mass of particlefor 1^st case

R₁= relative standard deviationfor 1^st case

蟽₁ = Standard deviationfor 1^st case

m₂ = mass of particle for 2^nd case

R₂= relative standard deviationfor 2^nd case

蟽₂ = Standard deviationfor 2^nd case

As per the given information

$蟽_{n 2} = \frac{蟽_{n 1}}{2}$

Determine relative standard deviation

$\begin{array}{rcl} K_{s} & = & m R^{2} \\ m & = & \frac{K_{s}}{R^{2}} \\ m_{1} & = & \frac{K_{s}}{{R_{1}}^{2}} \\ m_{1} & = & \frac{K_{s}}{{(\frac{蟽_{n 1}}{n})}^{2}} \\ m_{2} & = & \frac{K_{s}}{{R_{2}}^{2}} \\ m_{2} & = & \frac{K_{s}}{{(\frac{蟽_{n 2}}{n})}^{2}} \end{array}$

$m_{2} = \frac{K_{s}}{{(\frac{蟽_{n 1}}{2 n})}^{2}} A s 蟽_{n 2} = \frac{蟽_{n 1}}{2 n} m_{2} = \frac{K_{s}}{\frac{1}{4} {(\frac{蟽_{n 1}}{n})}^{2}} m_{2} = \frac{4 K_{s}}{{(\frac{蟽_{n 1}}{n})}^{2}} m_{2} = 4 m_{1} A s m_{1} = \frac{K_{s}}{{(\frac{蟽_{n 1}}{n})}^{2}}$