Error Analysis

We are given the measurements of a cylindrical wire with their uncertainties:

Mass ( $m$ ) = $0.3%20%5Cpm%200.003%20%5C%2C%20%5Ctext%7Bg%7D$
Radius ( $r$ ) = $0.5%20%5Cpm%200.005%20%5C%2C%20%5Ctext%7Bmm%7D$
Length ( $L$ ) = $6%20%5Cpm%200.06%20%5C%2C%20%5Ctext%7Bcm%7D$

We need to find the maximum percentage error in the measurement of its density ( $%5Crho$ ).

Key Concepts:

Density Formula: Density is mass per unit volume ( $%5Crho%20%3D%20%5Cfrac%7Bm%7D%7BV%7D$ ).
Volume of a Cylinder: The volume ( $V$ ) of a cylinder is $V%20%3D%20%5Cpi%20r%5E2%20L$ .
Density in terms of measured quantities: Substituting the volume formula, $%5Crho%20%3D%20%5Cfrac%7Bm%7D%7B%5Cpi%20r%5E2%20L%7D$ .
Propagation of Errors (Maximum Percentage Error): For a quantity $Q%20%3D%20%5Cfrac%7Bx%5Ea%20y%5Eb%7D%7Bz%5Ec%7D$ , the maximum percentage error in $Q$ is given by:

$%5Cleft(%20%5Cfrac%7B%5CDelta%20Q%7D%7BQ%7D%20%5Ctimes%20100%5C%25%20%5Cright)_%7B%5Ctext%7Bmax%7D%7D%20%3D%20%5Cleft(%20%7Ca%7C%20%5Cfrac%7B%5CDelta%20x%7D%7Bx%7D%20%2B%20%7Cb%7C%20%5Cfrac%7B%5CDelta%20y%7D%7By%7D%20%2B%20%7Cc%7C%20%5Cfrac%7B%5CDelta%20z%7D%7Bz%7D%20%5Cright)%20%5Ctimes%20100%5C%25$

Applying this to the density formula $%5Crho%20%3D%20%5Cfrac%7Bm%5E1%7D%7B%5Cpi%20r%5E2%20L%5E1%7D$ (note $%5Cpi$ is a constant with no error), the maximum percentage error in density is:

$%5Cleft(%20%5Cfrac%7B%5CDelta%20%5Crho%7D%7B%5Crho%7D%20%5Ctimes%20100%5C%25%20%5Cright)_%7B%5Ctext%7Bmax%7D%7D%20%3D%20%5Cleft(%20(1)%20%5Cfrac%7B%5CDelta%20m%7D%7Bm%7D%20%2B%20(2)%20%5Cfrac%7B%5CDelta%20r%7D%7Br%7D%20%2B%20(1)%20%5Cfrac%7B%5CDelta%20L%7D%7BL%7D%20%5Cright)%20%5Ctimes%20100%5C%25$

Calculations:

First, calculate the individual percentage errors for mass, radius, and length:

Percentage error in mass ( $%5Cfrac%7B%5CDelta%20m%7D%7Bm%7D%20%5Ctimes%20100%5C%25$ ):

$%5Cfrac%7B0.003%20%5C%2C%20%5Ctext%7Bg%7D%7D%7B0.3%20%5C%2C%20%5Ctext%7Bg%7D%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B3%7D%7B300%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B1%7D%7B100%7D%20%5Ctimes%20100%5C%25%20%3D%201%5C%25$

Percentage error in radius ( $%5Cfrac%7B%5CDelta%20r%7D%7Br%7D%20%5Ctimes%20100%5C%25$ ):

$%5Cfrac%7B0.005%20%5C%2C%20%5Ctext%7Bmm%7D%7D%7B0.5%20%5C%2C%20%5Ctext%7Bmm%7D%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B5%7D%7B500%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B1%7D%7B100%7D%20%5Ctimes%20100%5C%25%20%3D%201%5C%25$

Percentage error in length ( $%5Cfrac%7B%5CDelta%20L%7D%7BL%7D%20%5Ctimes%20100%5C%25$ ):

$%5Cfrac%7B0.06%20%5C%2C%20%5Ctext%7Bcm%7D%7D%7B6%20%5C%2C%20%5Ctext%7Bcm%7D%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B6%7D%7B600%7D%20%5Ctimes%20100%5C%25%20%3D%20%5Cfrac%7B1%7D%7B100%7D%20%5Ctimes%20100%5C%25%20%3D%201%5C%25$

Now, substitute these percentage errors into the formula for the percentage error in density:

$%5Cleft(%20%5Cfrac%7B%5CDelta%20%5Crho%7D%7B%5Crho%7D%20%5Ctimes%20100%5C%25%20%5Cright)_%7B%5Ctext%7Bmax%7D%7D%20%3D%20%5Cleft(%20%5Cfrac%7B%5CDelta%20m%7D%7Bm%7D%20%5Ctimes%20100%5C%25%20%5Cright)%20%2B%202%20%5Cleft(%20%5Cfrac%7B%5CDelta%20r%7D%7Br%7D%20%5Ctimes%20100%5C%25%20%5Cright)%20%2B%20%5Cleft(%20%5Cfrac%7B%5CDelta%20L%7D%7BL%7D%20%5Ctimes%20100%5C%25%20%5Cright)$

$%5Cleft(%20%5Cfrac%7B%5CDelta%20%5Crho%7D%7B%5Crho%7D%20%5Ctimes%20100%5C%25%20%5Cright)_%7B%5Ctext%7Bmax%7D%7D%20%3D%20(1%5C%25)%20%2B%202%20%5Ctimes%20(1%5C%25)%20%2B%20(1%5C%25)$

$%5Cleft(%20%5Cfrac%7B%5CDelta%20%5Crho%7D%7B%5Crho%7D%20%5Ctimes%20100%5C%25%20%5Cright)_%7B%5Ctext%7Bmax%7D%7D%20%3D%201%5C%25%20%2B%202%5C%25%20%2B%201%5C%25%20%3D%204%5C%25$

The maximum percentage error in the measurement of the density is 4%.

The final answer is $%7B4%7D$

High-Yield Trend

Chapter Questions
1 MCQs

Official Solution

Error Analysis

High-Yield Trend

Chapter Questions 1 MCQs

Official Solution

Chapter Questions
1 MCQs