Propagated uncertainty in measurements

Purpose: To learn how to calculate the propagated error in each density measurement to find out whether or not your measurements are within the experimental uncertainty of the accepted values.

The vernier caliper will be used to measure the height and diameter of the metal cylinders.

The scale will be used to measure the mass of the metal cylinders.

After we have collected the measurements of mass height and diameter of the metal cylinders. we are going to use the formula for density to calculate the propagated error in our measurements. Then we will calculate the density from our measurements and add the propagated uncertainty to see if our measurement for density is within the experimental uncertainty of the accepted values for the metal cylinders that are copper zinc and Iron.

 

 

 
Using these measurements we found out the propagated error for the density by adding the uncertainty in the mass with the uncertainty in diameter and the uncertainty in height. To find out the uncertainty in the mass we took the partial derivative of the mass in the density formula and pretended all the other variables were constants. After finding the partial derivative we multiplied the partial derivative with the accuracy of the instrument we used which was the scale, so we multiplied the partial derivative by .1. After getting the uncertainty of each measurement we add them all up to get the uncertainty for the density. Then we found the density of the metal using the density formula and added or subtracted the uncertainty for the density and the accepted value for the density should fall within the range of our density and uncertainty which in our case it did. Our uncertainty and density for the copper cylinder was 8.182 +or- .75089 g/cm^3 and the accepted value is 8.83 g/cm^3. for our zinc cylinder we got 6.60169 +or- .62572 g/cm^3 and the accepted value is 7.140 g/cm^3. The only density that didn't make sense was our iron we got 6.36379 +or- .5996 g/cm^3 which is not within the range of the accepted value 7.874 g/cm^3.

 After we learned how to calculate propagated error we did the second part of the experiment which involved calculating the mass of a hanging object held up by two strings we did this for two systems.

The first thing we did was take the measurement of the angles of the strings and the force of the tension on each string .















Next, we drew free body diagrams to find our the equations of the net force and solved to find the unknown mass for each system.

After we found the mass of each object we had to find the propagated error to account for the error in our instruments.

 

So we learned how to calculated propagated error to show that our values are within the accepted values and we applied what we learned in the second part by finding the propagated error of the unknown mass.