Physics 290 Homework Assignment 1
Measurement Error
Name__________________________________
Lab Time__________________
This short homework assignment is due at the start of your regularly scheduled lab the
first week of classes. Please give your completed homework to your lab instructor.
It is worth 10 points toward your grade.
1. Determine the area of a dollar bill in square millimeters, using a metric ruler.
(a) Measure the width and height in mm and write them in the space below in the form
x +/- ∆x, y +/- ∆y, including units (mm). Remember to round all numbers to one
uncertain digit!
Here, ∆x and ∆y are your best estimates of the uncertainty in these measurements. This
includes, for example, the accuracy with which you can guess at fractional mm given the
marks on the ruler. It might also include the estimated shrinkage of the ruler during the
manufacturing process, as a function of temperature, etc. You might get two different
metric rulers and compare their markings!
(b) Calculate the area A in mm2, and its uncertainty using the brute force method (that is,
calculate values of Amax with x + ∆x, y+∆y, Amin with x-∆x, y-∆y). Write the
intermediate results, and your final value for A in the space below, using the form A +/∆A mm2
(c) Calculate the uncertainty in the area ∆A using the formula below (explained in class)
and compare this with the value you obtained above. Round to one uncertain digit.
2
⎛ ∆x ⎞ ⎛ ∆y ⎞
∆A = A ⎜ ⎟ + ⎜ ⎟
⎝ x ⎠ ⎝ y ⎠
2
2. Determine the area in mm2 of a dollar bill, but using an inch ruler.
This is to test whether your estimated uncertainty in the measurements you performed
using the metric ruler is reasonable. Preferably, the ruler is subdivided into 1/16 inch. Use
the conversion factor 1 inch = 25.40 mm to convert inches to mm.
(a) Measure the width and height of a dollar bill in inches and write them in the space
below in the form x +/- ∆x, y +/- ∆y, including units. Again, ∆x, etc. are your estimated
uncertainties in inches for that measurement.
(b) Convert the values obtained above, including the uncertainties, into mm and write
them in the space below (1 inch = 25.4 mm)
(c) Calculate the area A of a dollar bill in mm2, and its uncertainty, using the brute force
method (x + ∆x, y+∆y, x-∆y, y-∆y, etc.). Write the intermediate results, and your value
for A in the space below, using the form A +/- ∆A mm2 (one uncertain digit).
(c) Calculate the uncertainty in the area A using the formula given in part 1 (c) and
compare this with the values you obtained above.
(d) Compare the measurements you obtained using the mm ruler and the inch ruler. Are
they the same? If not, are they consistent with each other, given the estimated
uncertainty? Explain