# 1.1 Accuracy, precision, and significant figures  (Page 3/12)

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## Percent uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement $A$ is expressed with uncertainty, $\mathrm{\delta A}$ , the percent uncertainty    (%unc) is defined to be

$\text{}\text{% unc =}\frac{\mathrm{\delta A}}{A}×\text{100%}\text{}\text{.}$

## Calculating percent uncertainty: a bag of apples

A grocery store sells $\text{5-lb}$ bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements:

• Week 1 weight: $\text{4.8 lb}$
• Week 2 weight: $\text{5.3 lb}$
• Week 3 weight: $\text{4.9 lb}$
• Week 4 weight: $\text{5.4 lb}$

You determine that the weight of the $\text{5-lb}$ bag has an uncertainty of $±0\text{.}4\phantom{\rule{0.25em}{0ex}}\text{lb}$ . What is the percent uncertainty of the bag’s weight?

Strategy

First, observe that the expected value of the bag’s weight, $A$ , is 5 lb. The uncertainty in this value, $\mathrm{\delta A}$ , is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:

$\text{}\text{% unc =}\frac{\mathrm{\delta A}}{A}×\text{100%}\text{}\text{.}$

Solution

Plug the known values into the equation:

Discussion

We can conclude that the weight of the apple bag is $5\phantom{\rule{0.25em}{0ex}}\text{lb}±8\text{%}$ . Consider how this percent uncertainty would change if the bag of apples were half as heavy, but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value.

## Uncertainties in calculations

There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents    can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation . For example, if a floor has a length of $4\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{m}$ and a width of $3\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{m}$ , with uncertainties of $2%\text{}$ and $1%\text{}$ , respectively, then the area of the floor is $\text{12}\text{.}0\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ and has an uncertainty of $3%\text{}$ . (Expressed as an area this is $0\text{.}\text{36}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ , which we round to $0\text{.}4\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ since the area of the floor is given to a tenth of a square meter.)

A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of $±0\text{.}\text{05}\phantom{\rule{0.25em}{0ex}}\mathrm{s}$ . Runners on the track coach’s team regularly clock 100-m sprints of $\text{11.49 s}$ to $\text{15.01 s}$ . At the school’s last track meet, the first-place sprinter came in at $\text{12}\text{.}\text{04 s}$ and the second-place sprinter came in at $\text{12}\text{.}\text{07 s}$ . Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?

No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times.

## Precision of measuring tools and significant figures

An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 millimeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be.

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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