# 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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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