# 1.3 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.

#### Questions & Answers

what is physics
what are the basic of physics
faith
tree physical properties of heat
tree is a type of organism that grows very tall and have a wood trunk and branches with leaves... how is that related to heat? what did you smoke man?
what are the uses of dimensional analysis
Dimensional Analysis. The study of relationships between physical quantities with the help of their dimensions and units of measurements is called dimensional analysis. We use dimensional analysis in order to convert a unit from one form to another.
Emmanuel
meaning of OE and making of the subscript nc
can I ask a question
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kinetic functional force
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A wave the movement of particles on rest position transferring energy from one place to another
Gabche
not wave. i need to know principal wave or waves.
Haider
principle wave is a superposition of wave when two or more waves meet at a point , whose amplitude is the algebraic sum of the amplitude of the waves
kindly define principal wave not principle wave (principle of super position) if u can understand my question
Haider
what is a model?
hi
Muhanned
why are electros emitted only when the frequency of the incident radiation is greater than a certain value
b/c u have to know that for emission of electron need specific amount of energy which are gain by electron for emission . if incident rays have that amount of energy electron can be emitted, otherwise no way.
Nazir
search photoelectric effect on Google
Nazir
what is ohm's law
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
hi
Gundala
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
just use v^2-u^2=2as
Gundala
how often does electrolyte emits?
alhassan
just use +€^3.7°√π%-4¢•∆¥%
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akinyemi
what is distribution of trade
what's acceleration
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
hi
Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
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acceleration it is the rate of change in velocity with time
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acceleration is change in velocity per rate of time
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what is ohm's law
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Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
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acceleration is the rate of change. of displacement with time.
the rate of change of velocity is called acceleration
Asma
how i don understand
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
How do you determine the magnitude of force
mass × acceleration OR Work done ÷ distance
Seema