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Percent uncertainty

One method of expressing uncertainty is as a percent of the measured value. If a measurement A size 12{A} {} is expressed with uncertainty, δA size 12{δA} {} , the percent uncertainty    (%unc) is defined to be

% unc = δA A × 100% . size 12{%" unc = " { {δA} over {A} } times "100"%} {}

Calculating percent uncertainty: a bag of apples

A grocery store sells 5-lb size 12{ "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: 4.8 lb size 12{ "4.8 lb" } {}
  • Week 2 weight: 5.3 lb size 12{ "5.3 lb" } {}
  • Week 3 weight: 4.9 lb size 12{ "4.9 lb" } {}
  • Week 4 weight: 5.4 lb size 12{ "5.4 lb" } {}

You determine that the weight of the 5-lb size 12{ "5-lb" } {} bag has an uncertainty of ± 0 . 4 lb size 12{ +- 0 "." 4`"lb"} {} . What is the percent uncertainty of the bag’s weight?

Strategy

First, observe that the expected value of the bag’s weight, A size 12{A} {} , is 5 lb. The uncertainty in this value, δA size 12{δA} {} , is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight:

% unc = δA A × 100% . size 12{%" unc = " { {δA} over {A} } times "100"%} {}

Solution

Plug the known values into the equation:

% unc = 0 . 4  lb 5  lb × 100% = 8 % . size 12{%" unc = " { {0 "." 4" lb"} over {5" lb"} } times "100"%=8%} {}

Discussion

We can conclude that the weight of the apple bag is 5 lb ± 8 % size 12{5`"lb" +- 8%} {} . 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.

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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 . 00 m size 12{4 "." "00"" m"} {} and a width of 3 . 00 m size 12{3 "." "00"" m"} {} , with uncertainties of 2% size 12{2%} {} and 1% size 12{1%} {} , respectively, then the area of the floor is 12 . 0 m 2 size 12{"12" "." 0" m" rSup { size 8{2} } } {} and has an uncertainty of 3% size 12{3%} {} . (Expressed as an area this is 0 . 36 m 2 size 12{0 "." "36"" m" rSup { size 8{2} } } {} , which we round to 0 . 4 m 2 size 12{0 "." 4" m" rSup { size 8{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 . 05 s size 12{ +- 0 "." "05"`s} {} . Runners on the track coach’s team regularly clock 100-m sprints of 11.49 s size 12{"11.49 s"} {} to 15.01 s size 12{"15.01 s"} {} . At the school’s last track meet, the first-place sprinter came in at 12 . 04 s size 12{"12" "." "04"" s"} {} and the second-place sprinter came in at 12 . 07 s size 12{"12" "." "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.

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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

definition of mass of conversion
umezurike Reply
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
Adeshina Reply
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
Tamdy Reply
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
the range of objects and phenomena studied in physics is
Bethel Reply
what is Linear motion
Hamza Reply
straight line motion is called linear motion
then what
Amera
linear motion is a motion in a line, be it in a straight line or in a non straight line. It is the rate of change of distance.
Saeedul
Hi
aliyu
your are wrong Saeedul
Richard
Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension
Jason
is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimensions. 
Praise
what is a classical electrodynamics?
Marga
what is dynamics
Marga
dynamic is the force that stimulates change or progress within the system or process
Oze
what is the formula to calculate wavelength of the incident light
David Reply
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
Hassan Reply
State the forms of energy
Samzy Reply
machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
Clement Reply
correct
Akinpelu
what is mechanical wave
Akinpelu Reply
a wave which require material medium for its propagation
syed
The S.I unit for power is what?
Samuel Reply
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
correct
Jaheim
kW
Akinpelu
OK that's right
Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
What is physics
aish Reply
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
Syafiqah Reply
reasonable
Femi
because it is balanced by the inward acceleration otherwise known as centripetal acceleration
MUSTAPHA
What is a wave
Mutuma Reply
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
mistakes thanks
Akinpelu
find the triple product of (A*B).C given that A =i + 4j, B=2i - 3j and C = i + k
Favour Reply
Practice Key Terms 6

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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