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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 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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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