Assessment Of Total Uncertainty In The Final Result

To assess the total uncertainty or error, it is necessary to evaluate the likely uncertainties in all the factors involved in that calculation. The maximum possible uncertainty or error in the final result can be found as follows. The proofs of these rules are given in appendix 2.

For addition and subtraction

Absolute uncertainties are added: for example, the distance x determined by the difference between two separate position measurements

For your information

These are not decoration pieces of glass but are the earliest known exquisite and sensitive thermometers, built by the academia del cimento (1657-1667)in Florence. They contained alcohol, some times colored red for easier reading.

For multiplication and division

Percentage uncertainties are added. For example the maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference V and resulting current flow I by using R = V/ I is found as follows:

V = 5.2 ± 0.1 V

I = 0.84 ± 0.05A

Hence total uncertainty in the value of resistance R when V is divided by I is 8%. The result is thus quoted as

Uncertainty of 8%

That is                                   R = 6.2 ± 0.5 ohms

The result is rounded off to two significant digits because both V and R have two significant figures and uncertainty, being an estimate only, is recorded only, is recorded by one significant figure.

For power factor

Interesting information

Some specific temperatures

Multiply the percentage uncertainty by that power. For example, in the calculation of the volume of a sphere using

%age uncertainty in V = 3 x % age uncertainty in radius r.

As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by a vernier calipers with least count 0.01 cm, then

The radius r is recorded as

R= 2.25 ± 0.01 cm

Absolute uncertainty = least count = ± 0.01 cm

 

Total percentage uncertainty in V = 3 x 0.4 = 1.2 %

Thus the result should be recorded as

For uncertainty in the average value of many measurements.

(i)                  Find the average value of measured values.

(ii)                Find deviation of each measured value from the average value.

(iii)               The mean deviation is the uncertainty in the average value

For example, the six readings of the micrometer screw gauge to measure the diameter of a wire in mm are

1.20, 1.22, 1.23, 1.19, 1.22, 1.21.

= 1.21 mm

The deviation of the readings, which are the difference without regards to the sign, between each reading  and average value are 0.01, 0.01, 0.02,0.02, 0.01, 0,

= 0.01 mm

Thus, likely uncertainty in the mean diameter 1.21 mm is 0.01 mm recorded as 1.21 ± 0.01 mm.

5. Assessment Of Total Uncertainty In The Final Result:  For the uncertainty in a timing experiment: The uncertainty in the time period of a vibrating body is found by dividing the least count of timing device by the number of vibrations. For example, the time of 30 vibrations of a simple pendulum recorded by a stopwatch accurate upto one tenth of a second is 54.6 s, the period

Thus, period T is quoted as T = 1.82 ± 0.003 s

Hence, it is advisable to count large number of swings to reduce timing uncertainty.

 

Example 1.1:the length, breadth and thickness of a sheet are 3.233m, 2.105 m and 1.05 m respectively. Calculate the volume of the sheet correct upto the appropriate significant digits.Solution: given length i= 3.233 mBreadth b = 2.105m As the factor 1.05 cm has minimum number of significant figures equal to three, therefore, volume is recorded upto 3 significant figures, hence,

 

Example 1.2: the mass of a metal box measured by a lever balance is 2.2kg. two silver coins of masses 10.01 g and 10.2 g measured by a beam balance are added to it. What is now the total mass of the box correct upto the appropriate precision.Solution: total mass when silver coins are added to box = 2.2 kg + 0.01001 kg + 0.01002 kg =2022003 kg Sine least precise is 202 kg, having one decimal place, hence total mass should be to one decimal place which is the appropriate precision. Thus the total mass = 202 kg.

 

 

Example 1.3: the diameter and length of a metal cylinder measured with the help of vernier calipers of least count 0.01 cm are 1.22 cm and 5.35 cm. calculate the volume V of the cylinder and uncertainty in it.Solution: given data is Diameter d = 1.22 cm with least count 0.01 cm Length I = 5.35 cm with least count 0.01 cm Absolute uncertainty in length = 0.01 cm Absolute uncertainty in diameter = 0.01 cm As volume is             total uncertainty in V = 2 ( %age uncertainty in diameter) + (%age uncertainty in length) = 2 x 0.8 + 0.2 = 1.8% 1.8% uncertainty