Errors and uncertainties

Distinguish between accuracy and precision.

Accuracy refers to the closeness of a measured value to the ‘true’ or ‘known’ value

Precision refers to how close a set of measured values are to each other

The mass of a coin is measured to be 12.5±0.1g. The diameter is 2.8±0.1cm and the thickness 2.1±0.1mm. Calculate the average density of the material from which the coin is made with its uncertainty. Give your answer in `kg.m^(-3)`

The average density is `rho = m/V`

`12.5+-0.1 g=(12.5+-0.1)xx10^(-3)kg`

`2.8+-0.1 cm=(2.8+-0.1)xx10^(-2)m`

`2.1+-0.1mm=(2.1+-0.1)xx10^(-3)m`

Thus,

`rho=m/V=m/(pi(d/2)^2h)=(4m)/(pidh)=(4xx12.5xx10^(-3))/(pixx(2.8xx10^(-2))^2xx2.1xx10^(-3))=9666.8kg.m^(-3)`

To calculate uncertainty

`(Deltarho)/rho= (Deltam)/m+(DeltaV)/V=(Deltam)/m+2(Deltad)/d+(Deltah)/h`

`rightarrow Deltarho = ((Deltam)/m+2(Deltad)/d+(Deltah)/h)xxrho=(0.1/12.5+2(0.1)/1.8+0.1/2.1)xx9666.8=1228.1 kg.m^(-3)`

Therefore,

`rho=(9.7+-1.2)xx10^3kg.m^(-3)`

A metal wire of length L has a circular cross-section of diameter d, as shown in figure below

The volume V of the wire is given by the expression

`V=(pid^2L)/4`

The diameter d, length L and mass M are measured to determine the density of the metal of the wire. The measured values are:

d = 0.38±0.01 mm,

L = 25.0±0.1 cm,

M = 0.225±0.001 g.

Calculate the density of the metal, with its absolute uncertainty. Give your answer to an appropriate number of significant figures.

Convert all measurements to the SI units. 

`d=0.38+-0.01mm=(0.38+-0.01)xx10^(-3)m`

`L=25.0+-0.1cm=(25.0+-0.1)xx10^(-2)m`

`M=0.225+-0.001g = (0.225+-0.001)xx10^(-3)kg`

The density of metal:

`rho=M/V=M/((pid^2L)/4)=(0.225xx10^(-3))/((pixx(0.38xx10^(-3))^2xx25xx10^(-2))/4)=7935.7kg.m^(-3)`

To calculate uncertainty

`(Deltarho)/rho= (DeltaM)/M+(DeltaV)/V=(Deltam)/m+2(Deltad)/d+(DeltaL)/L`

`rightarrow Deltarho = ((Deltam)/m+2(Deltad)/d+(DeltaL)/L)xxrho=(0.001/0.225+2(0.01)/0.38+0.1/25.0)xx7935.7=484.7 kg.m^(-3)`

Therefore,

`rho=(7.9+-0.5)xx10^3kg.m^(-3)`

An analogue voltmeter is used to take measurements of a constant potential difference across a resistor.

For these measurements, describe one example of:

i. a systematic error,

ii. a random error

i. zero error or wrongly calibrated scale 

ii. reading scale from different angles or wrongly interpolating between scale readings/divisions

The potential difference across a resistor is measured as 5.0 V±0.1V.

The resistor is labelled as having a resistance of 125Ω±3%.

i. Calculate the power dissipated by the resistor

ii. Calculate the percentage uncertainty in the calculated power.

iii. Determine the value of the power, with its absolute uncertainty, to an appropriate number of significant figures.

i. `P=V^2/R or P=VI and V=IR`

`P=5.0^2/125 or 5.0xx0.04 or (0.04)^2xx125=0.20W`

ii. `%V=2% or (DeltaV)/V=0.02`

`%P = (2xx2%)+3%or %P =(2xx0.02+0.03)xx100=7%`

iii. absolute uncertainty in `P=(7/100)xx0.20=0.014`

power = `0.02+-0.01 W or (2.0+-0.1)xx10^(-1)W`

The speed of a sound wave through a gas of pressure P and density ρ is given by the equation

`v=sqrt((kP)/rho` where k is constant.

An experiment is performed to determine k. The percentage uncertainties in v, P and ρ are ±4%, ±2% and ±3% respectively. Which of the following gives the percentage uncertainty in k?

A. ±5%

B. ±9%

C. ±13%

D. ±21%

The answer is C.

From `v=sqrt((kP)/rho`, we have `k=(v^2rho)/P`

the percentage uncertainty in k will be

`4xx2+2+3=+-13%`

In an experiment, a radio-controlled car takes 2.50 ± 0.05 s to travel 40.0 ± 0.1 m. 

What is the car’s average speed and the uncertainty in this value? 

A. 16 ± 1 m/s 

B. 16.0 ± 0.2 m/s 

C. 16.0 ± 0.4 m/s 

D. 16.00 ± 0.36 m/s

The correct answer is A.

Calculate the average speed: `v=d/t=(40.0m)/(2.5s)=16.0m.s^(-1)` 

Calculate the relative uncertainties: `(Deltad)/d=0.1/40=0.0025` and `(Deltat)/t=0.05/2.50=0.02`. 

Adding these, we get a total relative uncertainty of 0.0225. 

Therefore, the uncertainty in the car's speed is `16.0 m.s^(-1)xx0.0225=0.36m.s^(-1)`, approximately. 

So, the average speed of the car is `16.0+-0.4m.s^(-1)`.

The speed of a car is calculated from measurements of distance travelled and the time taken. 

Distance is measured as 200 m, with an uncertainty of ± 2 m.

The time is measured as 10.0 s, with an uncertainty of  ± 0.2 s.

What is the percentage uncertainty in the calculated speed?

A. ± 0.5 %

B. ± 1 %

C. ± 2 %

D. ± 3 %

The answer is D.

We have `"Speed"(v)=("Distance"(d))/("Time"(t)`

Percentage uncertainty in the calculated speed = `(Deltav)/vxx100%=((Deltad)/d +(Deltat)/t)xx100%=(2/200+0.2/10)xx100%=+-3%`

In an experiment to determine the acceleration of free fall g , the time t taken for a ball to fall through distance s is measured. The percentage uncertainty in the measurement of s is 2%. The percentage uncertainty in the measurement of t is 3%. 

The value of g is determined using the equation shown `g=(2s)/t^2`

What is the uncertainty in the calculated value of g?

A. 1%

B. 5%

C. 8%

D. 11%

The answer is C.

Identify the given percentage uncertainties: 2% for s and 3% for t.

Since t is squared in the equation, its percentage uncertainty is doubled: 2×3%=6%

Add the percentage uncertainties of s and t: 2%+6%=8%

Therefore, the percentage uncertainty in the value of g is 8%.

A micrometer screw gauge is used to measure the diameter of a small uniform steel sphere. The micrometer reading is 5.00 mm ± 0.01 mm. 

What will be the percentage uncertainty in a calculation of the volume of the sphere, using these values?

A. 0.2%

B. 0.4%

C. 0.6%

D. 1.2%

The answer is C.

Diameter of sphere: D=5.00 mm ± 0.01 mm 

We know that volume of sphere is given by 

`V=4/3piR^3=4/3pi(D/2)^3=(piD^3)/6`

Percentage uncertainty in measurement is given as 

`(DeltaV)/Vxx100%=3(DeltaD)/Dxx100%=3xx(0.01/5)xx100%=0.6%`