Errors and uncertainties

A cart accelerates from (20 ± 1) m.s^-1 to (30 ± 1) m.s^-1
What is the percentage uncertainty in the change of speed of the cart?

Change in speed: `"Δv" = 30 - 20 = 10" " "m" ."s"^-1`.
Each speed has an uncertainty of ±1.
Absolute uncertainties: `"Δ(Δv)" = 1 + 1 = 2`.

`"Percentage uncertainty" = (2 / 10) xx 100 = "20%"``

The mass of a coin is measured to be 12.5 ± 0.1 g. The diameter is 2.8 ± 0.1 cm and the thickness 2.1 ± 0.1 mm. 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"+-"0.1 mm"=("2.1"+-"0.1")xx10^(-3)` `"m"`
Thus,

`rho="m"/"V"="m"/(pi("d"/2)^2"h")=("4m")/(pi"dh")`

`(4xx12.5xx10^(-3))/(pixx(2.8xx10^(-2))^2xx2.1xx10^(-3))=9666.8 " kg"."m"^(-3)``
To calculate uncertainty

`(Deltarho)/rho= (Delta"m")/"m"+(Delta"V")/"V"=(Delta"m")/"m"+2(Delta"d")/"d"+(Delta"h")/"h"`

`rightarrow Deltarho = ((Delta"m")/"m"+2xx(Delta"d")/"d"+(Delta"h")/"h")xxrho`

`=(0.1/12.5+2xx(0.1)/1.8+0.1/2.1)xx9666.8=1228.1 " kg"."m"^(-3)`

Therefore,
`rho=(9.7+-1.2)xx10^3 " kg"."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"=(pi"d"^2"L")/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 the SI units. 
`"d"=0.38+-0.01" mm"=(0.38+-0.01)xx10^(-3)" m"`
`"L"=25.0+-0.1" cm"=(25.0+-0.1)xx10^(-2)" m"`
`"M"=0.225+-0.001" g" = (0.225+-0.001)xx10^(-3)" kg"`
The density of metal:

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

Percentage uncertainty:

`(Deltarho)/rho= (Delta"M")/"M"+(Delta"V")/"V"`

`=(Delta"m")/"m"+2xx(Delta"d")/"d"+(Delta"L")/"L"`

`rightarrow Deltarho = ((Delta"m")/"m"+2xx(Delta"d")/"d"+(Delta"L")/"L")xxrho`

`=(0.001/0.225+2xx(0.01)/0.38+0.1/25.0)xx7935.7=484.7 " kg"."m"^(-3)`

Therefore, `rho=(7.9+-0.5)xx10^3 " kg"."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:
a. Systematic error.
b. Random error.

a. The voltmeter might always show a reading that is too high or too low because it is not set up (calibrated) correctly. For example, it always shows 0.2 V more than the real value.
b. Each time you read the voltmeter, your eyes might look from a slightly different angle. This can make the needle look like it's pointing at a different number, causing small changes in the readings.

The potential difference across a resistor is measured as 5.0 ± 0.1 V.
The resistor is labelled as having a resistance of 125 ± 3% Ω.
a. Calculate the power dissipated by the resistor
b. Calculate the percentage uncertainty in the calculated power.
c. Determine the value of the power, with its absolute uncertainty, to an appropriate number of significant figures.

a. `"P"="V"^2/"R" or "P"="VI" and "V"="IR"`

`"P"=5.0^2/125 or 5.0xx0.04 or (0.04)^2xx125="0.2 W"`

b. `%"V"=2% or (Delta"V")/"V"=0.02`

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

c. Absolute uncertainty in `Delta"P"=(7/100)xx0.20=0.014`

`"P"=0.2+-0.01 " 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%.

Answer: C

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

The percentage uncertainty of k: 

`(Δ"k")/"k" = 2xx(Δv)/v + (Δρ)/ρ + (Δ"P")/"P"= 8% + 3% + 2% = 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^-1.
B. 16.0 ± 0.2 m.s^-1.
C. 16.0 ± 0.4 m.s^-1.
D. 16.00 ± 0.36 m.s^-1.

Answer: C

Average speed: `v="d"/"t"=(40.0)/(2.5)=16.0 " m"."s"^(-1)` 

Percentage uncertainties:

`(Delta"d")/"d"=0.1/40=0.0025` and `(Delta"t")/"t"=0.05/2.50=0.02`. 

Adding these, we get the percentage uncertainty of average speed: `(Deltav)/v=0.0225` 

So, the absolute uncertainty of average speed:  `Deltav=vxx0.0225= 16.0xx0.0225=0.36`
The average speed of car: `v=16.0+-0.36 " m"."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%".

Answer: D

We have `v="d"/"t"`

Percentage uncertainty of speed:

`(Deltav)/v=((Delta"d")/"d" +(Delta"t")/"t")=(2/200+0.2/10)=0.03`

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"=(2"s")/"t"^2`.

What is the uncertainty in the calculated value of g?
A. 1%.
B. 5%.
C. 8%.
D. 11%.

Answer: C

`(Δ"g")/"g" = (Δ"s")/"s" + 2xx(Δ"t")/"t"="2%"+2xx"3%"="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%.

Answer: C
Diameter of sphere: D = 5.00 ± 0.01 mm 

Volume of sphere: `"V"=4/3pi"R"^3=4/3pi("D"/2)^3=(pi"D"^3)/6`

Percentage uncertainty:

`(Delta"V")/"V"=3xx(Delta"D")/"D"=3xx(0.01/5)=0.006="0.6%"`