Back to Course
Units and Errors
Physical Quantities and SI Units
The International System of Units (SI) classifies physical quantities into Base Quantities, Derived Quantities, and Supplementary Units to provide a universal standard for measurement.
$$F = ma$$
Force is derived from Newton's Second Law, showing how derived units combine base units.
$F$=Force(newtons (N))
$m$=Mass(kilograms (kg))
$a$=Acceleration(metres per second squared (m/s²))
deriving Pressure units
→$[P] = [F]/[A] = [MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}] = \text{Pa}$
deriving Energy units
→$[E] = [F][d] = [MLT^{-2}][L] = [ML^2T^{-2}] = \text{J}$
Base Quantities: Seven fundamental building blocks: length (m), mass (kg), time (s), current (A), temperature (K), intensity (cd), and amount (mol).
Derived Quantities: Combinations of base units, such as Force ($1 \text{ N} = 1 \text{ kg m s}^{-2}$) or Pressure ($1 \text{ Pa} = 1 \text{ kg m}^{-1} \text{ s}^{-2}$).
Scientific Notation: Standardized format using powers of ten with exactly one non-zero digit to the left of the decimal.
The Seven SI Base Units
•
Length → metre (m)
•
Mass → kilogram (kg)
•
Time → second (s)
•
Electric Current → ampere (A)
•
Temperature → kelvin (K)
•
Luminous Intensity → candela (cd)
•
Amount of Substance → mole (mol)
Common SI Prefixes
From largest to smallest
•
giga (G) = 10⁹
•
mega (M) = 10⁶
•
kilo (k) = 10³
•
centi (c) = 10⁻²
•
milli (m) = 10⁻³
•
micro (µ) = 10⁻⁶
•
nano (n) = 10⁻⁹
•
pico (p) = 10⁻¹²
Supplementary units describe purely geometric quantities: the Radian for plane angles and the Steradian for solid angles.
$$\theta = \frac{s}{r}$$
The radian is defined by the ratio of arc length to radius.
$\theta$=Plane angle(radians (rad))
$s$=Arc length(meters (m))
$r$=Radius(meters (m))
$s = r$
→$\theta = 1 \text{ rad} \approx 57.3^\circ$
$s = 2\pi r$
→Full circle is $2\pi$ radians
Steradian (sr): Subtended at center of sphere by area $A = r^2$. Total solid angle of a sphere is $4\pi$ sr.
Precision, Accuracy, and Errors
Experimental results differ from true values due to Random Error, caused by unknown fluctuations, and Systematic Error, caused by faulty calibration or technique.
Accuracy: How close a measurement is to the true value; limited by systematic error.
Precision: How consistent repeated measurements are; limited by Least Count and random error.
Parallax Error: Always read measurements with eye perpendicular to scale to avoid systematic offset.
Quantifying the reliability of data through absolute, fractional, and percentage uncertainties.
$$\text{Percentage Uncertainty} = \frac{\Delta x}{x} \times 100\%$$
Relates the instrument's limitation to the magnitude of the measured quantity.
$\Delta x$=Absolute uncertainty (usually least count)(same as $x$)
$x$=Measured value(variable)
$x$ is very small
→Percentage uncertainty becomes very high for a fixed instrument.
using $x^n$
→$\%\text{ uncertainty in } x^n = |n| \times \%\text{ uncertainty in } x$
Products/Quotients: ADD percentage uncertainties. $\%\Delta(AB) = \%\Delta A + \%\Delta B$.
Sums/Differences: ADD absolute uncertainties. $\Delta(A \pm B) = \Delta A + \Delta B$.
Powers: Multiply percentage uncertainty by the power. $\%\Delta(x^n) = |n| \times \%\Delta x$.
Precision ≠ Accuracy: A vernier caliper (least count 0.01 cm) is more precise than a metre rule (0.1 cm), but measuring a small object like 0.45 cm gives 2% uncertainty — less accurate than 25.5 cm with a metre rule (0.4% uncertainty).
Rules of Significant Figures
Significant Figures include all accurately known digits plus the first Doubtful Digit, indicating the precision of a measurement.
Leading Zeros: Never significant (e.g., 0.002 has 1 sig fig).
Trapped Zeros: Always significant (e.g., 2.002 has 4 sig figs).
Trailing Zeros: Significant only if a decimal is present (e.g., 2.30 has 3, 230 has 2).
Scientific Notation Clarity: In $8.70 \times 10^4$ kg, the digits 8, 7, 0 are all significant (3 sig figs). Powers of ten are NOT counted as significant figures.
Ambiguous Integers: 8000 kg could have 1, 2, 3, or 4 sig figs depending on instrument precision. Writing $8.00 \times 10^3$ kg makes it unambiguous (3 sig figs).
Maintaining precision consistency during calculations requires specific rounding rules for addition/subtraction and multiplication/division.
Multiplication/Division: Result keeps the same number of sig figs as the Least Accurate Factor.
Addition/Subtraction: Result keeps the same number of Decimal Places as the least precise term.
When the digit to be dropped is exactly 5, follow the even-odd convention to avoid systematic bias in large datasets.
$$\text{If digit to drop is 5, round toward even}$$
The 'round half to even' rule prevents statistical bias when rounding many numbers.
$\text{even}$=Last retained digit is even (0, 2, 4, 6, 8)(N/A)
$\text{odd}$=Last retained digit is odd (1, 3, 5, 7, 9)(N/A)
2.345 to 3 sig figs
→Last retained digit is 4 (even), so round to 2.34
2.355 to 3 sig figs
→Last retained digit is 5 (odd), so round to 2.36
4.650 to 2 sig figs
→Last retained digit is 6 (even), so round to 4.6
4.750 to 2 sig figs
→Last retained digit is 7 (odd), so round to 4.8
Why This Rule: If we always rounded 5 up, averaging many rounded numbers would show upward bias. Even-odd balances this.
Application Strategy
Worked Examples
Problem Statement
Calculate the area of a rectangular plate with length 15.3 cm and width 12.80 cm to the correct number of significant figures.
Area = L × WSignificant figures in multiplication
Strategic Toolkit
$A = L \times W$
Step 1
Identify Inputs
L = 15.3 (3 sig figs), W = 12.80 (4 sig figs).
$L = 15.3 \text{ cm}, W = 12.80 \text{ cm}$
Step 2
Determine Rule
In multiplication, the result must match the number of sig figs of the least accurate factor (3 sig figs).
$\text{Result Sig Figs} = \min(3, 4) = 3$
Step 3
Calculate Raw Product
Multiply the raw values.
$15.3 \times 12.80 = 195.84 \text{ cm}^2$
Step 4
Apply Rounding
Round 195.84 to 3 significant figures. Since the first dropped digit is 8 (> 5), increase the last retained digit.
$195.84 \xrightarrow{\text{3 SF}} 196 \text{ cm}^2$
Problem Statement
Add three length measurements: 72.1 cm, 3.42 cm, and 0.003 cm. Express the result to appropriate precision.
Decimal places rule in additionPrecision vs significant figures
Strategic Toolkit
Result precision = least precise term
Step 1
Identify Decimal Places
Count decimal places in each measurement.
$72.1 \text{ (1 dec. place)}, 3.42 \text{ (2 dec. places)}, 0.003 \text{ (3 dec. places)}$
Step 2
Find Limiting Term
The result is limited by the measurement with fewest decimal places.
$\text{Limiting term: } 72.1 \text{ cm with 1 decimal place}$
Step 3
Add Raw Values
Sum without rounding first.
$72.1 + 3.42 + 0.003 = 75.523 \text{ cm}$
Step 4
Apply Decimal Place Rule
Round to 1 decimal place (same as limiting term).
$75.523 \xrightarrow{\text{1 dec. place}} 75.5 \text{ cm}$
Problem Statement
A rectangle has length $L = 10.0 \pm 0.1$ cm and width $W = 5.0 \pm 0.1$ cm. Calculate the percentage uncertainty in the area.
Uncertainty in productsPercentage uncertainty addition
Strategic Toolkit
$\%\text{ uncertainty in product} = \sum \%\text{ uncertainties}$
Step 1
Calculate Individual Uncertainties
Find percentage uncertainty for each measurement.
$\%\Delta L = \frac{0.1}{10.0} \times 100\% = 1\%$$\%\Delta W = \frac{0.1}{5.0} \times 100\% = 2\%$
Step 2
Apply Product Rule
For multiplication/division, add percentage uncertainties.
$\%\Delta A = \%\Delta L + \%\Delta W = 1\% + 2\% = 3\%$
Step 3
Calculate Area
Compute nominal area value.
$A = L \times W = 10.0 \times 5.0 = 50.0 \text{ cm}^2$
Step 4
Express Final Result
Combine area with percentage uncertainty.
$A = 50.0 \pm 3\% \text{ cm}^2 \approx 50.0 \pm 1.5 \text{ cm}^2$
Memorize formulas
Radian Measure
Plane angle formula — ratio of arc length to radius.
Formula
$\theta = \frac{s}{r}$
Steradian Measure
Solid angle formula for 3D surfaces.
Formula
$\Omega = \frac{A}{r^2}$
Percentage Uncertainty
Uncertainty expressed as percentage of measured value.
Formula
$\%\text{ Uncertainty} = \frac{\Delta x}{x} \times 100\%$
Uncertainty in Powers
Percentage uncertainty multiplies by the power for derived quantities.
Formula
$\%\text{ uncertainty in } x^n = |n| \times \%\text{ uncertainty in } x$
Uncertainty in Products/Quotients
For multiplication/division, add percentage uncertainties.
Formula
$\%\Delta\left(\frac{A}{B}\right) = \%\Delta A + \%\Delta B$
Uncertainty in Sums/Differences
For addition/subtraction, add absolute uncertainties.
Formula
$\Delta(A \pm B) = \Delta A + \Delta B$
Zero Error Correction
How to correct for positive and negative zero error.
Formula
$\text{Actual} = \text{Reading} - \text{Positive Error} = \text{Reading} + |\text{Negative Error}|$
Review
1
A student measures the thickness of a wire as 0.00340 m. How many significant figures are in this measurement?
Ans
Three
Review
Tap to reveal
Count only the significant digits
Ignore the leading zeros because they only place the decimal point. Then count the non-zero digits and the trailing zero after the decimal, since that zero is significant.
$0.00340$ has $3$ significant figures.
2
A student multiplies $3.24 \times 1.5$. How many significant figures should the final answer have?
Ans
2
Review
Tap to reveal
Use the least precise factor
For multiplication, the final answer must have the same number of significant figures as the factor with the fewest significant figures. Here, $1.5$ limits the result to 2 significant figures.
$3.24$ has $3$ sig figs and $1.5$ has $2$ sig figs, so the product is reported to $2$ sig figs.
3
The time period of a simple pendulum is $T = 2\pi\sqrt{l/g}$. If $l$ has a 4% uncertainty, what is the percentage uncertainty in $T$?
Ans
2%
Review
Tap to reveal
Apply the square-root power rule
The period depends on length as a square root, so the percentage uncertainty is multiplied by the power $1/2$. The constant factor $2\pi$ and the fixed value of $g$ do not change the uncertainty here.
$\%\Delta T = \frac{1}{2} \times 4\% = 2\%$
4
Which type of error can be reduced by taking the arithmetic mean of several observations?
Ans
Random Error
Review
Tap to reveal
Identify the error that fluctuates
A mean helps when the errors vary unpredictably above and below the true value, because the positive and negative deviations tend to cancel. A consistent bias cannot be removed this way.
Random error is reduced by averaging.
5
If $x$ has a $2\%$ uncertainty and $y$ has a $3\%$ uncertainty, what is the percentage uncertainty in $\frac{x^2}{y}$?
Ans
7%
Review
Tap to reveal
Apply the power rule first
The uncertainty in $x^2$ is twice the uncertainty in $x$, because a power multiplies percentage uncertainty by that power. So $x^2$ contributes $4\%$.
$\%\Delta(x^2) = 2 \times 2\% = 4\%$
Add the division uncertainties
For division, percentage uncertainties add. Combine the $4\%$ from $x^2$ with the $3\%$ from $y$ to get the total.
$\%\Delta\left(\frac{x^2}{y}\right) = 4\% + 3\% = 7\%$
6
A Vernier caliper has a positive zero error of 0.02 cm. If it reads 2.56 cm for the length of a rod, what is the corrected length?
Ans
2.54 cm
Review
Tap to reveal
Subtract a positive zero error
A positive zero error means the instrument overreads, so the displayed value is larger than the true value. Correct it by subtracting the error from the reading.
$2.56 - 0.02 = 2.54\text{ cm}$
7
A body starts from rest ($u = 0$). Using $s = \frac{v^2}{2a}$, if $v$ has 2% uncertainty and $a$ has 1% uncertainty, what is the percentage uncertainty in $s$?
Ans
5%
Review
Tap to reveal
Treat the squared velocity as the first source of uncertainty
Since $v$ is squared, its percentage uncertainty doubles before anything else is combined. The factor of 2 in the denominator is a constant and does not add uncertainty.
$\%\Delta(v^2) = 2 \times 2\% = 4\%$
Add the uncertainty from division by $a$
For a quotient, percentage uncertainties are added. Combine the $4\%$ from $v^2$ with the $1\%$ from $a$ to get the total for $s$.
$\%\Delta s = 4\% + 1\% = 5\%$
8
A density is given as $2.7 \text{ g/cm}^3$. What is this in SI base units (kg/m³)?
Ans
$2700 \text{ kg/m}^3$
Review
Tap to reveal
Convert grams and cubic centimetres separately
Replace the mass unit with kilograms and remember that the length conversion must be cubed because the denominator is in cubic centimetres.
$1\text{ g} = 10^{-3}\text{ kg},\quad 1\text{ cm}^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3$
Substitute the conversion factors
Combine the two conversions in the density expression and simplify the powers of ten.
$2.7\text{ g/cm}^3 = 2.7 \times \frac{10^{-3}}{10^{-6}}\text{ kg/m}^3 = 2.7 \times 10^3\text{ kg/m}^3$
9
A square has side length $L = 5.0 \pm 0.1$ cm. What is the percentage uncertainty in its area?
Ans
4%
Review
Tap to reveal
Find the percentage uncertainty in the side
The relative uncertainty in a measured length is the absolute uncertainty divided by the measured value, expressed as a percentage.
$\%\Delta L = \frac{0.1}{5.0} \times 100\% = 2\%$
Square the uncertainty for area
Because area depends on the square of the side, the percentage uncertainty doubles.
$\%\Delta A = 2 \times 2\% = 4\%$
10
Which prefix represents $10^{-9}$ and is the correct replacement for 'millimicro'?
Ans
nano (n)
Review
Tap to reveal
Match the power of ten
The SI prefix for $10^{-9}$ is nano. 'Milli' times 'micro' also gives $10^{-9}$, so the compound wording should be replaced by the standard prefix.
$10^{-9} \rightarrow \text{nano (n)}$
11
Two resistors $R_1 = 100 \pm 2 \text{ }\Omega$ and $R_2 = 200 \pm 5 \text{ }\Omega$ are connected in series. What is the absolute uncertainty in the total resistance?
Ans
$7 \text{ }\Omega$
Review
Tap to reveal
Add the resistances and their absolute errors
For quantities added in series, the absolute uncertainties add directly. Use the uncertainty values, not the percentage uncertainties.
$\Delta R = 2\,\Omega + 5\,\Omega = 7\,\Omega$
12
Express 0.000 045 6 in scientific notation. How many significant figures does it have?
Ans
3
Review
Tap to reveal
Shift the decimal to scientific notation
Move the decimal so the number is between 1 and 10, then count how many places it moved to determine the negative power of ten.
$0.0000456 = 4.56 \times 10^{-5}$
Count the significant digits
Only the non-zero digits count here, since the leading zeros are just placeholders. The digits 4, 5, and 6 give the total.
There are $3$ significant figures.
13
A student reads the length of a pencil using a ruler while looking at the scale from an angle. The reading is consistently 0.3 cm more than the true length. This error is classified as:
Ans
Systematic error due to parallax
Review
Tap to reveal
Recognize the consistent bias
If every reading is too high by the same amount, the error is not random. It is a repeatable shift caused by viewing the scale from the wrong angle, which is parallax.
Systematic error due to parallax
Question 1 of 13
0:00
1
A student measures the thickness of a wire as 0.00340 m. How many significant figures are in this measurement?