Measurements and their Errors

AQA Physics A-Level Study Notes

Section 1: Measurements and their Errors

3.1 Measurements and Their Errors

3.1.1 - Uses of SI Units and Their Prefixes

• SI Units: The fundamental units used in physics, essential for precise measurement. They comprise:

  • Mass (m): kg (kilograms)

  • Length (l): m (meters)

  • Time (t): s (seconds)

  • Amount of substance (n): mol (moles)

  • Temperature (T): K (kelvin)

  • Electric current (I): A (amperes)

• Derived Units: The SI units of quantities can be derived from their fundamental forms. For example, the formula for force is given by:

  • $F = ma$

  • The SI units for force (F) can be determined as:

  • $ext{Mass:} <br>ightarrow kg$

  • $ext{Acceleration:} <br>ightarrow m ext{s}^{-2}$

  • Thus, the units for force become:

    • $kg imes m imes s^{-2}$ (which is also known as N, Newtons)

• Example of Voltage (V) Calculation: To find the SI units of voltage:

  • Voltage defined as $V = rac{E}{Q}$
    where E is energy and Q is charge.

  • Energy can be expressed as $E = rac{1}{2}mv^2$, thus SI units:

  • For energy: $kg imes rac{m^2}{s^2}$

  • Charge can be defined as $Q = It$,

  • For charge: $As$ (ampere-seconds)

  • Therefore, simplifying:

    • $V = rac{As}{kg imes m imes s^{-2}}$ gives:

    • $V = rac{As imes s^2}{kg imes m} = rac{A imes s}{kg} imes m^2$

• SI Unit Prefixes: Additional prefixes used to denote multiples and submultiples of SI units include:

  • Tera (T): $10^{12}$

  • Giga (G): $10^{9}$

  • Mega (M): $10^{6}$

  • Kilo (k): $10^{3}$

  • Centi (c): $10^{-2}$

  • Milli (m): $10^{-3}$

  • Micro (µ): $10^{-6}$

  • Nano (n): $10^{-9}$

  • Pico (p): $10^{-12}$

  • Femto (f): $10^{-15}$

3.1.2 - Limitation of Physical Measurements

• Random Errors:

  • Affect the precision of measurements and lead to variations around a mean value.

  • Example: Electronic noise in instruments.

  • To minimize random errors, one should:

  • Take at least three measurements and calculate the mean; this helps identify anomalies.

  • Utilize technology such as computers and data loggers to reduce human error.

  • Employ high-resolution equipment, e.g. a micrometer offers a resolution of 0.1 mm compared to a ruler's 1 mm.

• Systematic Errors:

  • Impact accuracy and cause consistent deviations in one direction (either high or low).

  • Example: A scale that is not zeroed correctly (zero error) or parallax error when reading scales at an angle.

  • To minimize systematic errors, one might:

  • Calibrate devices against known values (e.g. weighing a 1 kg standard).

  • Correct for background radiation in experiments by measuring it first and excluding from results.

  • Read liquid levels at eye level to avoid parallax errors.

• Precision

  • Defined as the consistency of measurement, even if they deviate from the actual value.

• Repeatability

  • The ability for the original experimenter to replicate the experiment with the same setup and obtaining similar results.

• Reproducibility

  • If another experimenter, using different methods or equipment, achieves the same results.

• Resolution

  • The smallest distinguishable change in the quantity measured that results in a noticeable change in reading.

• Accuracy

  • Measures how close a value is to the actual, true value.

• Uncertainty

  • The extent to which a result may vary, described with bounds, e.g. for a temperature reading of 20°C ± 2°C, where true value could be within 18-22°C.

• Types of Uncertainty:

  • Absolute Uncertainty: Given as a fixed value, e.g. $ext{Voltage:} 7.0 ext{ V} ext{ ± 0.6 V}$.

  • Fractional Uncertainty: Expressed as a fraction of the measurement, e.g. $7.0 ext{ V} ext{ ± } rac{0.3}{7} ext{ V}$.

  • Percentage Uncertainty: Given as a percentage of the measurement, e.g. 7.0 ext{ V} ext{ ± } 8.6 ext{%}.

• Reducing Percentage and Fractional Uncertainty:

  • Measure larger quantities whenever possible.

• Reading vs. Measuring:

  • Readings involve one value, e.g. temperature from a thermometer.

  • Measurements compare two values, e.g. the difference read from a ruler.

• Uncertainty Calculation:

  • Uncertainty for a reading is ± half the smallest division.

  • Uncertainty for measurements takes into account both ends. For instance:

  • For a ruler, if each side $±0.5 ext{ mm}$, the total uncertainty is $±1 ext{ mm}$.

• Digital Values:

  • Often quoted or assumed as $±$ the last significant digit.

• Reducing Uncertainty Techniques:

  • Fix one end of the measuring device to reduce measured uncertainty.

  • Increase the number of measurements to refine results by averaging. For instance, measuring the time for multiple swings can yield:

  • $10 ext{ swings: } 6.2 ext{ ± } 0.1 ext{ s} <br>ightarrow 1 ext{ swing: } 0.62 ext{ ± } 0.01 ext{ s}$.

• Significant Figures in Uncertainty:

  • Uncertainties should match the number of significant figures of the data values.

• Combining Uncertainties:

  • For Addition/Subtraction: Add absolute uncertainties.

  • Example Calculation:

    • If a thermometer shows water temperature dropping from $298 ext{ ± 0.5 K}$ to $273 ext{ ± 0.5 K}:$

    • Temperature Difference Calculation:

      • $298 - 273 = 25 K$

      • Total Uncertainty: $0.5 + 0.5 = 1 K$

      • Final Difference: $25 ext{ ± } 1 K$

• For Multiplying/Dividing: Add percentage uncertainties.

  • Example Problem:

    • Given force of $91 ext{ ± 3 N}$ and mass of $7 ext{ ± 0.2 kg}$:

    • Acceleration Calculation: $a = rac{F}{m} = rac{91}{7} = 13 ext{ m s}^{-2}$

    • Calculate percentage uncertainty:

      • Uncertainty % = rac{3}{91} imes 100 = 3.3 ext{%}

      • Uncertainty % for mass = rac{0.2}{7} imes 100 = 2.9 ext{%}

      • Total = 3.3 ext{%} + 2.9 ext{%} = 6.2 ext{%}

      • Thus, $a = 13 ext{ ± } 0.8 m s^{-2}$ where 6.2 ext{%} ext{ of } 13 ext{ is } 0.8.

• Raising Measurements to a Power: Multiply percentage uncertainty by the power raised.

  • Example Problem:

    • For a circle with radius $5 ext{ ± 0.3 cm}$, percentage uncertainty in area:

    • Area formula: $A = ext{π} r^2$ thus:

      • Area Calculation: $A = ext{π} imes 25 = 78.5 ext{ cm}^2$.

      • % uncertainty in radius = rac{0.3}{5} imes 100 = 6 ext{%}.

      • % uncertainty in area = 6 ext{%} imes 2 = 12 ext{%}.

      • Final Area with Uncertainty: A = 78.5 ext{ ± } 12 ext{%} ext{ cm}^2.

• Uncertainties on Graphs:

  • Represent uncertainties via error bars. E.g., if uncertainty = 5 mm, represent with 5 squares error bar on either side of a data point.

  • A line of best fit must pass through all error bars, neglecting any anomalous points.

  • Gradient uncertainties: Calculated using the steepest and shallowest lines of worst fit that cover all error bars.

  • Gradient Uncertainty Calculation:

    • ext{Percentage Uncertainty} = rac{ ext{Best Gradient}}{100 ext{%}} imes | ext{Best Gradient} - ext{Worst Gradient}|.

3.1.3 - Estimation of Physical Quantities

• Orders of Magnitude:

  • Powers of ten indicating the scale of an object, useful for comparing sizes.

  • Example: Diameter of atomic nuclei is around $10^{-14} m$.

  • Comparison: $100 m$ is two orders of magnitude greater than $1 m$.

  • Nearest Order of Magnitude Calculation:

  • For diameter of a hydrogen atom, find approximate area assuming a spherical shape:

  • $ext{Diameter} = 1.06 imes 10^{-10} m$, thus radius $r = 0.53 imes 10^{-10} m$.

  • Use the area formula:

    • $A = ext{π}r^2 = ext{π} imes (0.53 imes 10^{-10})^2 = ext{π} imes 2.8 imes 10^{-21} m^2$,

    • Resulting area approximated to $8 imes 10^{-21} m^2$ (expressed to 1 s.f.).

  • Final answer rounded to nearest order of magnitude: $10^{-20} m^2$.

• Estimation in Physics:

  • Necessary to approximate physical quantities for comparisons or validation of calculated values.