Measurement, units, uncertainty and vectors

Measurement, units, uncertainty and vectors

Quantities and the meaning of a measurement

A measurement compares a physical quantity with an agreed unit. Reporting a bare number is usually insufficient: a length of12 mm differs from12 m. SI base quantities include length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity. Their base units are metre, kilogram, second, ampere, kelvin, mole and candela respectively. Derived quantities combine base quantities: velocity has unit m/s, acceleration m/s², force kg m/s² and density kg/m³. Write unit symbols carefully: m denotes metre, while M is a different prefix or symbol in other contexts. A space between a value and its unit improves clarity.

Choosing and reading instruments

A metre rule measures ordinary lengths, vernier calipers measure external or internal dimensions and depth, and a micrometer screw gauge measures small thicknesses or diameters. First identify the smallest readable increment, then check for a zero error. A positive zero error means the instrument reads too high when the true reading should be zero, so subtract it; a negative zero error is added in magnitude. Read the scale at right angles to reduce parallax. For a vernier reading, add the main-scale reading just before the vernier zero to the coincident vernier division multiplied by the least count. The exact least count depends on that instrument's scale.

Mass, volume and density

A balance measures mass by comparison with a calibrated reference or force-related mechanism. Weight is a force and has a different unit. The volume of a regular rectangular solid is length × width × height. An irregular insoluble solid can be measured by displacement in a suitable liquid, provided it is completely submerged and trapped air is avoided. Read a measuring cylinder at eye level and use the appropriate meniscus. Density is. To convert g/cm³ to kg/m³, multiply by1,000. For example,2.7 g/cm³ is2,700 kg/m³. Do not convert a cubic unit using the linear factor.

Accuracy, precision and error

Accuracy describes closeness to the accepted or reference value; precision describes how closely repeated measurements agree or the fineness of resolution, depending on context. Repeated readings can be precise yet inaccurate because an uncorrected zero error affects all of them. Random errors produce scatter and can be reduced in their effect on the mean through suitable repeated measurements. Systematic errors require identifying and correcting the cause. Absolute uncertainty has the same unit as the measured quantity. Fractional uncertainty is absolute uncertainty divided by the measured value, and percentage uncertainty is that fraction multiplied by100. A larger measured interval often reduces the percentage effect of a fixed reading uncertainty.

Significant figures and standard form

Use standard form a ×10ⁿ with1 ≤|a| <10 for a nonzero number. Thus0.00042 m is4.2 ×10⁻⁴ m and3,500,000 s is3.5 ×10⁶ s when two significant figures are intended. Leading zeros locate the decimal point but do not count as significant digits. Retain extra digits in intermediate calculations and round the final result to a precision justified by the data. A calculator can display many digits without making the original measurements more precise. In simple school treatments, percentage uncertainties add approximately for multiplication and division; use the uncertainty rule specified in the question rather than inventing greater confidence.

Dimensions as a checking method

Dimensions describe the base-quantity structure of a physical quantity. Velocity has dimensions LT⁻¹, acceleration LT⁻², force MLT⁻² and energy ML²T⁻². Every term added within a valid physical equation must have the same dimensions. In +½at², both terms on the right have dimensions of length. Dimensional agreement can rule out an incorrect formula but cannot prove a formula is completely correct: it cannot determine a pure number such as½. Trigonometric, exponential and logarithmic arguments require dimensionless quantities. Do not confuse a dimension with a chosen unit; centimetres and metres are different units with the same length dimension.

Scalars, vectors and resolution

A scalar has magnitude without a spatial direction, such as mass, time, temperature or distance. A vector has magnitude and direction, such as displacement, velocity, acceleration or force. Two5 N forces can produce results from zero to10 N depending on their directions. For perpendicular components x and y, resultant magnitude is√(x² +y²). If a vector makes angleθ with a chosen x-axis, its components are A cosθ along x and A sinθ perpendicular to x. Use signs to show whether components point along or against your positive axes. Sine and cosine swap roles if the angle is measured from the other axis.

Vector addition and relative velocity

Add vectors head to tail or add their corresponding signed components. A displacement6 m east followed by8 m north gives resultant magnitude10 m and direction tan⁻¹(8/6), about53.1° north of east. The distance travelled is14 m, demonstrating why a scalar route length differs from a vector change of position. Relative velocity compares two objects within the same frame: velocity of A relative to B is vA −vB. Two cars moving east at20 and15 m/s have relative speed5 m/s. If they move toward each other at those speeds on a straight line, their separation decreases at35 m/s.

Planning an experiment and presenting graphs

Choose a range of values that reveals the relationship and keep relevant control variables steady. Label each table column with the quantity and unit, not a unit repeated inconsistently in every entry. On a graph use sensible scales, plotted points and a best-fit line or curve rather than automatically joining every point. Find a gradient using a large triangle on the best-fit line. State what the gradient represents from the equation connecting the axes; the vertical intercept may reveal an offset. Repeated observations, a reasoned estimate of uncertainty and a physically explained conclusion are stronger evidence than a smooth-looking graph alone.

Self-check and interpretation

A micrometer gives5.68 mm and has a positive zero error of0.03 mm. Its corrected reading is5.65 mm. A length of50.0 cm with uncertainty±0.1 cm has percentage uncertainty0.2%, while a5.0 cm reading with the same uncertainty has2%. A force has components12 N east and5 N north, so its magnitude is13 N. Finally, an equation claiming distance equals acceleration multiplied by time is dimensionally wrong, the dimension of velocity. These checks involve different skills, so practise instrument correction, unit conversion, uncertainty and vector diagrams separately before combining them in a longer problem.