Formula sheet: Units and Measurements

Units and Measurements14 min readLesson 1 of 5

1.1 Units and the SI system A measurement is a number attached to a unit, and every calculation carries the units along. The SI is built on a small set of base quantities, and every other unit is assembled from them, which is why a dimensional check can catch many errors before a number is computed. The official base unit table is reproduced below. Each unit is now fixed by a defined constant rather than by a physical artefact, so length, mass and time trace back to the speed of light, the Planck constant and the caesium transition. Prefix Symbol Factor Prefix Symbol Factor tera T 10^12 centi c 10^-2 giga G 10^9 milli m 10^-3 mega M 10^6 micro mu 10^-6 kilo k 10^3 nano n 10^-9 deci d 10^-1 pico p 10^-12

Prefix	Symbol	Factor	Prefix	Symbol	Factor
tera	T	10^12	centi	c	10^-2
giga	G	10^9	milli	m	10^-3
mega	M	10^6	micro	mu	10^-6
kilo	k	10^3	nano	n	10^-9
deci	d	10^-1	pico	p	10^-12

Significant figures are the reliable digits of a measured value plus the first uncertain one. Every non-zero digit counts, zeros between non-zero digits count, and trailing zeros after a decimal point count. Zeros written before the leading non-zero digit carry no information, and trailing zeros in a whole number without a decimal point do not either unless the value is written in scientific notation. A change of unit never changes the count, so 2.308 cm, 0.02308 m and 23.08 mm each carry four significant figures. Situation Rule Example Multiplication or division Keep the smallest number of significant figures among the inputs 4.237 g / 2.51 cm^3 = 1.69 g cm^-3 Addition or subtraction Keep the smallest number of decimal places among the inputs 436.32 + 227.2 + 0.301 = 663.8 g Exact numbers Treated as having unlimited significant figures the 2 in 2 pi r, the 6 in 6 a^2 Rounding, digit above 5 Increase the last kept digit 2.746 to 2.75 Rounding, digit exactly 5 Keep the last kept digit even 2.745 to 2.74, 2.735 to 2.74

Situation	Rule	Example
Multiplication or division	Keep the smallest number of significant figures among the inputs	4.237 g / 2.51 cm^3 = 1.69 g cm^-3
Addition or subtraction	Keep the smallest number of decimal places among the inputs	436.32 + 227.2 + 0.301 = 663.8 g
Exact numbers	Treated as having unlimited significant figures	the 2 in 2 pi r, the 6 in 6 a^2
Rounding, digit above 5	Increase the last kept digit	2.746 to 2.75
Rounding, digit exactly 5	Keep the last kept digit even	2.745 to 2.74, 2.735 to 2.74

Δ a = ∣ a - a_{mean} ∣, a ˉ = \frac{1}{n} i = 1 \sum n a_{i}, mean absolute error = \frac{1}{n} i = 1 \sum n Δ a_{i}

Absolute error of one reading and the mean absolute error of a set

relative error = \frac{Δ a}{a ˉ}, percentage error = \frac{Δ a}{a ˉ} \times 100%

Relative and percentage error, both quoted with the mean value

1.2 How errors combine For a sum or a difference the absolute errors add, for a product or a quotient the relative errors add, and for a power the relative error is multiplied by the power. These three rules cover most numericals. Operation Result Error to quote Z = A + B or Z = A - B best value Delta Z = Delta A + Delta B Z = A B or Z = A / B best value Delta Z / Z = Delta A / A + Delta B / B Z = A^n best value Delta Z / Z = n (Delta A / A) Z = k A, k exact best value Delta Z = k Delta A

\frac{Δ Z}{Z} = \frac{Δ A}{A} + \frac{Δ B}{B}, Z = A^{n} B^{m} \Rightarrow \frac{Δ Z}{Z} = n \frac{Δ A}{A} + m \frac{Δ B}{B}

Propagation of relative error through a product and through powers

1.3 Dimensions A dimensional formula records the powers of the base dimensions in a quantity. Mechanics needs only mass, length and time, and only quantities with the same dimensions can be added, subtracted or equated. Quantity Dimensional formula SI unit Area [M^0 L^2 T^0] m^2 Volume [M^0 L^3 T^0] m^3 Density [M L^-3 T^0] kg m^-3 Speed, velocity [M^0 L T^-1] m s^-1 Acceleration [M^0 L T^-2] m s^-2 Force [M L T^-2] N Work, energy, torque [M L^2 T^-2] J Power [M L^2 T^-3] W Pressure, stress [M L^-1 T^-2] Pa Momentum, impulse [M L T^-1] kg m s^-1 Angular momentum, Planck constant [M L^2 T^-1] J s Frequency, angular velocity [M^0 L^0 T^-1] Hz, rad s^-1 Gravitational constant [M^-1 L^3 T^-2] N m^2 kg^-2 Surface tension [M L^0 T^-2] N m^-1

Quantity	Dimensional formula	SI unit
Area	[M^0 L^2 T^0]	m^2
Volume	[M^0 L^3 T^0]	m^3
Density	[M L^-3 T^0]	kg m^-3
Speed, velocity	[M^0 L T^-1]	m s^-1
Acceleration	[M^0 L T^-2]	m s^-2
Force	[M L T^-2]	N
Work, energy, torque	[M L^2 T^-2]	J
Power	[M L^2 T^-3]	W
Pressure, stress	[M L^-1 T^-2]	Pa
Momentum, impulse	[M L T^-1]	kg m s^-1
Angular momentum, Planck constant	[M L^2 T^-1]	J s
Frequency, angular velocity	[M^0 L^0 T^-1]	Hz, rad s^-1
Gravitational constant	[M^-1 L^3 T^-2]	N m^2 kg^-2
Surface tension	[M L^0 T^-2]	N m^-1

[x] = [L], [v_{0} t] = [L T^{- 1}] [T] = [L], [\frac{1}{2} a t^{2}] = [L T^{- 2}] [T^{2}] = [L]

The homogeneity test applied term by term to x = x0 + v0 t + (1/2) a t^2

1.4 Units that turn up in the exercises A light year is the distance light covers in one year, about 9.46 x 10^15 m. One astronomical unit is the mean Earth to Sun distance, about 1.496 x 10^11 m, and one parsec is about 3.26 light years.

Exam

Two checks settle most one mark questions. Terms that do not share dimensions make an equation wrong, so a mismatch rules it out at once. A dimensionally correct equation is not necessarily right, so the test can reject but never confirm.

in hinglish

Is chapter ka core simple hai: har measurement ek number plus unit hai, aur unit ko calculation ke saath carry karna padta hai. SI ke saat base units yaad rakho, aur baaki sab derived units unse bante hain. Significant figures mein reliable digits plus pehla uncertain digit aata hai; unit badalne se count nahi badalta. Multiplication mein sabse kam significant figures wala input decide karta hai, addition mein sabse kam decimal places wala. Error ke teen rules yaad rakho: sum ya difference mein absolute errors add hote hain, product ya quotient mein relative errors, aur power mein relative error power se multiply hota hai. Dimensions se equation check hoti hai aur relation milta hai, par dimensions match hona sahi hone ki guarantee nahi. Numerical mein pehle unit convert karo, phir significant figures mein round karo.