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Measurements and Uncertainties - Standard Units and Symbols

Grade 12IBPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Fundamental SI units are the building blocks of all physical measurements. There are seven base units: meter (mm) for length, kilogram (kgkg) for mass, second (ss) for time, ampere (AA) for electric current, kelvin (KK) for temperature, mole (molmol) for amount of substance, and candela (cdcd) for luminous intensity.

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Derived units are combinations of the seven base units. For example, the unit of force, the Newton (NN), is expressed in base units as kg⋅m⋅s−2kg \cdot m \cdot s^{-2}.

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Scientific notation is used to express very large or very small numbers in the form a×10ka \times 10^k, where 1≤∣a∣<101 \le |a| < 10 and kk is an integer.

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Metric prefixes are used to represent powers of ten. Common IB prefixes include pico- (10−1210^{-12}), nano- (10−910^{-9}), micro- (μ=10−6\mu = 10^{-6}), milli- (10−310^{-3}), centi- (10−210^{-2}), kilo- (10310^{3}), mega- (10610^{6}), giga- (10910^{9}), and tera- (101210^{12}).

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Orders of magnitude are used to estimate and compare sizes. An order of magnitude is the power of ten closest to the value. For example, the diameter of an atom is approximately 10−10 m10^{-10} \text{ m}, while the diameter of a nucleus is approximately 10−15 m10^{-15} \text{ m}.

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Dimensional analysis ensures that equations are physically consistent. Both sides of an equation must have the same base units. For example, in v=u+atv = u + at, every term has the unit m⋅s−1m \cdot s^{-1}.

📐Formulae

1 N=1 kg⋅ m⋅ s−21 \text{ N} = 1 \text{ kg} \cdot \text{ m} \cdot \text{ s}^{-2}

1 J=1 N⋅ m=1 kg⋅ m2⋅ s−21 \text{ J} = 1 \text{ N} \cdot \text{ m} = 1 \text{ kg} \cdot \text{ m}^2 \cdot \text{ s}^{-2}

1 W=1 J⋅ s−1=1 kg⋅ m2⋅ s−31 \text{ W} = 1 \text{ J} \cdot \text{ s}^{-1} = 1 \text{ kg} \cdot \text{ m}^2 \cdot \text{ s}^{-3}

1 Pa=1 N⋅ m−2=1 kg⋅ m−1⋅ s−21 \text{ Pa} = 1 \text{ N} \cdot \text{ m}^{-2} = 1 \text{ kg} \cdot \text{ m}^{-1} \cdot \text{ s}^{-2}

V=I⋅R  ⟹  1 V=1 A⋅Ω=1 kg⋅ m2⋅ s−3⋅ A−1V = I \cdot R \implies 1 \text{ V} = 1 \text{ A} \cdot \Omega = 1 \text{ kg} \cdot \text{ m}^2 \cdot \text{ s}^{-3} \cdot \text{ A}^{-1}

💡Examples

Problem 1:

Determine the SI base units for the universal gravitational constant GG using the formula F=Gm1m2r2F = G \frac{m_1 m_2}{r^2}.

Solution:

G=F⋅r2m1⋅m2=(kg⋅m⋅s−2)⋅m2kg⋅kg=m3⋅kg−1⋅s−2G = \frac{F \cdot r^2}{m_1 \cdot m_2} = \frac{(kg \cdot m \cdot s^{-2}) \cdot m^2}{kg \cdot kg} = m^3 \cdot kg^{-1} \cdot s^{-2}

Explanation:

By rearranging the gravitational force formula for GG and substituting the base units for force (kg⋅m⋅s−2kg \cdot m \cdot s^{-2}), distance (mm), and mass (kgkg), we simplify the expression to find the base units.

Problem 2:

Convert a density of 1.2 g⋅cm−31.2 \text{ g} \cdot \text{cm}^{-3} into the standard SI unit of kg⋅m−3kg \cdot m^{-3}.

Solution:

1.2 g⋅cm−3=1.2×10−3 kg(10−2 m)3=1.2×10−310−6 kg⋅ m−3=1.2×103 kg⋅ m−31.2 \text{ g} \cdot \text{cm}^{-3} = 1.2 \times \frac{10^{-3} \text{ kg}}{(10^{-2} \text{ m})^3} = 1.2 \times \frac{10^{-3}}{10^{-6}} \text{ kg} \cdot \text{ m}^{-3} = 1.2 \times 10^3 \text{ kg} \cdot \text{ m}^{-3}

Explanation:

To convert gg to kgkg, we multiply by 10−310^{-3}. To convert cm−3cm^{-3} to m−3m^{-3}, we note that (10−2 m)3=10−6 m3(10^{-2} \text{ m})^3 = 10^{-6} \text{ m}^3. Dividing 10−310^{-3} by 10−610^{-6} results in a factor of 10310^3.

Problem 3:

Estimate the order of magnitude for the number of seconds in a human lifespan of 7575 years.

Solution:

75 years≈7.5×101 years×3.65×102 days/year×2.4×101 hours/day×3.6×103 s/hour≈2.37×109 s75 \text{ years} \approx 7.5 \times 10^1 \text{ years} \times 3.65 \times 10^2 \text{ days/year} \times 2.4 \times 10^1 \text{ hours/day} \times 3.6 \times 10^3 \text{ s/hour} \approx 2.37 \times 10^9 \text{ s} Order of magnitude: 109 s10^9 \text{ s}

Explanation:

The total seconds are calculated by multiplying 75×365×24×360075 \times 365 \times 24 \times 3600. The result is roughly 2.42.4 billion seconds. Since 2.372.37 is less than 100.5≈3.1610^{0.5} \approx 3.16, the power of ten remains 10910^9.