Why Work, Energy and Torque Share the Same Dimensions

From the working of a mobile phone to the orbit of a satellite, physics explains nature through a small number of laws — and every law is tested by measurement. A measurement needs a number and a unit, and the units themselves carry information that can catch a mistake in an equation before any experiment is done.

This lesson covers the scope of physics and its links with technology and society, the SI base and derived units, and dimensional analysis.

Physics studies the basic laws of nature, from the smallest particles to the whole universe, and its ideas feed directly into technology, which in turn shapes society and raises new questions for physics.

Scope of physics:

- Classical physics — mechanics, heat, optics, electricity and magnetism, mainly at everyday scales
- Modern physics — quantum mechanics for the atomic world, and relativity for very high speeds and strong gravity
- Physics spans lengths from about $10^{-15}$ m, the size of a nucleus, to about $10^{26}$ m, the size of the observable universe

Physics and technology:

- Thermodynamics underlies engines, electromagnetic induction underlies generators, and electromagnetic waves underlie radio and mobile communication
- Nuclear fission powers nuclear reactors, and semiconductor physics makes computers and phones possible

The four fundamental forces:

- Gravitational — the weakest, yet it acts over vast distances and keeps planets in orbit
- Electromagnetic — acts between charges and lies behind everyday forces such as friction and tension
- Strong nuclear — the strongest, holding the nucleus together over a very short range
- Weak nuclear — responsible for beta decay

An everyday example. Satellite navigation in phones and taxis across Indian cities works only because physics corrects for satellite clocks running at slightly different rates from clocks on the ground, an effect predicted by relativity.

The substance. Physics and technology feed each other — precise clocks and powerful telescopes built from physics reveal new effects that physics must then explain.

The SI system has seven base units — metre, kilogram, second, ampere, kelvin, mole and candela — and every other unit, such as the newton or the joule, is derived by combining them.

The seven base quantities:

- Length — metre (m); mass — kilogram (kg); time — second (s)
- Electric current — ampere (A); thermodynamic temperature — kelvin (K)
- Amount of substance — mole (mol); luminous intensity — candela (cd)

Common derived units:

- Force: $1\text{ N}=1{\text{ kg m s}}^{-2}$
- Work and energy: $1\text{ J}=1\text{ N m}=1{\text{ kg m}}^2{\text{ s}}^{-2}$
- Power: $1\text{ W}=1{\text{ J s}}^{-1}$; pressure: $1\text{ Pa}=1{\text{ N m}}^{-2}$; charge: $1\text{ C}=1\text{ A s}$
- The radian and steradian are used for plane and solid angles

Worked example 1 — density of water.

$$1{\text{ g cm}}^{-3}=\frac{10^{-3}\text{ kg}}{10^{-6}{\text{ m}}^3}=10^3{\text{ kg m}}^{-3}$$

Worked example 2 — an electricity unit in joules.

$$1\text{ kW h}=1000\text{ W}\times 3600\text{ s}=3.6\times 10^6\text{ J}$$

An everyday example. One unit on an Indian electricity bill is exactly this kilowatt hour, so a household using 150 units in a month has used $150\times 3.6\times 10^6=5.4\times 10^8$ J of energy.

The substance. Many everyday units are not SI units — the kilowatt hour, the litre and the degree Celsius must be converted before they are used in SI formulas.

Dimensional analysis writes every quantity in terms of the base dimensions of mass [M], length [L] and time [T], and uses the principle of homogeneity — every term in a valid equation must have the same dimensions — to check equations and derive relationships.

Some dimensional formulae:

- Velocity $[\mathrm{L}{\mathrm{T}}^{-1}]$ and acceleration $[\mathrm{L}{\mathrm{T}}^{-2}]$
- Force $[\mathrm{ML}{\mathrm{T}}^{-2}]$ and energy $[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}]$
- Pressure $[\mathrm{M}{\mathrm{L}}^{-1}{\mathrm{T}}^{-2}]$ and power $[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-3}]$

**Worked example 1 — checking $v^2=u^2+2as$.**

$$[v^2]=[{\mathrm{L}}^2{\mathrm{T}}^{-2}][u^2]=[{\mathrm{L}}^2{\mathrm{T}}^{-2}][as]=[\mathrm{L}{\mathrm{T}}^{-2}][\mathrm{L}]=[{\mathrm{L}}^2{\mathrm{T}}^{-2}]$$

Every term matches, so the equation is dimensionally correct.

Worked example 2 — the period of a pendulum. Suppose the period t depends on mass m, length l and gravitational acceleration g, as $t=k{m}^a{l}^b{g}^c$:

$$[\mathrm{T}]=[\mathrm{M}{]}^a[\mathrm{L}{]}^b[\mathrm{L}{\mathrm{T}}^{-2}{]}^c=[{\mathrm{M}}^{\mathrm{a}}{\mathrm{L}}^{\mathrm{b}+\mathrm{c}}{\mathrm{T}}^{-2c}]$$

Matching powers gives $a=0$, $c=-\frac{1}{2}$ and $b=\frac{1}{2}$, so $t=k\sqrt{l/g}$ — the mass drops out completely. With $k=2\pi$ from experiment, a 1.0 m pendulum with $g=9.8$ m s${}^{-2}$ has

$$t=2\pi \sqrt{\frac{1.0}{9.8}}=2.0\text{ s}$$

Limitations:

- It cannot find dimensionless constants such as $2\pi$
- It fails when more than three base quantities are involved, or when trigonometric, exponential or logarithmic functions appear

An everyday example. Engineers at the hydraulic research station in Pune test scale models of dams and harbours, using dimensional analysis to scale their results up to full-size structures.

The substance. Dimensional analysis can prove an equation wrong but never prove it right — $s=ut+a{t}^2$ balances dimensionally, yet the correct equation has a factor of $\frac{1}{2}$.