मात्रक, विमाएँ और शुद्धगतिकी
The SI — seven base units, and the 2019 redefinition
| Quantity | Unit | Fixed since 2019 by |
|---|---|---|
| length | metre (m) | the speed of light, c |
| mass | kilogram (kg) | the Planck constant, h |
| time | second (s) | the caesium-133 hyperfine frequency |
| electric current | ampere (A) | the elementary charge, e |
| temperature | kelvin (K) | the Boltzmann constant, k |
| amount of substance | mole (mol) | the Avogadro constant |
| luminous intensity | candela (cd) | a fixed luminous efficacy |
Why 2019 matters. Until 20 May 2019 the kilogram was a metal cylinder kept near Paris, and its mass drifted. Now every base unit is defined by fixing the value of a constant of nature, so the definitions can be realised in any laboratory.
मात्रक, विमाएँ और शुद्धगतिकी
Dimensions — what they can prove, and what they cannot
| Quantity | Dimensions | Quantity | Dimensions |
|---|---|---|---|
| force | [M L T⁻²] | pressure, stress | [M L⁻¹ T⁻²] |
| work, energy, torque | [M L² T⁻²] | surface tension | [M T⁻²] |
| power | [M L² T⁻³] | viscosity η | [M L⁻¹ T⁻¹] |
| momentum, impulse | [M L T⁻¹] | G | [M⁻¹ L³ T⁻²] |
| Planck's h, angular momentum | [M L² T⁻¹] | frequency, angular velocity | [T⁻¹] |
Pairs with the same dimensions — a paper's favourite question: work and torque, impulse and momentum, Planck's constant and angular momentum, stress, pressure and Young's modulus, frequency and angular velocity.
What dimensional analysis can do
What it cannot
The trap: a dimensionally correct equation may still be wrong. Homogeneity is necessary, not sufficient.
मात्रक, विमाएँ और शुद्धगतिकी
Motion in a straight line — the equations and the graphs
For uniform acceleration
Reading the graphs
| Graph | Its slope is | The area under it is |
|---|---|---|
| position–time | velocity | — |
| velocity–time | acceleration | displacement |
| acceleration–time | — | change in velocity |
Galileo's odd-number rule. A body starting from rest with uniform acceleration covers distances in successive equal intervals in the ratio 1 : 3 : 5 : 7 … — because the total distance grows as t², and the differences of the squares 1, 4, 9, 16 are the odd numbers.
Free fall: a = g ≈ 9.8 m s⁻², the same for every body in a vacuum — Galileo against Aristotle.
Relative velocity: v_AB = v_A − v_B. Two trains approaching at 60 and 40 km/h close at 100 km/h; moving the same way, at 20 km/h.
The trap: the area under a velocity–time graph gives displacement, which can be zero on a round trip; the distance is the area taken without sign.
मात्रक, विमाएँ और शुद्धगतिकी
Projectiles and circular motion
A projectile launched with speed u at angle θ to the horizontal:
Uniform circular motion
The traps
मात्रक, विमाएँ और शुद्धगतिकी
Points to Remember