Unit & Dimension flash cards

1. 1.What is a physical quantity?

   A quantity that can be measured and expressed in terms of a number (magnitude) and a unit. Example: length = $5\,\text{m}$ means magnitude 5 and unit metre.

   Hint: Measurement = number × unit.

2. 2.Distinguish fundamental (base) and derived physical quantities.

   Base quantities are independent and not defined in terms of others (e.g. length, mass, time). Derived quantities are built from base ones (e.g. speed = length/time, force = mass × acceleration).

   Hint: Some stand alone; others are combinations.

3. 3.List the seven SI base quantities and their base units.

   Length — metre (m); Mass — kilogram (kg); Time — second (s); Electric current — ampere (A); Temperature — kelvin (K); Amount of substance — mole (mol); Luminous intensity — candela (cd).

   Hint: Length, mass, time, current, temperature, amount, luminous intensity.

4. 4.What are the two supplementary SI units?

   Plane angle — radian (rad); Solid angle — steradian (sr). Both are dimensionless ratios.

   Hint: One for flat angle, one for cone/solid angle.

5. 5.Define the dimensions of a physical quantity.

   The powers to which the base quantities must be raised to represent that quantity. Written using symbols $M$ (mass), $L$ (length), $T$ (time), $I$ or $A$ (current), $K$ or $\theta$ (temperature), $N$ (mol), $J$ (luminous intensity).

   Hint: Exponents of base quantities.

6. 6.What is a dimensional formula? Give an example.

   An expression showing which base quantities and what powers make up a physical quantity. Example: Force $= [M L T^{-2}]$; Energy $= [M L^2 T^{-2}]$.

   Hint: The $M^aL^bT^c$ expression.

7. 7.What is a dimensional equation?

   An equation that equates a physical quantity to its dimensional formula. Example: $[F] = [M L T^{-2}]$.

   Hint: Quantity = its dimensions.

8. 8.Dimensional formula of area and volume?

   Area $= [L^2] = [M^0 L^2 T^0]$; Volume $= [L^3] = [M^0 L^3 T^0]$.

   Hint: length squared and length cubed.

9. 9.Dimensional formula of velocity and acceleration?

   Velocity $= [L T^{-1}]$; Acceleration $= [L T^{-2}]$.

   Hint: displacement/time and velocity/time.

10. 10.Dimensional formula of force?

    Force $=$ mass $\times$ acceleration $= [M][L T^{-2}] = [M L T^{-2}]$. SI unit: newton (N).

    Hint: $F = ma$.

11. 11.Dimensional formula of work, energy and torque?

    All are $[M L^2 T^{-2}]$. Work/energy = force × distance; torque = force × perpendicular distance.

    Hint: Force × length — same for all three.

12. 12.Dimensional formula of power?

    Power $= \dfrac{\text{energy}}{\text{time}} = \dfrac{[M L^2 T^{-2}]}{[T]} = [M L^2 T^{-3}]$. SI unit: watt (W).

    Hint: Energy per unit time.

13. 13.Dimensional formula of pressure and stress?

    Both $= \dfrac{\text{force}}{\text{area}} = \dfrac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}]$.

    Hint: Force per unit area.

14. 14.Dimensional formula of momentum and impulse?

    Both $= [M L T^{-1}]$. Momentum = mass × velocity; impulse = force × time — dimensionally identical.

    Hint: $p = mv$; impulse = $F\,\Delta t$.

15. 15.Dimensional formula of density?

    Density $= \dfrac{\text{mass}}{\text{volume}} = \dfrac{[M]}{[L^3]} = [M L^{-3}]$.

    Hint: Mass per unit volume.

16. 16.Dimensional formula of frequency and angular velocity?

    Both $= [T^{-1}] = [M^0 L^0 T^{-1}]$. Frequency = 1/period; angular velocity = angle(dimensionless)/time.

    Hint: Per second — inverse time.

17. 17.Dimensional formula of gravitational constant $G$?

    From $F = \dfrac{G m_1 m_2}{r^2}$, $G = \dfrac{F r^2}{m_1 m_2} = \dfrac{[M L T^{-2}][L^2]}{[M^2]} = [M^{-1} L^3 T^{-2}]$.

    Hint: Rearrange Newton's gravitation law.

18. 18.Dimensional formula of Planck's constant $h$?

    From $E = h\nu$, $h = \dfrac{E}{\nu} = \dfrac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}]$. Same as angular momentum.

    Hint: $E = h\nu$; $h = E/\nu$.

19. 19.Dimensional formula of Boltzmann constant $k_B$?

    From $E = k_B T$, $k_B = \dfrac{E}{T} = \dfrac{[M L^2 T^{-2}]}{[K]} = [M L^2 T^{-2} K^{-1}]$.

    Hint: Energy per unit temperature.

20. 20.Dimensional formula of the universal gas constant $R$?

    From $PV = nRT$, $R = \dfrac{PV}{nT} = \dfrac{[M L^{-1} T^{-2}][L^3]}{[\text{mol}][K]} = [M L^2 T^{-2} K^{-1} \text{mol}^{-1}]$.

    Hint: $PV = nRT$.

21. 21.Dimensional formula of coefficient of viscosity $\eta$?

    From $F = \eta A \dfrac{dv}{dx}$, $\eta = [M L^{-1} T^{-1}]$. SI unit: $\text{Pa·s}$ (poiseuille).

    Hint: Force / (area × velocity gradient).

22. 22.Dimensional formula of surface tension?

    Surface tension $= \dfrac{\text{force}}{\text{length}} = \dfrac{[M L T^{-2}]}{[L]} = [M L^0 T^{-2}] = [M T^{-2}]$.

    Hint: Force per unit length (or energy per area).

23. 23.Dimensional formula of gravitational field / acceleration due to gravity $g$?

    $g = [L T^{-2}]$ — same as any acceleration.

    Hint: It is an acceleration.

24. 24.Dimensional formula of angular momentum $L$?

    $L = m v r = [M][L T^{-1}][L] = [M L^2 T^{-1}]$. Same as Planck's constant.

    Hint: $L = mvr$ or $I\omega$.

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