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 $=5$ m.

   Hint: Number + unit.

2. 2.Distinguish fundamental and derived quantities.

   Fundamental (base) quantities are independent and not defined in terms of others (e.g. length, mass, time). Derived quantities are expressed using base quantities (e.g. speed, force, density).

   Hint: Base = independent; derived = combinations.

3. 3.Name the seven SI base quantities and their 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: L, M, T, I, $\Theta$, N, J.

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

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

   Hint: radian and steradian.

5. 5.Define 1 radian.

   The angle subtended at the centre of a circle by an arc equal in length to the radius: $\theta=\dfrac{\text{arc}}{\text{radius}}$.

   Hint: Arc = radius.

6. 6.Define 1 steradian.

   The solid angle subtended at the centre of a sphere by a surface area equal to $r^2$ (radius squared): $\Omega=\dfrac{A}{r^2}$.

   Hint: Area = $r^2$.

7. 7.What is a unit of measurement?

   An arbitrarily chosen internationally accepted reference standard of the same kind used to compare and express a physical quantity.

   Hint: Reference standard.

8. 8.Why is the SI system preferred over CGS or FPS?

   SI is coherent, rational (one unit per quantity), metric (decimal), internationally accepted, and covers mechanical, electrical, and optical quantities.

   Hint: Coherent, decimal, universal.

9. 9.State the properties of a good unit.

   It should be well-defined, of suitable size, invariable (constant with time/place/conditions), easily reproducible, and internationally accepted.

   Hint: Constant + reproducible + convenient.

10. 10.What are dimensions of a physical quantity?

    The powers to which the base quantities are raised to represent that quantity. Written using $[M],[L],[T]$ etc.

    Hint: Powers of base quantities.

11. 11.What is a dimensional formula?

    An expression showing how and which base quantities represent a physical quantity, e.g. force $=[M^1L^1T^{-2}]$.

    Hint: Expression in $M,L,T$.

12. 12.What is a dimensional equation?

    An equation obtained by equating a physical quantity with its dimensional formula, e.g. $[F]=[M^1L^1T^{-2}]$.

    Hint: Quantity = its dimensional formula.

13. 13.Give the dimensional formula of area and volume.

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

    Hint: $L^2$ and $L^3$.

14. 14.Give the dimensional formula of velocity and acceleration.

    Velocity $=[M^0L^1T^{-1}]$; Acceleration $=[M^0L^1T^{-2}]$.

    Hint: $LT^{-1}$, $LT^{-2}$.

15. 15.Give the dimensional formula of force.

    $[F]=[M^1L^1T^{-2}]$ since force $=$ mass $\times$ acceleration.

    Hint: $ma$.

16. 16.Give the dimensional formula of work / energy.

    $[W]=[M^1L^2T^{-2}]$ since work $=$ force $\times$ displacement.

    Hint: $Fd$.

17. 17.Give the dimensional formula of power.

    $[P]=[M^1L^2T^{-3}]$ since power $=$ work$/$time.

    Hint: Energy per time.

18. 18.Give the dimensional formula of pressure and stress.

    Both $=[M^1L^{-1}T^{-2}]$ since pressure $=$ force$/$area.

    Hint: $F/A$.

19. 19.Give the dimensional formula of momentum and impulse.

    Both $=[M^1L^1T^{-1}]$. Momentum $=mv$; impulse $=$ force $\times$ time.

    Hint: $mv$ and $Ft$ are same.

20. 20.Give the dimensional formula of density.

    $[\rho]=[M^1L^{-3}T^0]$ since density $=$ mass$/$volume.

    Hint: $M/L^3$.

21. 21.Give the dimensional formula of gravitational constant $G$.

    $[G]=[M^{-1}L^3T^{-2}]$, from $F=\dfrac{Gm_1m_2}{r^2}$.

    Hint: Rearrange Newton's law of gravitation.

22. 22.Give the dimensional formula of frequency.

    $[\nu]=[M^0L^0T^{-1}]$ since frequency $=1/$period.

    Hint: Inverse of time.

23. 23.Give the dimensional formula of surface tension.

    $[M^1L^0T^{-2}]$ since surface tension $=$ force$/$length.

    Hint: $F/L$.

24. 24.Give the dimensional formula of coefficient of viscosity.

    $[\eta]=[M^1L^{-1}T^{-1}]$.

    Hint: From $F=\eta A\,dv/dx$.

25. 25.Give the dimensional formula of Planck's constant $h$.

    $[h]=[M^1L^2T^{-1}]$, from $E=h\nu$ so $h=E/\nu$.

    Hint: Energy $\times$ time.

26. 26.Give the dimensional formula of angular velocity.

    $[\omega]=[M^0L^0T^{-1}]$ since $\omega=\theta/t$ and angle is dimensionless.

    Hint: Same as frequency dimension.

27. 27.Give the dimensional formula of torque.

    $[\tau]=[M^1L^2T^{-2}]$ since torque $=$ force $\times$ perpendicular distance.

    Hint: Same as energy dimension, different quantity.

28. 28.Give the dimensional formula of the gas constant $R$.

    $[R]=[M^1L^2T^{-2}K^{-1}mol^{-1}]$, from $PV=nRT$.

    Hint: Energy per mole per kelvin.

29. 29.Give the dimensional formula of specific heat capacity.

    $[M^0L^2T^{-2}K^{-1}]$, from $Q=mc\Delta T$.

    Hint: Energy per unit mass per kelvin.

30. 30.Name three physical quantities that are dimensionless.

    Strain, refractive index, relative density, plane angle, solid angle, and any pure ratio or number like $\pi$.

    Hint: Ratios of like quantities.

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