Physical Quantities & Measurements

1.1 Dimensions of Physical Quantities

• Definition

  • Dimension = qualitative description of a physical quantity in terms of basic (fundamental) quantities, independent of the unit system.
  • Notation: enclose the quantity or symbol in square brackets, e.g. $[v]$ or $[velocity]$.
  • Example: $100\,\text{cm}=1\,\text{m}=2\,\text{mi}=3\,\text{ly}$ all share the dimension $[L]$ (length).

• Fundamental (base) dimensions

  • Mass $[M]$
  • Length $[L]$
  • Time $[T]$
  • Electric current $[I]$
  • Thermodynamic temperature $[\Theta]$
  • Amount of substance $[N]$
  • Luminous intensity $[J]$

• Dimensional analysis: purposes

  • Derive the SI unit of any derived quantity.
  • Check or derive equations by treating dimensions as algebraic symbols.
  • Verify dimensional homogeneity: $\text{dimension LHS}=\text{dimension RHS}$.
  • Reminder: pure numerical factors (e.g. $\tfrac12,\,\tfrac34$) carry no dimension.
  • Addition/Subtraction rule: quantities may be added or subtracted only if their dimensions match.

• Common derived quantities

  • Velocity: $[v]=[L][T]^{-1};\;\text{unit}=\text{m s}^{-1}$
  • Acceleration: $[a]=[L][T]^{-2};\;\text{unit}=\text{m s}^{-2}$
  • Linear momentum: $[p]=[M][L][T]^{-1};\;\text{unit}=\text{kg m s}^{-1}$
  • Density: $[\rho]=[M][L]^{-3};\;\text{unit}=\text{kg m}^{-3}$
  • Force: $[F]=[M][L][T]^{-2};\;\text{unit}=\text{N}=\text{kg m s}^{-2}$
  • Pressure: $[P]=[M][L]^{-1}[T]^{-2};\;\text{unit}=\text{Pa}=\text{kg m}^{-1} \text{s}^{-2}$

• Illustrative homogeneity checks

  • Kinematic expression $s=ut+\tfrac12 at^{2}$
  • $[s]=[L]$; $[u t]=[L][T]^{-1}[T]=[L]$; $[\tfrac12 a t^{2}]=[L][T]^{-2}[T]^{2}=[L]$
  • All terms $\rightarrow [L]$ ⇒ equation homogeneous.
  • $v^{2}=u^{2}+2as$: $[v^{2}]=[L]^{2}[T]^{-2}$ equals $[u^{2}]$; $[2as]=[L][T]^{-2}[L]=[L]^{2}[T]^{-2}$ ⇒ homogeneous.
  • Period of a simple pendulum: $T=2\pi\sqrt{\tfrac{l}{g}}$
  • $[l]=[L]$, $[g]=[L][T]^{-2}$ ⇒ radicand $=[T]^{2}$ ⇒ $T=[T]$; homogeneous.

• Multiple-choice recall

  • Dimension of temperature is $[\Theta]$ (option D, symbol $\theta$ sometimes used).
  • Refractive index is dimensionless.

1.2 Scalars and Vectors

• Scalar quantity
  • Defined by magnitude only.
  • Examples: mass, distance, speed, work, pressure, electric current, temperature, energy.
• Vector quantity
  • Defined by both magnitude and direction.
  • Examples: displacement, velocity, acceleration, force, momentum, impulse, torque, electric field $\vec E$, magnetic field $\vec B$.
  • Graphical representation: arrow; length → magnitude, arrowhead → direction.
• Mathematical operations
  • While adding, subtracting, or multiplying vectors, always account for both magnitude and direction.

Resolving a Vector Into Perpendicular Components

• Consider vector $\vec A$ making an angle $\theta$ with the positive $x$-axis.
  • Horizontal (x) component: $A_x = A\cos\theta$ (adjacent).
  • Vertical (y) component: $A_y = A\sin\theta$ (opposite).
  • Direction of $\vec A$ expressed as $\theta=\tan^{-1}\bigl(\tfrac{A<em>y}{A</em>x}\bigr)$, measured from $x$-axis.

Determining a Resultant Vector (Component Method)

Flow map / strategy:

1. Resolve every vector into its $x$ and $y$ components.
   • Keep correct signs (+ toward +axes, − toward −axes).
2. Add components separately:
   $\Sigma F<em>x = \sum A</em>x, \qquad \Sigma F<em>y = \sum A</em>y$.
3. Form the resultant vector $\vec R$:
   • Magnitude $R = \sqrt{(\Sigma F<em>x)^2 + (\Sigma F</em>y)^2}$.
   • Direction $\theta = \tan^{-1}\bigl( \tfrac{\Sigma F<em>y}{\Sigma F</em>x} \bigr)$ measured from $x$-axis.
4. Never mix $x$ and $y$ values directly (e.g. do not add them).
5. Present direction clearly (e.g. $6.6^{\circ}$ above −x-axis, or “due north”, etc.).

Worked Vector Examples

• Example – component identification

  • Given vector of magnitude $10\,\text{units}$ at $45^{\circ}$ in 4th quadrant (x positive, y negative).
  • $A_x = +10\cos45^{\circ}=+10/\sqrt2$.
  • $A_y = -10\sin45^{\circ}=-10/\sqrt2$.
  • Correct multiple-choice: $-10\cos45^{\circ}$ or $-10\sin45^{\circ}$ depending on the given orientation (transcript answer: $-10\cos45^{\circ}$ for x component shown).

• Car velocity problem

  • Speed $30\,\text{m s}^{-1}$, direction “north $60^{\circ}$ west”.
  • Due north (y): $30\cos60^{\circ}=15\,\text{m s}^{-1}$.
  • Due west (x negative): $30\sin60^{\circ}=-25.98\,\text{m s}^{-1}$.

• Force on particle S

  • Force magnitude $100\,\text{N}$ at given angle (from diagram).
  • Components extracted using $\sin$/$\cos$ according to its orientation; transcript solution yields $F<em>x, F</em>y$.

• Three-force system (Fig. 7.1) – two methods (table or graphical) both yield same resultant magnitude; transcript lists $|\vec R|\approx$ value, confirm with calculation.

Multiple-Choice Highlights (Sample Questions & Answers)

• Correct dimension/quantity mismatches: any listed except given wrong pair.
• $1\,\text{N}=1\,\text{kg m s}^{-2}$.
• Pressure dimension $[M][L]^{-1}[T]^{-2}$ and unit $\text{kg m}^{-1}\text{s}^{-2}$ (option B).
• Scalar pair: area & speed.
• Vector pair: stress & magnetic field.

Tutorial-Style Practice (with provided answers)

1. Determine dimensions/units

   • Linear momentum $[M][L][T]^{-1}\;\;(\text{kg m s}^{-1})$
   • Kinetic energy $[M][L]^{2}[T]^{-2}\;(\text{kg m}^{2}\text{s}^{-2})$
   • Elastic potential energy same as kinetic energy.
   • (Extra given answers: $\text{kg m s}^{-2}$ – actually matches force, transcript typo; $\text{kg m}^{-3}$ matches density.)

2. Show mechanical wave equation $v=A\omega\sin(\omega t+kx)$ is homogeneous

   • $[v]=[L][T]^{-1}$; $[A]=[L]$; $[\omega]=[T]^{-1}$ ⇒ RHS $=[L][T]^{-1}$; sine function dimensionless; homogeneous.

3. Homogeneity checks

   • $v=u+2as$ not homogeneous (dimension mismatch) – validate.
   • $v=u+at$ homogeneous.
   • Work $W=Fs$ homogeneous.

4. Resolve each listed vector; numerical answers supplied in bracket.

   • Examples include forces, accelerations, electric fields; answers: $7.66\,\text{N},\,6.43\,\text{N}$ etc.

5. Magnetic field system (Fig. 1.1)

   • $B<em>1=20\,\text{T},\;B</em>2=40\,\text{T},\;B_3=80\,\text{T}$
   • Results: $B<em>x=-75.14\,\text{T},\;B</em>y=+8.78\,\text{T}$
   • Resultant $|\vec B_R|=75.65\,\text{T},\,\theta=-6.66^{\circ}\text{ above –x axis}$.

6. Electric field system (Fig. 1.2)

   • $E<em>1=15\,\text{N C}^{-1}, E</em>2=38\,\text{N C}^{-1}, E_3=17\,\text{N C}^{-1}$
   • Resultant $29.98\,\text{N C}^{-1},\;62.96^{\circ}\text{ above –x axis}$.

7. Sailboat displacement (Fig. 1.3)

   • Legs: $2.00\,\text{km}$ east, $3.50\,\text{km}$ southeast, unknown.
   • Final displacement $5.80\,\text{km}$ east of start.
   • Third leg magnitude $0.81\,\text{km}$ at $61.84^{\circ}$ (direction per diagram).

Study Tips & Connections

• Dimensional analysis links directly to error-checking in experimental physics and to model building in engineering (e.g., Buckingham Π-theorem).
• Master vector components early; they recur in kinematics, dynamics, electromagnetism and fluid mechanics.
• Treat sine/cosine arguments as dimensionless ⇒ any phase term $\omega t,\,kx$ must itself be dimensionless, a universal consistency test.
• Remember scalars can still be negative (e.g., work, potential energy) even though they lack direction.
• When carrying significant figures & uncertainty (topic slated for laboratory work), keep at most one uncertain digit in the final quoted value.