MET01403 Physics – Chapter 1: Units, Trigonometry & Vectors

Course & Assessment Structure

• Course Code: MET01403 – Physics (Semester 1)

• Instructor: Madam Noor Salehah Jefferi

• Weightings

  • Assignment – 15 %

  • Project – 15 %

  • Mid-test – 10 %

  • Tutorial – 10 %

  • Final Exam – 50 %

Why Study Physics?

• "Mother of all sciences"

  • Foundation for chemistry, life sciences, architecture, engineering, and many other fields.

• Definition: Physics = study of the behaviour and structure of matter & energy and their interactions.

• Goals

  • Quantitative & qualitative description of the physical world.

  • Process: Observation → Explanation (laws & theories).

  • Requires creativity and imagination; not merely memorisation of facts/formulas.

Sub-Topic 1.1 – Standard Length, Mass & Time

• Physical Quantities

  • Measurable properties of the physical world.

  • Need a unit to communicate meaningfully.

• SI Base Units

  • Length → metre (m)

  • Mass → kilogram (kg)
    • International prototype kilogram (cylinder in France) defines the standard.

  • Time → second (s)

Basic vs. Derived Quantities

• Basic (Fundamental) Quantity → defined directly by a standard (m, kg, s …).

• Derived Quantity → combination of base quantities. Example:

  • Speed $= \frac{\text{metre}}{\text{second}} = \text{m s}^{-1}$

Common Derived Units (from transcript)

• Velocity: $v = \frac{d}{t} \;\Rightarrow\; \text{m s}^{-1}$

• Acceleration: $a = \frac{\Delta v}{t} \;\Rightarrow\; \text{m s}^{-2}$

• Momentum: $p = m v \;\Rightarrow\; \text{kg m s}^{-1}$

• Force: $F = m a \;\Rightarrow\; \text{kg m s}^{-2} = \text{N}$

Sub-Topic 1.2 – Uncertainty in Measurement & Significant Figures

• Physics is experimental → every measurement contains error/uncertainty.

• Experimental Uncertainty

  • Example: Measured board width $= (23.2 \pm 0.1)\,\text{cm}$ where $\pm 0.1\,\text{cm}$ is the estimated precision.

  • Percent Uncertainty
    $\%\text{uncertainty} = \left( \frac{0.1}{23.2} \right) \times 100 \approx 0.4\%$

Significant-Figure Rules

1. Non-zero digits are significant.

2. Zeros between significant digits are significant.

3. Leading zeros (to the left of first non-zero) are not significant.

4. Trailing zeros in a decimal number are significant.

• Ambiguity solution → scientific notation:

  • $8900\,\text{g}$

  • $8.9 \times 10^{3}\,\text{g}$ (2 s.f.)

  • $8.90 \times 10^{3}\,\text{g}$ (3 s.f.)

  • $8.900 \times 10^{3}\,\text{g}$ (4 s.f.)

Sub-Topic 1.3 – Unit Conversion

• Units cancel algebraically (treat them like variables).

• Example: Convert $120\,\text{ft}$ to centimetres.
  \begin{aligned}
  120\,\text{ft} &= 120\,\text{ft} \times \frac{30.48\,\text{cm}}{1\,\text{ft}} \
  &= 3657.6\,\text{cm}
  \end{aligned}

• Always keep track of units to avoid mistakes.

Sub-Topic 1.4 – Coordinate Systems

• Cartesian (rectangular)

  • Two perpendicular axes: $x$ & $y$.

  • Point identified by ordered pair $(x, y)$.

• Polar

  • Radial distance $r$ (from origin) and angle $\theta$ (from +x-axis).

  • Conversion:
    $x = r\cos\theta,\qquad y = r\sin\theta$
    $r = \sqrt{x^{2}+y^{2}},\qquad \theta = \tan^{-1}!\left(\frac{y}{x}\right)$

Sub-Topic 1.5 – Trigonometry Refresher

• For right triangle with sides $x$ (adjacent), $y$ (opposite), hypotenuse $r$ and interior angle $\theta$ (at the origin):

  • $\sin\theta = \frac{y}{r}$

  • $\cos\theta = \frac{x}{r}$

  • $\tan\theta = \frac{y}{x}$

• Pythagorean theorem: $r^{2} = x^{2} + y^{2}$

• Essential for resolving vectors and coordinate conversions.

Sub-Topic 1.6 – Vectors

Scalars vs. Vectors

• Scalar: Completely specified by magnitude (single number).

• Vector: Requires magnitude and direction.

  • Notation in text: bold italic with arrow, e.g.
    $\vec{A}$ or A.

  • Magnitude written as $A = |\vec{A}|$.

Graphical Representation

• Drawn as arrows; length → magnitude, arrowhead → direction.

• Equality: Two vectors are equal if they have identical magnitude & direction, independent of initial point.

• Negative of a vector: Same magnitude, opposite direction.

Vector Addition

1. Head-to-Tail Method

   • Place tail of $\vec{B}$ at head of $\vec{A}$; resultant $\vec{C} = \vec{A} + \vec{B}$ from tail of $\vec{A}$ to head of $\vec{B}$.

   • Order independent: $\vec{A}+\vec{B} = \vec{B}+\vec{A}$.

2. Parallelogram Method

   • Draw $\vec{A}$ & $\vec{B}$ from common origin; complete parallelogram; diagonal is the sum.

3. Adding >2 Vectors

   • Continue head-to-tail

   • Resultant independent of order (commutative & associative).

Vector Subtraction

• $\vec{A}-\vec{B} = \vec{A} + (-\vec{B})$ where $-\vec{B}$ is $\vec{B}$ reversed.

• Two common diagrams (head-to-tail vs. head-to-head) both yield same resultant.

Multiplying a Vector by a Scalar $c$

• Result: $c\vec{A}$ with magnitude $|c|A$.

  • If c>0: same direction.

  • If c<0: opposite direction.

  • Example: $2\vec{A}$ is twice as long; $-3\vec{A}$ is triple length, opposite direction.

Components of a Vector

• For vector $\vec{A}$ making angle $\theta$ with +x-axis:

  • $A_x = A\cos\theta$

  • $A_y = A\sin\theta$

• Vector expressed as $\vec{A} = A<em>x\hat{i} + A</em>y\hat{j}$.

• Components can be positive or negative (quadrant dependent).

Example: Addition of Two Perpendicular Vectors

• Displacements: $2.00\,\text{km}$ North and $1.00\,\text{km}$ East.

  • Graphical resultant points northeast.

  • Magnitude: $r = \sqrt{(1.00)^2 + (2.00)^2} = 2.24\,\text{km}$.

  • Direction: $\theta = \tan^{-1}!\left( \frac{2.00}{1.00} \right) = 63.4^{\circ}\text{ N of E}$.

Practical / Philosophical Takeaways

• Measurement uncertainties necessitate error analysis; no experiment is perfectly exact.

• Clear communication of precision through significant figures and unit consistency is vital.

• Vectors & trigonometry underpin virtually all areas of physics (kinematics, forces, fields, waves…).

• Conceptual understanding (imagination, creativity) plus mathematical formalism = productive scientific reasoning.