Physics

Examination Terminology and Investigative Assessments

The Scientific Method and Experimental Design

• Galileo Galilei (16th Century): Known as the "Father of modern scientific methodology." Established the foundational approach still used today.

• Four Pillars of the Scientific Method:

  1. Observation: Viewing a natural event that requires investigation.

  2. Hypothesis: Proposing a potential explanation for the observation.

  3. Experiment: Collecting quantitative data to test generalizations. Only one variable should be changed at a time.

  4. Conclusion: Validating the hypothesis through calculations. Modern conclusions are often expressed as equations. Galileo’s example: $s = \frac{1}{2}\,kt^2$

• Variables in Experimental Research:

  • Independent Variable: The factor being intentionally altered or investigated.

  • Dependent Variable: The factor affected by the change in the independent variable.

  • Control Variable: Factors kept constant to ensure a fair test.

  • Simple Pendulum Example: To test the period ($T$), if length ($l$) is the independent variable, mass ($m$) and angular displacement must be controlled.

Principles of Graphical Analysis

• Plotting Standards:

  • Title: Must state the variables and the purpose of the plot.

  • Axes: The first variable mentioned in the title goes on the y-axis; the second on the x-axis. Both must be labeled with quantity and units.

  • Scale: Use easy ratios ($1:1$, $1:2$, $1:5$). Scales should cover at least two-thirds of the sheet. Use a "broken scale" if points are cluttered near the origin.

  • Data Points: Plot as an "x" or a circled dot using a sharp pencil.

  • Lines of Best Fit: Use a transparent ruler for straight lines. Sketch smooth curves for non-linear data. The line should balance the mean deviation of points on either side to reduce random error impact.

• Calculating the Gradient (Slope, m):

  • Select two points ($x_1$, $y_1$) and ($x_2$, $y_2$) far apart on the line of best fit.

  • Points must lie on the intersections of the graph grid.

  • Construct a large right-angled triangle.

  • Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • Unit of slope: $\frac{\text{unit of y-axis}}{\text{unit of x-axis}}$

Quantitative Accuracy: Significant Figures and Precision

• Measurement Precision: A reading is generally taken to the nearest half of the smallest scale interval (e.g., $23.5\,mm$ on a mm ruler).

• Multiplication and Division: The final result should match the least number of significant figures in the calculation.

  • Example: $\frac{4.2 \times 375 \times 41.3}{27} = 2409.166...$ Result: $2400$ (to 2 sig. figs due to 4.2 and 27).

• Addition and Subtraction: Based on decimal precision (place value).

  • Example: $21.3 + 325 + 2132.1 = 2478.4$. Result: $2478$. Since 325 is the least precise (rounding to the units place), the answer must round to the units place.

Classification and Resolution of Errors

• Random Errors:

  • Definition: Errors with an equal chance of being higher or lower than the true value.

  • Causes: Poor judgment or fluctuating environmental conditions.

  • Parallax Error: Occurs when the line of sight is not perpendicular to the scale/pointer.

  • Reduction Methods: Use mirrors behind meter scales (align pointer with its image), take eye-level readings, or calculate the mean of multiple readings.

• Systematic Errors:

  • Definition: Errors that consistently shift results in one direction (always too high or too low).

  • Causes: Inaccurate hardware or flawed experimental systems (e.g., zero error on a meter, poor calibration).

  • Reduction Methods: These are NOT reduced by averaging. You must discover the error value and add/subtract it from results, or adjust the instrument.

Precision Measurement Instruments

• Vernier Calliper:

  • Measures to one extra decimal place compared to a main scale.

  • Ideal for lengths between $1\,cm$ and $10\,cm$.

  • Usage: Read main scale before the zero of the vernier scale; find where vernier and main scale lines align for the final decimal.

• Micrometer Screw Gauge:

  • Used for lengths under $1\,cm$ (e.g., wire diameter).

  • One thimble revolution = $0.5\,mm$ movement.

  • Reading: Sum the sleeve scale (main) and thimble scale values.

• Instrument Suitability:

  • Metre Rule: Range (height of a table), accuracy to $mm$.

  • Vernier Calliper: Internal diameter of pipes (jaws provide access).

  • Micrometer: Sensitivity/accuracy for engineering (very small diameters).

  • Clinical Thermometer: High sensitivity/accuracy, limited range ($35^{\circ}C$ to $42^{\circ}C$).

  • Laboratory Thermometer: Wider range for general school experiments.

Fundamental and Derived Physical Quantities

• SI Base Units:

  • Mass: kilogram ($kg$)

  • Length: metre ($m$)

  • Time: second ($s$)

  • Electric Current: ampere ($A$)

  • Temperature: kelvin ($K$)

  • Amount of Substance: mole ($mol$)

  • Luminous Intensity: candela ($cd$)

• Derived Units (Combinations of Base Units):

  • Force: $N = kg\,m\,s^{-2}$

  • Work/Energy: $J = N\,m = kg\,m^2\,s^{-2}$

  • Power: $W = J\,s^{-1} = kg\,m^2\,s^{-3}$

  • Pressure: $Pa = N\,m^{-2} = kg\,m^{-1}\,s^{-2}$

  • Charge: $C = A\,s$

  • Voltage: $V = J\,C^{-1} = kg\,m^2\,A^{-1}\,s^{-3}$

  • Resistance: $\Omega = V\,A^{-1} = kg\,m^2\,A^{-2}\,s^{-3}$

  • Frequency: $Hz = s^{-1}$

Prefixes and Scientific Notation

• Standard Form (Index Notation): Expressed as $M \times 10^p$. Mantissa ($M$) has one non-zero digit before the decimal.

• Conversion and Prefixes:

  • tera (T): $10^{12}$

  • giga (G): $10^9$

  • mega (M): $10^6$

  • kilo (k): $10^3$

  • milli (m): $10^{-3}$

  • micro (\mu): $10^{-6}$

  • nano (n): $10^{-9}$

  • pico (p): $10^{-12}$

• Rule for Conversion: Moving to a larger unit results in a smaller number; moving to a smaller unit results in a larger number. Each prefix jump represents 3 decimal places.

Density and Volumetric Analysis

• Definition: Density is mass per unit volume ($\rho = \frac{m}{V}$).

• Measurement:

  • Mass: Obtained via an electronic balance.

  • Liquid Volume: Via a measuring cylinder.

  • Regular Solid Volume: Calculate from dimensions.

  • Irregular Solid Volume: Displacement method (increase in liquid volume in a cylinder).

• Units: Expressed in $kg\,m^{-3}$ or $g\,cm^{-3}$.

Scalars and Vectors

• Scalar: Magnitude only (e.g., mass, time, speed, energy, current).

• Vector: Magnitude and direction (e.g., force, displacement, velocity, momentum).

• Vector Summation Methods:

  • Parallelogram Law: Draw vectors from the same point; the diagonal represents the resultant.

  • Polygon (Head-to-Tail) Method: Arrange vectors in a chain; the resultant closes the polygon.

  • Parallel/Anti-parallel: Sum algebraically (Right = positive, Left = negative).

  • Perpendicular: Use Pythagoras ($R = \sqrt{x^2 + y^2}$) and Trigonometry ($\theta = \tan^{-1}(\frac{y}{x})$).

• Resolution into Components: A vector $F$ at angle $\theta$ to the horizontal can be split into:

  • Horizontal: $F\cos(\theta)$

  • Vertical: $F\sin(\theta)$

Force, Mass, and Universal Gravitation

• Force Definitions: A pull or push that alters a body's motion or shape.

  • Gravitational: Attractive force between masses.

  • Electrostatic: Attraction/repulsion between charges.

  • Magnetic: Attraction/repulsion between magnetic poles.

  • Nuclear: Strong binding force in the nucleus.

  • Elastic: Restoring force in stretched/compressed bodies.

  • Mechanical: Contact forces like friction (opposes motion).

• Mass vs. Weight:

  • Mass: Quantity of matter (constant regardless of location).

  • Weight: Force of gravity ($W = mg$). It varies with gravitational field strength ($g$).

  • Earth's $g \approx 10\,N\,kg^{-1} = 10\,m\,s^{-2}$.

Moments of Force and Mechanical Equilibrium

• Moment (T): The turning effect of a force. Formula: $T = F \times d_{\perp}$ (Force times perpendicular distance from pivot). Unit: $N\,m$.

• Equilibrium Conditions:

  1. Translational: Sum of forces in any direction = sum of forces in the opposite direction.

  2. Rotational (Principle of Moments): Sum of clockwise moments = sum of anticlockwise moments about any point.

• Center of Gravity (CoG): The point where the resultant gravitational force acts.

• Equilibrium Types:

  • Stable: CoG rises when displaced; restoring moment returns body.

  • Unstable: CoG falls when displaced; toppling moment removes body.

  • Neutral: CoG height remains constant.

• Stability Enhancements: Lower the CoG, widen the base, or increase the weight (for stable bodies).

Deformation of Solids and Hooke’s Law

• Hooke’s Law: Force is directly proportional to extension ($F = ke$), provided the proportional limit is not exceeded.

• Key Limits:

  • Proportional Limit (P): The point where force and extension are no longer linear.

  • Elastic Limit (E): The point beyond which permanent deformation occurs (Plasticity).

• Energy: The area under a force-extension graph represents the work done (stored elastic potential energy).

• Materials: Springs typically obey Hooke's law; elastic bands do not and dissipate heat energy during load/unload cycles.

Kinematics: The Study of Motion

• Definitions:

  • Displacement: Distance in a specific direction (vector).

  • Velocity: Rate of change of displacement ($v = \frac{\Delta s}{\Delta t}$).

  • Acceleration: Rate of change of velocity ($a = \frac{\Delta v}{\Delta t}$).

• Graph Analysis:

  • Displacement-Time (s-t): Gradient = Velocity.

  • Velocity-Time (v-t): Gradient = Acceleration. Area under the curve = Distance / Displacement.

Comprehensive Physics Formulae Compendium

• Mechanics:

  • Density: $\rho = \frac{m}{V}$

  • Acceleration: $a = \frac{v - u}{t}$

  • Force: $F = ma = \frac{m(v - u)}{t}$

  • Weight: $W = mg$

  • Pressure: $p = \frac{F}{A}$; Fluid Pressure: $p = h\rho g$

  • Momentum: $p = mv$

  • Moment: $T = Fd$

  • Power: $P = \frac{W}{t} = Fv$

  • Efficiency: $\left( \frac{\text{Useful Output}}{\text{Total Input}} \right) \times 100$

  • Kinetic Energy: $E_k = \frac{1}{2}\,mv^2$

  • Potential Energy: $\Delta E_p = mg\Delta h$

• Thermal Physics:

  • Specific Heat: $E_H = mc\Delta T$

  • Latent Heat: $E_H = ml$

  • Ideal Gas Law: $\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}$

• Waves and Optics:

  • Wave Speed: $v = f\lambda$

  • Lens Equation: $\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$

  • Refraction: $n = \frac{\sin(\theta_1)}{\sin(\theta_2)} = \frac{v_1}{v_2}$

  • Magnification: $m = \frac{v}{u} = \frac{h_i}{h_o}$

• Electricity and Atoms:

  • Charge: $Q = It$

  • Energy: $E = VQ = VIt = I^2Rt$

  • Transformer: $\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$

  • Mass-Energy: $\Delta E = \Delta m c^2$