Physics for Cambridge IGCSE™ Study Notes

How to use this book

• FOCUS POINTS: Each topic begins with a bullet point summary of content and a short outline of the topic.

• Test yourself: Questions integrated throughout the topic to check understanding.

• Revision checklist: At the end of each topic, a summary to recap learning and confirm understanding of key concepts.

• Exam-style questions: Follow each topic to familiarize with question styles and consolidate learning. Past paper questions are also provided at the back of the book.

• Shaded text (Yellow): Indicates Supplement content only.

• Unshaded text: Covers the Core syllabus.

• Extended syllabus: Requires studying both Core and Supplement sections.

Additional Support Features

• Key definitions: Provide explanations of key words as required by the syllabus.

• Practical work: Boxes identifying key practical skills to understand and apply.

• Worked example: Step-by-step guidance on calculations with follow-up practice questions.

• Going further: Content that extends beyond the Cambridge syllabus for enrichment.

• Mathematics for Physics section: A reference for key mathematical skills. Answers are online with the Teacher's Guide.

• Practical Skills Workbook: Available for further support in developing practical skills.

Scientific Enquiry

• Purpose: Carry out experiments and investigations to develop skills scientists use.

• Types of Investigations:

  • Simple experiments: Designed to measure quantities like temperature or electric current.

  • Longer investigations: Designed to establish or verify relationships between physical quantities.

• Origin of Investigations: May arise from class topics, teacher suggestions, or personal ideas.

• Time Commitment: Typically requires at least one hour, often longer, of laboratory time.

Aspects of an Investigation

1. Selecting and safely using suitable techniques, apparatus, and materials:

   • Choice depends on the investigation.

   • Always perform a risk assessment first, with teacher assistance to identify and address hazards.

2. Planning your experiment:

   • Making predictions and hypotheses: Informed guesses to focus the investigation.

   • Identifying variables:

     • Controlled variables: Kept constant to avoid affecting experimental results.

     • Independent variable: The variable that is changed.

     • Dependent variable: The variable that is measured.

   • Deciding on range of values: For the independent variable.

   • Recording and analyzing results: Methods to fulfill investigation aims.

   • Apparatus precision: Tools must have enough precision (e.g., metre ruler to $0.5$mm).

   • Explaining experimental procedure: Include clearly labelled diagrams, difficulties encountered, and precautions for accuracy.

3. Making and recording observations and measurements:

   • Obtain necessary data safely and accurately.

   • Familiarize yourself with apparatus use before taking measurements.

   • Ordered recording: Use tables for multiple measurements with column headings like 'Mass of load/kg'.

   • Repeat measurements: Record each value, then calculate an average.

   • Significant figures: Numerical values should be given to the appropriate number of significant figures for the measuring device.

   • Graphs: If plotting a graph, need at least eight data points over as large a range as possible; label axes with name and unit.

   • Equipment: Do not dismantle until analysis is complete and no repeats are needed.

4. Interpreting and evaluating observations and data:

   • Helps establish relationships between quantities.

   • Calculations/Graphs: May need to calculate specific values or plot a graph, draw a line of best fit, and calculate a gradient.

   • Anomalous results: Explain and describe how they were dealt with.

   • Graph commentary: Comment on shape, point alignment, and trends.

   • Conclusions: Draw conclusions justified by the data, which can be numerical values (with units), known laws, relationships, or statements related to the experiment's aim.

   • Data quality: Comment on quality and compare results to expectations within accuracy limits.

5. Evaluating methods and suggesting possible improvements:

   • Acknowledge that experiments are rarely flawless.

   • Sources of error: Identify random, systematic, and measurement errors.

   • Unsuitable apparatus: Mention any equipment that proved inadequate.

   • Modifications for accuracy: Discuss how the experiment could be improved (e.g., using a more appropriate ammeter scale).

   • Suggested improvements: (e.g., reducing thermal energy losses, changing control variables).

Suggestions for Investigations

1. Stretching of a rubber band (Topic 1.5.1).

2. Stretching of a copper wire – wear eye protection (Topic 1.5.1).

3. Toppling (Topic 1.5.1).

4. Friction – factors affecting (Topic 1.5.1).

5. Model wind turbine design (Topic 1.7.3).

6. Speed of a bicycle and its stopping distance (Topic 1.7.1).

7. Energy transfer using different insulating materials (Topic 2.3.1).

8. Cooling and evaporation (Topic 2.2.3).

9. Pitch of a note from a vibrating wire (Topic 3.4).

10. Variation of the resistance of a thermistor with temperature (Topic 4.2.4).

11. Variation of the resistance of a wire with length (Topic 4.2.4)

12. Heating effect of an electric current (Topic 4.2.2).

13. Strength of an electromagnet (Topic 4.1).

14. Efficiency of an electric motor (Topic 4.2.5).

Ideas and Evidence in Science

• Scientific controversy: Arises from different interpretations of evidence.

• Historical Examples:

  • Geocentric vs. Heliocentric models:

    • Ancient Greeks believed Earth was central, requiring complex planetary rotation to match observations.

    • 1543: Nicolaus Copernicus proposed planets revolved around the Sun (heliocentric).

    • Tycho Brahe's observations were used by Johannes Kepler to show elliptical (not circular) planetary paths.

    • Galileo's telescope observations supported the Copernican view, leading to his imprisonment.

    • $~50$ years later: Isaac Newton introduced gravity, explaining all motion and leading to full acceptance of the Copernican model.

    • Early 20th century: Einstein's theories of relativity refined Newton's mechanics. Hubble Space Telescope data continues to confirm Einstein's ideas.

  • Other scientific theories: Often require new data, technological inventions, or a supportive social/intellectual climate for acceptance.

  • Health and medicine: Long development times for diseases like cancer meant delayed recognition of dangers of X-rays and radioactive materials (Topic 5.2.5).

  • Quantum mechanics: Early 20th-century attempts to reconcile wave and particle theories of light.

  • Current issues: Collecting evidence on health risks of microwaves in mobile phone networks. Public and manufacturers may be reluctant to accept adverse findings due to social/economic interests, as seen with global warming.

1.1 Physical Quantities and Measurement Techniques

FOCUS POINTS

• Describe how to measure length, volume and time intervals using simple devices.

• Know how to determine the average value for a small distance and a short time interval.

• Understand the difference between scalar and vector quantities, and give examples of each.

• Calculate or determine graphically the resultant of two perpendicular vectors.

Overview

• Introduces describing space and time with numbers and basic physics units.

• Learn to use simple devices for measuring length, area, and volume.

• Emphasizes accurate time measurement and appropriate timer selection.

• Discusses error reduction by averaging or measuring multiples.

• Introduces scalars (magnitude only) and vectors (magnitude and direction).

• Explains combining two perpendicular vectors graphically or by calculation.

Units and Basic Quantities

• Pre-measurement requirement: A standard or unit must be chosen.

• Measurement process: Size found with an instrument scaled in the chosen unit.

• Three basic quantities: Length, mass, and time.

• Derived units: Units for other quantities are based on these three.

• SI (Système International d’Unités) system:

  • Set of metric units widely used.

  • Decimal system: units divided or multiplied by $10$ (e.g., Figure 1.1.1, aircraft flight deck).

Powers of Ten Shorthand (Standard Notation)

• A concise way to write large or small numbers.

• Example format: $4000 = 4 \times 10 \times 10 \times 10 = 4 \times 10^3$

• Power: Indicates how many times a number is multiplied by $10$ (if >$0$) or divided by $10$ (if <$0$).

• $1$: Written as $10^0$.

• Examples:

  • $4000 = 4 \times 10^3$

  • $400 = 4 \times 10^2$

  • $40 = 4 \times 10^1$

  • $4 = 4 \times 10^0$

  • $0.4 = 4/10 = 4/10^1 = 4 \times 10^{-1}$

  • $0.04 = 4/100 = 4/10^2 = 4 \times 10^{-2}$

  • $0.004 = 4/1000 = 4/10^3 = 4 \times 10^{-3}$

Length

• SI Unit: Metre (m).

• Definition: Distance travelled by light in a vacuum during a specific time interval.

• Historic definition: Previously distance between marks on a metal bar.

• Submultiples:

  • $1$ decimetre (dm) = $10^{-1}$m

  • $1$ centimetre (cm) = $10^{-2}$m

  • $1$ millimetre (mm) = $10^{-3}$m

  • $1$ micrometre (µm) = $10^{-6}$m

  • $1$ nanometre (nm) = $10^{-9}$m

• Multiples:

  • $1$ kilometre (km) = $10^3$m (approx. $5/8$ mile)

  • $1$ gigametre (Gm) = $10^9$m = $1$ billion metres

• Ruler measurement:

  • Correct reading (e.g., $76$mm or $7.6$cm).

  • Parallax error: Eye must be directly over the mark on the scale; ruler thickness can cause error (Figure 1.1.2).

• Average value for small distances: Measure multiples and divide (e.g., distance of five waves in a ripple tank, then divide by $5$).

Significant Figures

• Purpose: Indicates the perceived accuracy of a measurement, arising from apparatus limitations and experimenter skill.

• Rule: Do not give more figures than justified.

• Examples:

  • $4.5$: two significant figures.

  • $0.0385$: three significant figures (3 most significant, 5 least, due to potential estimation).

• Calculations: Answer should have the same number of significant figures as the least precise measurement used.

• Rounding rules:

  • If the next figure to the right is <$5$, leave the least significant figure as is (e.g., $3.41$ becomes $3.4$ for two sig figs).

  • If the next figure is $\ge 5$, increase the least significant figure by $1$ (round up) (e.g., $3.418$ becomes $3.42$ for three sig figs).

• Standard notation: Number of significant figures is the number of digits before the power of ten (e.g., $2.73 \times 10^3$ has three significant figures).

Area

• Square/Rectangle area: length $\times$ breadth.

• SI unit of area: Square metre ($\text{m}^2$).

• Conversions:

  • $1\text{cm}^2 = (1/100)\text{m} \times (1/100)\text{m} = 1/10000\text{m}^2 = 10^{-4}\text{m}^2$

• Triangle area: $\frac{1}{2} \times \text{base} \times \text{height}$.

• Circle area: $\pi\text{r}^2$, where $\pi \approx 22/7$ or $3.14$.

• Circle circumference: $2\pi\text{r}$.

• Worked Example: Calculating area of triangles (Figure 1.1.4).

  • Triangle ABC: $\frac{1}{2} \times 4\text{cm} \times 6\text{cm} = 12\text{cm}^2$

  • Triangle PQR: $\frac{1}{2} \times 5\text{cm} \times 4\text{cm} = 10\text{cm}^2$

Volume

• Definition: Amount of space occupied.

• SI unit of volume: Cubic metre ($\text{m}^3$).

• Commonly used unit: Cubic centimetre ($\text{cm}^3$) due to $\text{m}^3$ being large.

• Conversions:

  • $1\text{cm}^3 = (1/100)\text{m} \times (1/100)\text{m} \times (1/100)\text{m} = 1/1000000\text{m}^3 = 10^{-6}\text{m}^3$

  • $1\text{ml} = 1\text{cm}^3$

  • $1000\text{cm}^3 = 1\text{dm}^3 = 1$ litre

• Regularly shaped object (rectangular block): volume = length $\times$ breadth $\times$ height (Figure 1.1.5).

• Cylinder volume: $\pi\text{r}^2\text{h}$.

• Liquid volume (measuring cylinder) (Figure 1.1.6):

  • Cylinder must be upright.

  • Eye level with the bottom of the curved liquid surface (meniscus).

  • For mercury, meniscus is curved oppositely; read the top.

• Worked Example: Calculating volume.

  • Block: $40\text{cm} \times 12\text{cm} \times 5\text{cm} = 2400\text{cm}^3 = 2.4 \times 10^{-3}\text{m}^3$

  • Cylinder: $\pi \times (1.0\text{cm})^2 \times 5.0\text{cm} = 16\text{cm}^3 = 1.6 \times 10^{-5}\text{m}^3$

Time

• SI Unit: Second (s).

• Historic definition: Based on Earth's revolution on its axis.

• Modern definition: Time interval for a certain number of energy changes in the caesium atom.

• Time-measuring devices: Rely on constantly repeating oscillation.

  • Traditional clocks: oscillating balance wheel.

  • Digital clocks: vibrating quartz crystal.

  • Pendulum clocks: swinging pendulum.

• Choosing a timer: Must be precise enough for the task.

  • Stopwatch: Adequate for pendulum period (Figure 1.1.7) (seconds).

  • Millisecond clock: Needed for very short intervals (e.g., speed of sound).

  • Digital clocks triggered by electronic signals: Useful for very short intervals (microphones, photogates, mechanical switches).

  • Tickertape timers or dataloggers: For short time intervals in motion experiments.

• Improving accuracy: Measure longer time intervals.

  • Pendulum: Time several oscillations, then divide by the number of oscillations for average period.

  • Tickertape timers: Use ten ticks instead of single ticks.

Practical Work: Period of a Simple Pendulum (Figure 1.1.7)

• Safety: Soft landing for masses, keep feet clear.

• Apparatus: Small metal ball (bob) on a string, suspended from a support stand.

• Procedure:

  • Displace bob slightly and release so it oscillates through a small angle.

  • Measure time for several complete oscillations (A to O to B to O to A is one oscillation).

  • Repeat timing a few times and calculate average.

• Calculations:

  • Period ($T$) = time for one oscillation.

  • Frequency ($f$) = number of complete oscillations per second = $1/T$.

• Observations: Comment on amplitude change over time.

• Investigation planning: Effect on $T$ of (i) longer string and (ii) larger bob.

• Length of pendulum: Measured from the point of suspension to the center of the bob.

Systematic Errors

• Definition: Errors introduced by the system itself, causing readings to be consistently too high or too low.

• Example (Ruler) (Figure 1.1.8):

  • A ruler with a space '$x$' before the zero mark will cause all readings to be too small by '$x$'.

  • Height = scale reading $+ x$.

• Avoiding systematic errors:

  • Use rulers with zero at the end.

  • Hold ruler vertically when determining height.

Going Further: Vernier Scales and Micrometers

• Ruler precision: Approximately $0.5$mm.

• Vernier calipers (Figure 1.1.9):

  • For more precise length measurements (e.g., $0.01$cm).

  • Vernier scale: Small sliding scale, $9$mm long divided into ten equal divisions ($1$ division = $0.9$mm = $0.09$cm).

  • Reading: Match zero of mm scale and zero of vernier scale. Reading (e.g., $1.36$cm) calculated from main scale reading + (vernier mark coinciding $\times$ vernier division value).

  • Used on barometers, travelling microscopes, spectrometers.

• Micrometer screw gauge (Figure 1.1.11):

  • Measures very small objects (e.g., $0.001$cm).

  • One revolution of drum moves jaws by one division on shaft (usually $0.5$mm or $0.05$cm).

  • If drum has $50$ divisions, one drum division = $0.05/50 = 0.001$cm.

  • Friction clutch: Ensures consistent force when gripping objects.

  • Reading: Shaft scale reading + (drum scale reading $\times$ drum division value) (e.g., $2.5$mm on shaft + $33$ divisions = $0.25$cm + $33(0.001)$cm = $0.283$cm).

  • Zero error: Check reading when jaws are closed and adjust measurements accordingly.

Scalars and Vectors

• Scalar quantity: Has magnitude (size) only.

  • Examples: Time, distance, speed, mass, pressure, energy, temperature.

• Vector quantity: Has both magnitude (size) and direction.

  • Examples: Force, weight, velocity, acceleration, momentum, gravitational field strength, electric field strength.

  • Representation: Straight line with length representing magnitude and arrow showing direction.

• Adding quantities:

  • Scalars: By ordinary arithmetic.

  • Vectors: Geometrically, considering both magnitude and direction.

• Resultant of two perpendicular vectors (Figure 1.1.12):

  • Let $\text{F}<em>X$ and $\text{F}</em>Y$ be two perpendicular vectors.

  • Magnitude of resultant $F = \sqrt{\text{F}<em>X^2 + \text{F}</em>Y^2}$.

  • Angle $\theta$ between $\text{F}<em>X$ and $F$: $\tan\theta = \text{F}</em>Y / \text{F}_X$.

• Worked Example: Resultant of $3.0$N and $4.0$N perpendicular forces.

  • $F = \sqrt{3.0^2 + 4.0^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0$N.

  • $\tan\theta = 4.0/3.0 = 1.33$, so $\theta = 53^{\circ}$.

  • Resultant: $5.0$N at $53^{\circ}$ to the $3.0$N force.

• Graphical method for vectors:

  • Draw vectors to scale on graph paper at right angles.

  • Complete the rectangle.

  • Draw diagonal from origin (represents resultant $F$).

  • Measure length of $F$ and convert using scale for magnitude.

  • Measure angle $\theta$ with a protractor for direction.

1.2 Motion

FOCUS POINTS

• Define speed and velocity and use the appropriate equations to calculate these and average speed.

• Draw, plot and interpret distance–time or speed–time graphs for objects at different speeds and use the graphs to calculate speed or distance travelled.

• Define acceleration and use the shape of a speed–time graph to determine constant or changing acceleration and calculate the acceleration from the gradient of the graph.

• Know the approximate value of the acceleration of freefall, g, for an object close to the Earth’s surface.

• Describe the motion of objects falling with and without air/liquid resistance.

Overview

• Concepts of speed and acceleration are daily phenomena (e.g., sports, vehicles).

• Define speed in terms of distance and time.

• Use graphs (distance-time, speed-time) to calculate speed, distance, and analyze acceleration.

• Acceleration occurs in falling objects due to gravity.

• All objects near Earth experience constant gravitational acceleration directed towards the Earth's center.

Speed

• Key Definition: Speed = distance travelled per unit time.

• Equation: $\text{v} = \text{s} / \text{t}$ (where $s$ is distance, $t$ is time).

• Average speed: total distance travelled / total time taken.

  • Example: Car travels $300$km in $5$h, average speed = $60$km/h.

  • Actual speed can vary (e.g., speedometer changes).

• Instantaneous speed: Distance moved in a very short time interval.

  • Can be found by multiflash photography (Figure 1.2.1: golfer's club-head speed approx. $200$km/h).

Velocity

• Key Definition: Velocity = distance travelled in unit time in a given direction (or change in displacement per unit time).

• Equation: velocity = speed in a given direction = (distance moved in a given direction) / time taken.

• Uniform/Constant velocity: Steady speed in a straight line.

• Non-uniform velocity: Motion in a curved path (direction changes).

• Units: Same as speed (km/h, m/s).

  • Example: $60$km/h = $60000$m / $3600$s = $17$m/s.

• Displacement: Distance moved in a stated direction (a vector).

• Scalar vs. Vector: Speed is scalar, velocity is vector.

Acceleration

• Key Definition: Acceleration = change in velocity in unit time.

• Equation: $\text{a} = \Delta\text{v} / \Delta\text{t}$ (where $\Delta\text{v}$ is change in velocity, $\Delta\text{t}$ is time taken for change).

• Units: m/s${^2}$.

  • Example: Velocity increases by $2$m/s in $1$s, acceleration is $2$m/s${^2}$.

• Scalar vs. Vector: Acceleration is a vector; magnitude and direction should be stated.

  • For straight-line motion, magnitude of velocity = speed, magnitude of acceleration = change of speed in unit time.

• Positive acceleration: Velocity increases.

• Negative acceleration: Velocity decreases (also called deceleration or retardation).

  • Example: Car speeds (0-30 m/s in 6s) show constant acceleration of $5$m/s${^2}$. (Speed increases by $5$m/s every second).

Speed-Time Graphs

• Plotting speed against time.

• Constant speed (Figure 1.2.2):

  • Graph is a horizontal straight line (e.g., AB at $20$m/s).

  • Speed is constant over time interval.

  • Gradient is zero, so acceleration is zero.

• Constant acceleration (Figure 1.2.3a):

  • Graph is a straight line sloping upwards (e.g., PQ).

  • Gradient is constant, representing constant acceleration (e.g., $20/5 = 4$m/s${^2}$).

  • Speed increases by same amount each second.

• Acceleration, constant speed, deceleration, at rest (Figure 1.2.3b):

  • OA: acceleration.

  • AB: constant speed.

  • BC: deceleration (speed decreasing; steeper gradient than OA means greater deceleration).

  • CD: at rest (speed and acceleration zero; horizontal line on x-axis).

• Changing acceleration (Figure 1.2.3c):

  • Graph is a curve (e.g., OX).

  • Gradient changes, so acceleration changes.

  • Upward curve indicates increasing speed; if gradient decreases, acceleration is decreasing.

• Calculating acceleration from gradient: Gradient of a speed-time graph = acceleration of the object.

Distance-Time Graphs

• Plotting distance against time.

• Constant speed (Figure 1.2.4a):

  • Graph is a straight line sloping upwards (e.g., OL).

  • Gradient of a distance-time graph = speed of the object (e.g., $40$m/$4$s = $10$m/s).

  • Equal distances covered in equal times.

• At rest, constant speed, at rest, constant speed (Figure 1.2.4b):

  • OA: at rest (distance does not change).

  • AB: constant speed.

  • BC: at rest.

  • CD: constant speed.

  • Steeper gradient indicates higher speed (e.g., faster in AB than CD).

• Non-constant speed (Figure 1.2.5):

  • Graph is a curve.

  • Upward curve of increasing gradient: accelerating.

  • Upward curve of decreasing gradient: decelerating.

  • Speed at any point = gradient of the tangent at that point (e.g., tangent at T: $40$m/$2$s = $20$m/s).

Area Under a Speed-Time Graph

• Measures: Distance travelled.

• Constant speed (Figure 1.2.2):

  • Distance = average speed $\times$ time (e.g., $20$m/s $\times 5$s = $100$m).

  • Corresponds to the area of rectangle OABC.

• Constant acceleration (Figure 1.2.3a):

  • Distance = area of rectangle OPRS + area of triangle PQR.

  • Distance = $(20\text{m/s} \times 5\text{s}) + (\frac{1}{2} \times 5\text{s} \times 20\text{m/s}) = 100\text{m} + 50\text{m} = 150\text{m}$.

  • Unit of time must be consistent on both axes.

• Non-constant acceleration (Figure 1.2.3c):

  • Distance travelled = shaded area OXY.

Equations for Constant Acceleration (Straight Line Motion)

• First equation: $\text{v} = \text{u} + \text{at}$

  • $u$ = initial speed, $v$ = final speed, $a$ = constant acceleration, $t$ = time.

• Second equation: $\text{s} = (\text{u} + \text{v})/2 \times \text{t}$

  • Average Speed = $(u + v)/2$.

  • $s$ = distance moved.

• Going Further: Third equation: $\text{s} = \text{ut} + \frac{1}{2}\text{at}^2$

  • Derived by substituting $v = u + at$ into the second equation.

• Going Further: Fourth equation: $\text{v}^2 = \text{u}^2 + 2\text{as}$

  • Derived by eliminating $t$ from the first and third equations.

• Usage: If any three of $u$, $v$, $a$, $s$, $t$ are known, the others can be found.

• Worked Example: Sprint cyclist (u=0, a=1m/s${^2}$, t=20s).

  • Final speed $v = 0 + 1 \times 20 = 20$m/s.

  • Distance travelled $s = (0 + 20)/2 \times 20 = 200$m.

Falling Bodies

• Air resistance: Coins fall faster than paper in air due to air resistance affecting lighter bodies more significantly.

• Vacuum: In a vacuum, all bodies fall at the same rate (Figure 1.2.6).

• Galileo's experiment (Leaning Tower of Pisa) (Figure 1.2.7): Dropped objects of different masses reportedly hit the ground almost simultaneously, despite mass differences.

Practical Work: Motion of a Falling Object (Figure 1.2.8)

• Safety: Place soft material on floor, keep feet clear.

• Apparatus: $100$g mass, retort stand, tickertape timer, tickertape.

• Procedure:

  • Investigate $100$g mass falling $~2$m using tickertape timer (vibrates $50$ times/s, dots at $1/50$s intervals).

  • Ignore start of tape where dots are too close.

  • Repeat with $200$g mass and compare.

• Observations: Spacing between dots increases, indicating increasing speed.

• Calculations: Estimate fall time ($34$ dots/$50$ dots per second = $0.68$s).

• Timer choice: Stopwatch not chosen for accuracy of short fall time.

• Mass comparison: Expect similar fall times for $100$g and $200$g masses (ignoring air resistance).

Acceleration of Free Fall

• Definition: Uniform acceleration of bodies falling freely under gravity when air resistance is negligible.

• Symbol: $g$.

• Value: Varies slightly globally, average is approx. $9.8$m/s${^2}$ (or $10$m/s${^2}$ for estimates).

• Effect: Velocity of free-falling body increases by approx. $10$m/s every second.

  • Example: Ball shot upwards at $30$m/s decelerates by $10$m/s every second, reaching highest point in $3$s.

• In equations of motion: $g$ replaces $a$.

  • Falling bodies: $a = g = +9.8$m/s${^2}$.

  • Rising bodies: $a = -g = -9.8$m/s${^2}$ (decelerating).

Going Further: Measuring $g$ (Figure 1.2.9)

• Apparatus: Steel ball-bearing, electromagnet, electronic timer, impact switch.

• Procedure:

  • Two-way switch (down) releases ball and starts clock simultaneously.

  • Ball opens 'trap-door' on impact switch, stopping clock.

• Calculation: Using $s = ut + \frac{1}{2}at^2$

  • $u = 0$ (starts from rest), $a = g$.

  • $s = \frac{1}{2}gt^2$ or $g = 2s/t^2$.

• Air resistance: Negligible for dense objects falling short distances.

• Rough estimate: Time rubber ball fall from building (needs fast reactions for stopwatch).

Distance-Time Graphs for a Falling Object

• Without air resistance (constant acceleration $g$): $s = \frac{1}{2}gt^2$

  • Distance $s$ against time $t$ (Figure 1.2.10a): Gradually increasing slope as speed increases steadily.

  • Distance $s$ against time-squared $t^2$ (Figure 1.2.10b): Straight line through origin (since $s \propto t^2$ and $g$ is constant).

Going Further: Projectiles

• Multiflash photography (Figure 1.2.11): Shows that dropped balls and horizontally thrown projectiles have equal vertical accelerations due to gravity.

• Key principle: Horizontal and vertical motions are independent and can be treated separately.

• Example: Ball thrown horizontally from cliff.

  • Vertical motion: $u=0$, $a=g=+9.8$m/s${^2}$, $t=3$s.

  • Cliff height $s = ut + \frac{1}{2}at^2 = 0 \times 3 + \frac{1}{2}(+9.8)3^2 = 44$m.

• Range of projectiles (e.g., cricket balls, shells):

  • Depends on: speed of projection (greater speed = greater range) and angle of projection.

  • Maximum range (neglecting air resistance): Achieved at $45^{\circ}$ (Figure 1.2.12).

Air Resistance: Terminal Velocity

• Mechanism: Air resistance (fluid friction) increases with speed, reducing acceleration.

• Terminal velocity: When air resistance = object's weight, resultant force is zero. Object falls at constant velocity.

• Factors affecting terminal velocity: Size, shape, and weight of the object.

  • Small, dense object (steel ball): High terminal velocity, accelerates for considerable distance.

  • Light object (raindrop), large surface area (parachute): Low terminal velocity, accelerates over short distance.

  • Skydiver: Terminal velocity > $50$m/s before parachute opens (Figure 1.2.13).

• Liquid behavior: Objects falling in liquids behave similarly.

• Absence of air resistance: Object falls with constant acceleration (Figure 1.2.10a shows distance-time graph).

1.3 Mass and Weight

FOCUS POINTS

• Define mass and weight and know that weights (and therefore masses) may be compared using a balance or force meter.

• Define gravitational field strength and know that this is equivalent to the acceleration of free fall.

• Understand that weight is the effect of a gravitational field on mass.

• Describe, and use the concept of, weight as the effect of a gravitational field on a mass.

Overview

• Astronauts on the Moon bounce due to lower gravity.

• Introduces mass as a fundamental property (quantity of matter).

• Weight is the force of gravity on mass, proportional to mass and gravitational field strength.

• Mass is constant, but weight varies with gravity (e.g., less on Moon).

Mass

• Key Definition: Mass = measure of the amount of matter in an object at rest relative to an observer.

• Standard unit: Kilogram (kg).

• Historical definition: Previously a piece of platinum-iridium alloy.

• Modern definition: Based on a fundamental physical constant.

• Gram: $1$g = $1/1000$kg = $10^{-3}$kg = $0.001$kg.

• Common confusion: Weight often used when mass is meant.

• Measurement: Found using a balance (e.g., beam balance, lever balance, digital top-pan balance (Figure 1.3.1)). This is referred to as 'weighing', which causes confusion between mass and weight.

Weight

• Key Definition: Weight = a gravitational force on an object that has mass.

• Cause: Pull of the Earth (gravity).

• Effect: Causes unsupported bodies to fall.

• Variation: Greater at poles than equator because Earth is flatter at the poles (nearer to Earth's center).

• **