Comprehensive Physics Notes – Measurement Techniques, Motion, Mass & Weight, Density, and Forces

Physical Quantities and Measurement Techniques

Units, Standards & SI System

Measurement begins with an agreed unit (standard); instruments carry scales calibrated in that unit.
Three fundamental mechanical quantities: length (m), mass (kg), time (s).
SI is decimal/metric; prefixes change size by factors of $10^{\pm n}$ (e.g. $1\,\mathrm{km}=10^3\,\mathrm{m}$ ).

Powers of Ten & Standard Form

Large/small numbers written as $a\times10^n$ with 1\le a < 10.
- $4000 = 4\times10^3$ , $0.004 = 4\times10^{-3}$ .
Powers are positive for multiplication by 10, negative for division by 10; $10^0=1$ .

Significant Figures (s.f.)

Digits believed to be correct plus one doubtful digit.
Rules: 2.5 (2 s.f.), 0.0385 (3 s.f.), $4.55\times10^4$ (3 s.f.).
Rounding: if next digit <5 round down; ≥5 round up.
Result of a calculation should not contain more s.f. than the data.

Measuring Length

Rulers/Tapes read eye on mark; avoid parallax.
Sub-multiples: $1\,\mathrm{dm}=10^{-1}\,\mathrm{m},\;1\,\mathrm{cm}=10^{-2}\,\mathrm{m},\;1\,\mathrm{mm}=10^{-3}\,\mathrm{m},\;1\,\mu\mathrm{m}=10^{-6}\,\mathrm{m},\;1\,\mathrm{nm}=10^{-9}\,\mathrm{m}$ .
Multiples: $1\,\mathrm{km}=10^3\,\mathrm{m},\;1\,\mathrm{Gm}=10^9\,\mathrm{m}$ .
Average small lengths: measure several wavelengths etc., divide by count.

Measuring Area

Rectangle: $A = l\times b$ ; Square 1 cm side = $1\,\mathrm{cm^2}=10^{-4}\,\mathrm{m^2}$ .
Triangle: $A=\tfrac12\,\text{base}\times\text{height}$ .
Circle: $A=\pi r^2$ , circumference $C=2\pi r$ ( $\pi\approx\tfrac{22}{7}=3.14$ ).

Measuring Volume

Rectangular block: $V=l\times b\times h$ .
Cylinder: $V=\pi r^2 h$ .
$1\,\mathrm{cm^3}=1\,\mathrm{mL},\;1000\,\mathrm{cm^3}=1\,\mathrm{dm^3}=1\,\mathrm{L}$ ; $1\,\mathrm{cm^3}=10^{-6}\,\mathrm{m^3}$ .
Use measuring cylinder: read bottom of meniscus; mercury read top (inverse curve).

Measuring Time

Base unit second now defined from caesium-133 transitions.
Devices based on periodic processes: balance wheel, quartz crystal, pendulum, photogates, ticker-timer, datalogger.
For accuracy: choose timer with resolution << interval; time many oscillations and average.

Systematic & Parallax Errors

Zero error adds/subtracts constant (e.g. ruler with gap before 0).
Hold ruler vertical; keep eye perpendicular to scale.

Vernier Calipers

Vernier 9 mm long split into 10 → $0.09\,\mathrm{cm}$ per div., hence precision $0.01\,\mathrm{cm}$ .
Reading = main-scale + vernier coincidence. Example: 1.36 cm.

Micrometer Screw Gauge

Shaft scale $0.5\,\mathrm{mm}$ pitch; drum 50 divisions → $0.01\,\mathrm{mm}=1\,\mu\mathrm{m}$ precision.
Reading = shaft + drum; zero error must be corrected.

Scalars vs Vectors

Scalar: magnitude only (distance, speed, mass, energy, temperature).
Vector: magnitude & direction (displacement, velocity, force, weight, acceleration, momentum, field strengths).
Represent vector by arrow; add using parallelogram or $\sqrt{Fx^2+Fy^2}$ for perpendicular components.

Motion

Speed & Velocity

$\text{speed}= \dfrac{\text{distance}}{\text{time}}$ .
Average speed $=\dfrac{\text{total distance}}{\text{total time}}$ .
Velocity = rate of change of displacement (vector).

Acceleration

$a = \dfrac{\Delta v}{t}$ , units $\mathrm{m/s^2}$ .
Positive for speeding up; negative (retardation) for slowing.

Graphical Analysis

Speed–time graph: gradient → acceleration; area under curve → distance.
Distance–time graph: gradient → speed.
- Steeper = faster; horizontal = rest.
Constant acceleration → straight line on $v–t$ ; changing acceleration → curve.

Kinematic Equations (constant a)

$v=u+at$
$s=\tfrac12(u+v)t$
$v^2=u^2+2as$ (derived, transcript section imminent).

Free Fall & g

Near Earth, $g\approx9.8\,\mathrm{m/s^2}$ downward.
In vacuum all objects accelerate equally (Galileo, coin–feather tube).
Use $a=\pm g$ in equations; positive downward.

Distance–time for Free Fall

Parabolic curve; slope (speed) increases uniformly.

Projectiles

Horizontal & vertical motions independent; same vertical acceleration as dropped body.

Mass & Weight

Definitions

Mass (m): quantity of matter; inertia; unit kg.
Weight (W): gravitational force on mass; vector; unit N.
Relation: $W = mg$ .

Gravitational Field Strength

$g = \dfrac{W}{m}$ , units $\mathrm{N\,kg^{-1}}$ ; numerically equal to $\mathrm{m\,s^{-2}}$ .
Varies slightly over Earth; Moon $\approx1.6\,\mathrm{N\,kg^{-1}}$ .

Inertia

Resistance to change of motion; proportional to mass.

Measuring Mass & Weight

Beam/lever/electronic balance: compares weight against known masses (works anywhere because $W\propto m$ ).
Spring balance / force meter measures weight directly in N.

Density

Formula & Units

$\rho = \dfrac{m}{V}$ .
SI: $\mathrm{kg\,m^{-3}}$ ; practical: $\mathrm{g\,cm^{-3}}$ ( $1\,\mathrm{g\,cm^{-3}} = 10^3\,\mathrm{kg\,m^{-3}}$ ).

Typical Values

Aluminium 2.7; Water 1.0; Iron 7.9; Lead 11.3; Gold 19.3 $\mathrm{g\,cm^{-3}}$ . Air 1.3 $\mathrm{kg\,m^{-3}}$ .

Measuring Density

Regular solid: measure $l,b,h$ via ruler; weigh on balance.
Irregular solid: find mass; volume by displacement:
- Method 1: difference in cylinder readings.
- Method 2: overflow can + measuring cylinder.
Liquid: measure mass of empty beaker, then filled; volume via burette/measuring cylinder.
Gas (air): evacuate flask, weigh full/empty; determine volume with water fill.

Floatation Principle

Object sinks if $\rho{object} > \rho{fluid}$ ; rises/floats otherwise.

Forces

Types & Free-Body Diagrams

Contact: friction, tension, normal reaction, thrust, upthrust.
Action-at-distance: gravitational, electrostatic, magnetic.
Represent each by arrow; resultant obtained by vector addition.

Elastic Deformation

Hooke’s Law: within limit of proportionality $F \propto x$ → $F = kx$ .
Spring constant k: $k=\dfrac{F}{x}$ ; units $\mathrm{N\,m^{-1}}$ .
Load–extension graph: straight line through origin up to point E; beyond E, material yields; permanent set OS remains on unloading.
Limit of proportionality: point where linearity ceases.

Resultant Forces

Colinear forces: algebraic sum (consider direction).
Perpendicular forces: $R=\sqrt{Fx^2+Fy^2}$ ; direction $\tan\theta=Fy/Fx$ .
Parallelogram law: diagonal gives resultant.

Newton’s Laws

First Law: body remains at rest or moves with constant velocity unless acted on by resultant force.
Second Law: $\vec F = m\vec a$ (resultant force equals rate of change of momentum; simplified for constant mass).
- $1\,\mathrm{N}=1\,\mathrm{kg\,m\,s^{-2}}$ defined as force giving $1\,\mathrm{kg}$ an acceleration $1\,\mathrm{m\,s^{-2}}$ .
Third Law: For every action force there is an equal and opposite reaction acting on a different body.

Friction & Drag

Opposes relative motion; converts kinetic energy to heat.
Static friction (max) > dynamic/sliding friction.
Drag in fluids increases with speed; acts opposite motion.

Terminal Velocity

Falling object: weight $W$ downward, drag $D$ upward increasing with speed.
When $D=W$ , resultant $0$ → constant speed $v_t$ .
Dense small object: large $vt$ ; parachutist with canopy: small $vt$ .

Driving Safety

Stopping distance $=$ thinking + braking distances.
- Thinking $\propto$ speed $v$ because $d=vt$ .
- Braking distance increases with $v^2$ for constant braking force.
Factors increasing stopping distance: higher speed, tiredness, alcohol/drugs, poor visibility (reaction time); wet/icy roads, worn brakes/tyres, heavy load (braking force).

Circular Motion (intro)

Object moving at speed $v$ in circle radius $r$ experiences inward (centripetal) force.
Magnitude $F=\dfrac{mv^2}{r}$ (equation to be derived later).
$F$ increases with bigger $m$ , higher $v$ , smaller $r$ .
Examples: satellites (gravity supplies $F$ ), hammer throw (string tension), car cornering (friction between tyres & road).

Worked-Example Highlights

Copper Density: $\rho=9\,\mathrm{g\,cm^{-3}}$ ; mass of $5\,\mathrm{cm^3}$ is $45\,\mathrm{g}$ ; volume of $63\,\mathrm{g}$ is $7\,\mathrm{cm^3}$ .
Spring Constant: 2.0 N stretches $0.01\,\mathrm{m}$ → $k=200\,\mathrm{N\,m^{-1}}$ ; $x=0.08\,\mathrm{m}$ ⇒ $F=16\,\mathrm{N}$ .
Block on Table: friction $5\,\mathrm{N}$ ; increase push to 9 N → resultant 4 N; $a=2\,\mathrm{m\,s^{-2}}$ for 2 kg block.

Ethical / Practical Notes

Safety in labs: eye protection when stretching springs; soft landing for falling masses; secure runways for trolleys.
Real-world implications: road-safety campaigns stress reaction time and tyre condition; engineering structures use density & vector resultants; sports (skydiving, athletics) rely on motion & forces understanding.

Formulae Collection (LaTeX)

$\text{Speed } v = \dfrac{s}{t}$
$\text{Average speed}=\dfrac{\Sigma s}{\Sigma t}$
$a = \dfrac{v-u}{t}$
$v = u + at$
$s = \tfrac12 (u+v)t$
$F = ma$
$W = mg$
$g = \dfrac{W}{m}$
$\rho = \dfrac{m}{V}$
$F = kx$
$F_{\text{centripetal}} = \dfrac{mv^2}{r}$

Concept Web (Connections)

Motion graphs feed into forces via $F=ma$ (slope gives $a$ ).
Mass links inertia (1st law) & weight ( $mg$ ).
Density underpins buoyancy & choice of materials.
Significant figures critical in reporting all measurements.