Area: $\text{Area} = \text{length} \times \text{breadth}$, SI unit: square metre (m$^2$).
Volume: Amount of space occupied. SI unit: cubic metre (m$^3$). Commonly used: cubic centimetre (cm$^3$).
Volume of a cylinder: $V = \pi r^2 h$.
Measuring cylinder (for liquids and irregular shapes by displacement). Always measure from the bottom of the meniscus.
Time: Stopwatches, clocks. Unit: second (s). Time-measuring devices use oscillations.
Mass: Balances. Unit: Kilogram (kg).
Precision: Emphasize precision and avoiding parallax errors.
1.2 Motion
Speed:
Definition: Distance travelled per unit time.
Formula:
$$v=\frac{s}{t}$$
$v$: speed (m/s)
$s$: distance travelled (m)
$t$: time (s)
Average Speed: Total distance travelled / Total time taken.
Velocity:
Definition: Distance travelled in unit time in a given direction. Change in displacement per unit of time.
Formula:
$$\text{Velocity}=\frac{\text{Distance moved in a given direction}}{\text{Time taken}}=\frac{\text{Displacement}}{\text{Time}}$$
Velocity is a vector quantity; speed is a scalar quantity.
Uniform velocity: steady speed in a straight line.
Acceleration:
Definition: Rate of change of velocity. Change of velocity in unit time.
Formula:
$$a=\frac{\Delta v}{t}=\frac{v-u}{t}$$
$a$: acceleration (m/s$^2$)
$\Delta v$: change in velocity (m/s)
$v$: final velocity (m/s)
$u$: initial velocity (m/s)
$t$: time taken (s)
Acceleration is positive if velocity increases, negative if velocity decreases (deceleration/retardation).
Graphs of Motion:
Distance-Time Graphs:
Gradient = Speed.
Horizontal line = stationary.
Straight sloping line = constant speed.
Curved line = changing speed (accelerating: upward curve of increasing gradient; decelerating: upward curve of decreasing gradient).
Speed-Time Graphs:
Gradient = Acceleration.
Area under graph = Distance travelled.
Horizontal line = constant speed (zero acceleration).
Straight sloping line = constant acceleration/deceleration.
Equations for constant acceleration:
$$v=u+at$$
$$s=\frac{(u+v)}{2} \cdot t$$
Freefall & Terminal Velocity:
Freefall: Motion under gravity only (negligible air resistance). Objects accelerate uniformly due to weight.
Acceleration of Free Fall ($g$): $\approx 9.8 \text{ m/s}^2$ on Earth's surface.
Air Resistance (Drag): Frictional force opposing motion through air/fluids. Increases with speed. Has a greater effect on light bodies.
Terminal Velocity: Constant speed reached when the force of air resistance equals the weight of the falling object, resulting in zero resultant force and zero acceleration. Depends on object's size, shape, and weight.
1.3 Mass and weight
Mass:
Definition: Measure of the quantity of matter in an object at rest relative to the observer. Property of an object that resists change in motion.
Scalar quantity.
Unit: Kilogram (kg). (1 g = 0.001 kg)
Weight:
Definition: Gravitational force acting on an object that has mass.
Vector quantity (acts downwards).
Unit: Newton (N).
Formula:
$$W=mg$$
$W$: weight (N)
$m$: mass (kg)
$g$: gravitational field strength (N/kg or m/s$^2$)
Weight varies with location due to differences in gravitational field strength.
Gravitational Field Strength ($g$): Force per unit mass. On Earth's surface, $g = 9.8 \text{ N/kg}$ or $9.8 \text{ m/s}^2$.
1.4 Density
Definition: Mass per unit volume.
Formula:
$$\rho=\frac{m}{V}$$
$\rho$: density (kg/m$^3$ or g/cm$^3$)
$m$: mass (kg or g)
$V$: volume (m$^3$ or cm$^3$)
Calculation Methods:
Regular Shape: Measure mass (m) using a balance, and volume (V) by direct measurement of dimensions.
Irregular Shape: Measure mass (m) using a balance. Measure volume (V) using displacement methods (e.g., measuring cylinder, displacement can).
Liquid: Measure mass of empty container, then with liquid. Subtract to find liquid mass. Divide by known volume.
Floating & Sinking: Objects float or sink based on their density relative to the liquid's density. A higher-density object sinks in a lower-density liquid and vice versa.
1.5 Forces
Force:
Definition: A push or a pull that acts on an object due to interaction with another object.
Effects: Can change an object's speed, direction, shape, or size. Examples include thrust, gravitational attraction, compression.
Unit: Newton (N).
Resultant Force: The single force that has the same effect as all the individual forces acting on an object. Found by vector addition. A resultant force changes the velocity of an object.
Newton's Laws of Motion:
First Law (Law of Inertia): An object remains at rest or continues to move at a constant speed in a straight line unless acted upon by a resultant force.
Second Law of Motion: The acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass.
$$F=ma$$
$F$: resultant force (N)
$m$: mass (kg)
$a$: acceleration (m/s$^2$)
Also defined as force equals the rate of change of momentum ($F=\frac{\Delta p}{\Delta t}$).
Third Law of Motion: For every action, there is an equal and opposite reaction. Forces always occur in pairs.
Hooke's Law (Extension in Springs): The extension of a spring is directly proportional to the stretching force up to the limit of proportionality.
$$F=kx$$
$F$: force (N)
$k$: spring constant (N/m)
$x$: extension (m)
Load-Extension Graphs: Show relationship between applied force and extension. Non-linear graphs beyond the limit of proportionality indicate permanent deformation.
Friction:
Definition: The force that opposes the motion of one surface over another.
Friction converts kinetic energy into thermal energy.
Circular Motion (Centripetal Force): In a circular motion, an object moves in a curved path due to a force directed towards the centre of the circle. This force is called centripetal force.
Despite constant speed, circular motion involves acceleration because velocity direction changes continuously.
Factors affecting centripetal force: Speed (v), Radius (r), Mass (m). Increasing v or m, or decreasing r increases centripetal force.
Moments:
Definition: The turning effect of a force around a pivot point.
Formula:
$$\text{Moment} =F \times d$$
$F$: force (N)
$d$: perpendicular distance from the pivot to the line of action of the force (m)
Unit: Newton-metre (Nm).
Principle of Moments: For an object to be in equilibrium (balanced), the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point.
Conditions for Equilibrium: Sum of all forces equals zero (static equilibrium), and sum of all moments equals zero (rotational equilibrium).
Centre of Gravity:
Definition: The single point where the entire weight of an object appears to act.
Stability: An object is stable if its centre of gravity remains over its base of support. Increasing base area and lowering centre of gravity improves stability.
Types of Equilibrium: Stable, Unstable, Neutral.
1.6 Momentum
Momentum ($p$):
Definition: The product of an object's mass ($m$) and its velocity ($v$).
Formula:
$$p=mv$$
Vector quantity: has both magnitude and direction.
Unit: kilogram metre per second (kg m/s) or newton second (Ns).
Impulse ($J$):
Definition: The change in momentum ($\Delta p$) of an object when a force acts on it over a period of time ($\Delta t$).
Formula:
$$J=F\Delta t=\Delta p$$
Impulse is a vector quantity, same direction as the force.
Principle of Conservation of Momentum: The total momentum of a closed system of objects remains constant if no external forces act on it. Momentum is conserved in collisions (elastic and inelastic) and explosions. $p_{\text{initial}} = p_{\text{final}}$.
Energy Transfers: Mechanical Working, Electrical Working, Waves (Electromagnetic and Sound), Heating.
Kinetic Energy (KE):
Definition: Energy possessed by an object due to its motion.
Formula:
$$KE=\frac{1}{2}mv^2$$
$m$: mass (kg)
$v$: speed (m/s)
Potential Energy (PE) / Gravitational Potential Energy (GPE):
Definition: Energy an object has due to its position or condition.
Formula:
$$GPE=mgh$$
$m$: mass (kg)
$g$: gravitational field strength (N/kg or m/s$^2$)
$h$: height (m)
Principle of Conservation of Energy: Energy cannot be created or destroyed, only transformed from one form to another. The total energy in a closed system remains constant.
Work ($W$):
Definition: Energy transferred when a force ($F$) displaces a body through a distance ($d$) in the direction of the force.
Greater area over which a force acts results in less pressure.
Liquid Pressure:
Pressure in a liquid increases with depth because of the greater weight of liquid above.
Pressure at one depth acts equally in all directions.
Pressure depends on the density of the liquid.
Formula:
$$\Delta p=\rho g\Delta h$$
$\Delta p$: change in pressure (Pa)
$\rho$: density of the liquid (kg/m$^3$)
$g$: gravitational field strength (N/kg or m/s$^2$)
$\Delta h$: change in depth (m)
Unit 2: Thermal Physics
2.1 Kinetic particle model of matter
States of Matter:
Matter consists of tiny particles (molecules, atoms).
Solids: Definite shape and volume, particles close together in fixed positions, vibrate, strong forces, incompressible.
Liquids: Definite volume, take shape of container, particles further apart and can slide, weaker forces, incompressible.
Gases: No definite shape or volume, particles move much further apart and freely, negligible forces, highly compressible.
Brownian Motion: Random and erratic motion of particles in fluids (liquids and gases) caused by collisions with smaller, faster-moving particles of the fluid. Provides evidence for the kinetic model of matter.
Pressure and Kinetic Energy (Gases):
Gas particles move randomly at high speeds.
Collisions with container walls change momentum, creating force and pressure.
Increasing temperature increases collision frequency and energy, thus increasing pressure (at constant volume).
Reducing volume increases particle concentration, leading to more collisions and higher pressure (at constant temperature).
2.2 Thermal properties and temperature
Temperature and Kinetic Energy:
Particles in solids vibrate in fixed positions. Heating increases vibrations and average kinetic energy.
Cooling reduces vibrations until absolute zero (0 K or $-273^\circ\text{C}$), where all motion stops.
Absolute Zero and Kelvin Temperature Scale:
Absolute zero: Lowest possible temperature.
Kelvin scale temperatures are derived by adding 273 to Celsius temperatures:
$$T(K) = \theta(^\circ\text{C}) + 273$$
In Kelvin scale, temperatures are always positive and directly proportional to the average kinetic energy of particles.
Boyle's Law: For a fixed mass of gas at constant temperature, the product of pressure and volume is constant.
$$p_1V_1 = p_2V_2$$
Graphing pressure ($p$) against the reciprocal of volume ($\frac{1}{V}$) gives a straight line.
Thermal Expansion:
Solids and Liquids: When heated, particles vibrate more, pushing apart slightly, causing expansion.
Gases: Heating increases particle speed and collisions with container walls, causing container expansion to maintain pressure.
Applications:
Bimetallic Strips: Made from metals with different expansion rates (e.g., copper and iron). Used in fire alarms and thermostats.
Shrink-fitting: Cooling components contracts them, fitting tightly into other parts upon warming.
Lid Removal: Expanding metal lids with hot water loosens them from glass jars.
Precautions: Expansion joints are spaces left between rail tracks and pipes to allow for thermal expansion without damage.
Internal Energy and Heating: Internal energy increases when an object is heated.
Specific Heat Capacity ($c$):
Definition: The energy required per unit mass per unit temperature increase.
Formula:
$$\Delta E = mc\Delta\theta$$
$\Delta E$: heat energy (J)
$m$: mass (kg)
$c$: specific heat capacity (J/(kg$^\circ\text{C}$))
$\Delta\theta$: temperature change ($^\circ\text{C}$)
Materials with higher specific heat capacities require more heat energy per unit mass for the same temperature change.
Heat Transfer and Equilibrium: Heat transfers from higher to lower temperature bodies until thermal equilibrium is reached.
Change of State:
Heating can change a solid to a liquid (melting) and a liquid to a solid (freezing). Pure substances melt and freeze at specific temperatures.
Melting: Solid to liquid. Particles overcome intermolecular forces.
Solidification (Freezing): Liquid to solid. Particles lose potential energy to surroundings.
Vaporisation: Liquid to gas (vapour). Requires substantial energy to overcome intermolecular forces.
Condensation: Gas to liquid. Gas particles lose potential energy.
Evaporation:
Occurs at any temperature below the boiling point, only at the surface.
Higher temperatures, larger surface areas, and wind/draughts increase the rate.
Energy is transferred from the liquid to the surroundings, cooling the liquid.
Differences between Boiling and Evaporation:
Feature
Boiling
Evaporation
Temperature
Occurs at a specific boiling temperature.
Occurs at any temperature below boiling point.
Process
Bubbles of vapour form within the liquid.
Occurs at the surface of the liquid.
Energy Req.
Requires sufficient heat to reach boiling point.
Requires less heat and occurs due to energetic particles escaping.
Speed
Rapid.
Slower.
Throughout Liquid?
Happens throughout the entire volume.
Happens only at the liquid's surface.
2.3 Transfer of thermal energy
Conduction:
Definition: Heat transfer through matter from hot to cold without moving matter.
Metals conduct heat well (e.g., copper, aluminum) due to fast-moving free electrons and lattice vibrations.
Insulators (wood, plastic) are poor conductors; non-metals transfer heat through slower atomic/molecular vibrations.
Liquids and gases conduct heat slowly because particles are further apart.
Convection:
Definition: Heat transfer method in fluids (liquids and gases) by movement of the matter itself.
Convection Currents: Warm fluids expand, become less dense, and rise. Cooler, denser fluids sink and replace the rising warm fluid, creating a current.
Radiation:
Definition: Method of thermal energy transfer which occurs without matter, even in a vacuum. Emits as electromagnetic waves, travels at speed of light.
Absorption and Reflection: Black surfaces absorb more radiation than shiny white ones. Shiny white surfaces are poor absorbers/emitters and good reflectors.
Emission: Dull black surfaces emit more radiation than shiny surfaces when hot. All bodies emit radiation above absolute zero.
Greenhouse Effect: Greenhouse gases trap heat similar to glass in a greenhouse, crucial for climate stability. Increased carbon dioxide and methane absorb more infrared which cannot escape.
Unit 3: Light, Sound & The Electromagnetic Spectrum
3.1 General properties of waves
Waves:
Definition: Disturbances that transfer energy from one place to another without transferring matter.
Particles in the medium oscillate or vibrate about a fixed point.
Two Types of Progressive Waves:
Transverse Waves:
Definition: Oscillations of particles are perpendicular to the direction of energy transfer.
Examples: Light waves, all electromagnetic waves, waves on a string, water waves.
Features: Crests (peaks) and Troughs (valleys).
Longitudinal Waves:
Definition: Oscillations of particles are parallel to the direction of energy transfer.
Examples: Sound waves.
Features: Compressions (regions of high pressure/density) and Rarefactions (regions of low pressure/density).
Wave Characteristics:
Wavelength ($\lambda$): Distance between 2 successive crests/troughs (or compressions/rarefactions).
Frequency ($f$): Number of complete waves created per second, measured in Hertz (Hz).
Wave speed ($v$): Distance moved by a crest or any point on the wave in 1 second.
Amplitude ($a$): Height of a crest or depth of a trough from the undisturbed or mean position.
Phase: Particles in ‘phase’ have the same speed and direction of vibration.
Wave Equation: Relates wave speed, frequency, and wavelength.
$$v=f \times \lambda$$
Faster vibration produces a shorter wavelength; higher frequency results in a smaller wavelength.
Wavefronts and Rays:
Wavefront: A straight line where the wave has the same phase at all points.
Ray: Line drawn at right angles to a wavefront showing the direction of travel.
Wave Behaviour:
Reflection: Bouncing back of a wave when it hits a boundary.
Law of Reflection: Angle of incidence ($i$) = Angle of reflection ($r$).
Wavelength and speed remain unchanged.
Refraction: Bending of a wave as it passes from one medium to another, due to a change in speed and wavelength.
When waves move from less dense to more dense regions, they bend towards the normal (light waves slow down due to smaller wavelengths).
When waves move from dense to less dense regions, they bend away from normal (light waves speed up due to bigger wavelengths).
Diffraction: Spreading out of waves as they pass through a gap or around an obstacle, creating circular wavefronts.
Most noticeable when the gap size is similar to the wavelength.
Longer wavelengths diffract more.
Ripple Tank: Used to demonstrate wave phenomena (reflection, refraction, diffraction) with water waves.
3.2 Light
Light Rays and Beams: Light travels in a path called a ray. A beam is a stream of light shown by several rays (can be parallel, diverging, or converging).
Speed of Light: Approximately $3 \times 10^8 \text{ m/s}$ in a vacuum (or 300,000 km/s).
Reflection of Light against a Plane Mirror:
The normal is perpendicular to the mirror at the point where the incident ray strikes.
The angle of incidence ($i$) is between the incident ray and the normal.
The angle of reflection ($r$) is between the reflected ray and the normal.
Law of Reflection: Angle of incidence ($i$) = Angle of reflection ($r$).
Real and Virtual Images:
Real Image: Formed when light rays actually converge at a point. Can be produced on a screen. (e.g., image formed by a converging lens when object is beyond focal point).
Virtual Image: Formed when light rays appear to diverge from a point. Cannot be formed on a screen. (e.g., image in a plane mirror, image formed by a diverging lens).
Refractive Index ($n$): A measure of how much a medium slows down light.
Definition: Ratio of light speed in air (or vacuum) to light speed in the medium.
Formula:
$$n=\frac{\text{speed of light in air}}{\text{speed of light in medium}}=\frac{c}{v}$$
Snell's Law:
$$n=\frac{\sin i}{\sin r}$$
(where $i$ is angle in air, $r$ is angle in medium).
Higher refractive index means greater bending of light as it slows down more.
Critical Angle ($c$) and Total Internal Reflection (TIR):
When light passes from an optically denser to an optically less dense medium:
Critical Angle ($c$): The angle of incidence in the denser medium for which the angle of refraction in the less dense medium is $90^\circ$.
Formula:
$$\sin c = \frac{1}{n}$$
Total Internal Reflection (TIR): Occurs when the angle of incidence in the denser medium is greater than the critical angle ($c$), causing all light to reflect internally.
Converging (Convex) Lens: Thickest in the center, bends light inwards to a real principal focus (F).
Diverging (Concave) Lens: Thinnest in the center, spreads light out, light appears to diverge from a virtual principal focus.
Optical Center (C): Center of a lens.
Principal Axis: Line through C at right angles to the lens.
Focal Length ($f$): Distance from C to F.
Ray Diagrams (for thin lenses):
A ray parallel to the principal axis is refracted through the principal focus (F).
A ray through the optical center (C) is undeviated (not refracted).
A ray through the principal focus (F) is refracted parallel to the principal axis.
The intersection of rays after refraction gives the location of the image.
Magnification ($M$):
$$M = \frac{\text{image size}}{\text{object size}} = \frac{\text{distance of image from lens}}{\text{distance of object from lens}}$$
Image Properties at different object positions:
Object Position
Image Position
Image Nature
Image Size
At 2F
At 2F
Real, inverted
Same size
Between 2F and F
Beyond 2F
Real, inverted
Larger
At F
At infinity
Real, inverted
Infinitely large
Between F and lens
On same side of lens
Virtual, upright
Larger
Applications of Lenses in Vision Correction:
Short-Sightedness (Myopia): Eye focuses light in front of retina. Corrected with a diverging (concave) lens.
Long-Sightedness (Hypermetropia): Eye focuses light behind retina. Corrected with a converging (convex) lens.
Dispersion of Light:
Refraction by a Prism: A prism bends light due to refraction at each surface, causing a combined change in direction (deviation).
Dispersion: The splitting of white light into its constituent colours (spectrum) when it passes through a prism. Occurs because the refractive index of glass varies with the wavelength of light.
Visible Spectrum: ROYGBIV (Red, Orange, Yellow, Green, Blue, Indigo, Violet). Red light (longest wavelength, lowest frequency) is refracted least; Violet light (shortest wavelength, highest frequency) is refracted most.
Monochromatic Light: Light of a single frequency (and thus a single colour).
3.3 Electromagnetic spectrum
Electromagnetic (EM) Waves:
Nature: Transverse waves.
Speed: All EM waves travel at the speed of light in a vacuum ($3.0 \times 10^8 \text{ m/s}$).
Medium: Do not require a medium to travel (can travel through a vacuum).
Energy: Energy of EM waves increases with frequency and decreases with wavelength.
Follow the wave equation $v=f\lambda$.
EM Spectrum Order (from longest wavelength/lowest frequency/lowest energy to shortest wavelength/highest frequency/highest energy):
Radio Waves
Microwaves
Infrared (IR)
Visible Light (ROYGBIV)
Ultraviolet (UV)
X-rays
Gamma Rays
Uses of EM Waves:
Radio waves: Broadcasting, communication (radio, TV), astronomy, RFID.
X-rays: Medical imaging (bone fractures), security screening, industrial inspection.
Gamma rays: Cancer detection and treatment, sterilization, material inspection.
Dangers of EM Waves:
High-frequency EM waves (UV, X-rays, Gamma rays) are ionizing radiation, meaning they can cause damage to living cells and DNA, leading to mutations and cancer. High-intensity infrared can cause burns and eye damage.
Communication Systems:
Analogue Signals: Continuous, varying amplitude and frequency. Limited by bandwidth and signal degradation.
Digital Signals: Discrete, binary (0s and 1s).
Advantages of Digital: Higher transmission rates, can be regenerated without loss of quality (minimal noise), increased range, error checking possible.
Infrared Optical Fibers: Use total internal reflection to transmit infrared/light for long-distance data transmission, offering high bandwidth and low signal loss.
3.4 Sound
Sound Waves:
Nature: Longitudinal waves.
Medium: Require a medium (solid, liquid, or gas) to travel; cannot travel through a vacuum.
Production: Produced by vibrating sources.
Compressions and Rarefactions: Regions of high (densely packed molecules) and low (less densely packed molecules) pressure, respectively.
Speed of Sound:
Varies depending on the medium: fastest in solids ($\approx 5100 \text{ m/s}$ in steel), slower in liquids ($\approx 1500 \text{ m/s}$ in water), slowest in gases ($\approx 330-350 \text{ m/s}$ in air at room temperature).
Affected by temperature (faster in warmer air).
Measurement Methods:
Echo Method:
$$v=\frac{2d}{t}$$
(sound travels to wall and back).
Direct Method:
$$v=\frac{d}{t}$$
(using two microphones).
Echoes: Reflected sound waves. Used to measure distances.
Frequency and Wavelength:
Frequency ($f$) of a sound wave is the number of complete wave cycles per second (Hz). Higher frequencies mean higher-pitch sounds.
Wavelength ($\lambda$) is the distance between two consecutive compressions or rarefactions.
The speed of sound ($v$) is $v=f\lambda$.
Limits of Hearing: Humans can only hear sound frequencies ranging from about 20 Hz (low pitch) to 20,000 Hz (high pitch). The upper limit decreases with age.
Musical Notes:
Pitch: Determined by the frequency of the sound wave.
Loudness: Determined by the amplitude of vibrations.
Quality (timbre): Unique shape or texture, caused by the instrument's construction.
Ultrasound:
Definition: Sound waves with frequencies above the upper limit of human hearing ($>20,000 \text{ Hz}$).
Uses: Medical imaging (e.g., prenatal scans), industrial applications for precision and non-destructive testing (e.g., checking for cracks in steel bolts), echo sounding.
Unit 6: Space Physics
6.1 Earth and the Solar System
Motion of the Earth:
Rotates on its axis once every 24 hours (causes day and night). Axis is tilted at $\approx 23.5^\circ$.
Orbits the Sun once every 365 days (causes seasons due to tilt).
Rising and Setting of the Sun: Earth's rotation causes the Sun to appear to move east to west daily.
Motion of the Moon:
Satellite of Earth, orbiting approximately every month. Average distance $\approx 400,000 \text{ km}$.
Revolves on its axis, always showing the same side to Earth.
Reflects sunlight, has no atmosphere, weaker gravitational field (one-sixth of Earth).
Phases of the Moon: Moon's appearance changes during its monthly orbit (New Moon, Crescent, First Quarter, Full Moon, Waning phases, Last Quarter, Old Crescent).
Eclipses:
Solar Eclipse: Moon is between Sun and Earth (Moon blocks Sun).
Lunar Eclipse: Earth is between Sun and Moon (Earth blocks Sun's light from Moon).
Orbital speed:
Formula:
$$v=\frac{2\pi r}{T}$$
$v$: average orbital speed (m/s)
$r$: average radius of the orbit (m)
$T$: orbital period (time for one orbit) (s)
The Solar System:
Consists of the Sun (a star), eight planets, dwarf planets, moons, asteroids, comets, and natural satellites.
Inner Planets (Terrestrial): Mercury, Venus, Earth, Mars. Small, similar size, solid and rocky with layered structures, high density. Formed close to the Sun where it was too hot for gases to condense.
Outer Planets (Gas/Ice Giants): Jupiter, Saturn, Uranus, Neptune. Much larger and colder, mainly consist of gases, low density, many moons and rings. Formed in cooler regions where gases could condense.
Asteroids: Pieces of rock of various sizes, mostly between Mars and Jupiter. Orbit the Sun.
Comets: Dust embedded in ice (water and methane). Orbit the Sun in highly elliptical paths. Develop a bright long tail when approaching the Sun due to radiation pressure.
Elliptical Orbits:
Planets, dwarf planets, and comets orbit the Sun in an ellipse, with the Sun at one focus. Comets have highly elliptical orbits.
Conservation of Energy: In an elliptical orbit, total energy (KE + GPE) is conserved.
When closer to the Sun: GPE decreases, KE increases, so speed increases.
When further from the Sun: GPE increases, KE decreases, so speed decreases.
Origin of the Solar System (Nebula Theory): Formed $\approx 4.5$ billion years ago from gravitational attraction pulling together clouds of hydrogen gas and dust (nebulae). Planets formed from the disc of matter left over from the nebula that formed the Sun.
Light-Year:
Definition: The distance light travels in a vacuum in one year.
Light from the Sun takes approximately 8.33 minutes to reach Earth.
Gravitational Field Strength ($g$):
Definition: The force of gravity acting per unit mass.
Formula:
$$g=\frac{W}{m}$$
$g$: gravitational field strength (N/kg or m/s$^2$)
$W$: weight (N)
$m$: mass (kg)
Varies with mass and radius of the celestial body.
6.2 Stars and the Universe
The Sun as a Star:
A medium-sized star, primarily composed of hydrogen and helium.
Emits energy in the infrared, visible, and ultraviolet regions of the electromagnetic spectrum.
Source of Energy: Energy from nuclear fusion in its core (hydrogen nuclei fuse to form helium, releasing vast amounts of energy).
Scale of the Universe: Vast distances measured in light-years.
Galaxies: Large collections of stars, gas, and dust (e.g., Milky Way, $\approx 100,000$ light-years in diameter, $\approx 800$ billion stars).
Origin and Life Cycle of Stars:
Formation:
Nebula: Interstellar clouds of dust and gas collapse under gravitational attraction.
Protostar: Mass increases and core temperature rises.
Main Sequence Star: Hydrogen fuses into helium when the core is hot enough. Stable phase where forces of gravity inward balance with thermal pressure outward. Lasts up to 10 billion years for Sun-like stars.
Life Cycle (depending on initial mass):
Low Mass Stars (e.g., Sun): Nebula $\rightarrow$ Protostar $\rightarrow$ Main Sequence Star $\rightarrow$ Red Giant (hydrogen depletes, core collapses, outer layers expand and cool; helium fuses into carbon) $\rightarrow$ White Dwarf (core collapses after helium used, outer layers expelled as planetary nebula) $\rightarrow$ Black Dwarf (white dwarf cools).
High Mass Stars (more than 8 times the Sun’s mass): Nebula $\rightarrow$ Protostar $\rightarrow$ Main Sequence Star (shorter stable phase) $\rightarrow$ Red Supergiant (core collapses after helium fusion; fusion of carbon into heavier elements until iron forms) $\rightarrow$ Supernova explosion (releases energy and heavy elements) $\rightarrow$ Neutron Star (dense core, may act as a pulsar) OR Black Hole (extremely dense core with gravity so strong light cannot escape).
Galactic Redshift:
Definition: The observed increase in the wavelength of light from distant galaxies, shifting it towards the red end of the spectrum.
Cause: The galaxies are moving away from us (Doppler effect).
Evidence for Expanding Universe: The greater the redshift, the faster the galaxy is moving away, and the further away it is.
Hubble's Law: Relates the recession velocity of a galaxy to its distance from Earth.
$$v=H_0d$$
$v$: recession velocity (km/s)
$H_0$: Hubble constant (km/s/Mpc), measures the rate of the Universe's expansion.
$d$: distance (Mpc)
The gradient of a velocity-distance graph gives the Hubble constant.
The Big Bang Theory:
Core Idea: The Universe originated from an extremely hot and dense state around 14 billion years ago and has been expanding and cooling ever since.
Evidence:
Redshift of distant galaxies: Indicates the Universe is expanding.
Cosmic Microwave Background (CMB) Radiation: Faint radiation detected from all directions in space, considered the "afterglow" of the Big Bang.
Age of the Universe: Estimated to be approximately 14 billion years, derived from Hubble's Law.