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Units and Prefixes
Every measurement or quantity has a unit, such as meters for distance or seconds for time. When
dealing with very large or very small numbers, we use prefixes.
Prefixes generally increase or decrease in factors of 1,000, except for centimeters and
decimeters.
To convert units, determine whether you need a larger number multiply or a smaller number
divide by the conversion factor.
Forces and Vectors
A force is any push or pull. Forces can be contact forces objectstouching or non-contact forces
magnetism,electrostaticforces,gravity.
Contact forces: Normal contact force, friction, air resistance, tension.
Even contact forces are due to the electrostatic repulsion between electrons.
We represent forces with vectors, which are arrows showing direction and magnitude.
Magnitude: The size of the force, indicated by the length of the arrow.
When two forces act on an object, there's a resultant force.
Found by adding the vectors.
If forces are in opposite directions, one is negative.
If vectors are at right angles, use Pythagoras' theorem to find the resultant.
Trigonometry SOHCAHTOA can be used to find angles, often using the tan function.
If forces are balanced adduptozero, the object won't accelerate, and its velocity remains
constant canbezero. This is Newton's First Law of Motion.
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Newton's First Law of Motion: An object at rest stays at rest and an object in motion
stays in motion with the same speed and in the same direction unless acted upon by
an unbalanced force.
Scalars vs. Vectors
A measurement or quantity with only magnitude nodirection. Examples: Speed,
Scalars distance.
A measurement or quantity with both magnitude and direction. Examples:
Vectors
Displacement distancewithdirection, velocity speedvector.
Weight is the force due to gravity on an object.
Calculated by multiplying mass inkg by gravitational field strength g, which is
9.8 N/kg oftenroundedto10N/kgonEarth.
Weight = mass × g
When holding an object, you exert an upward force equal to its weight to keep it balanced.
Lifting an object at a constant speed requires a force equal to its weight.
Work done is the energy transferred by a force.
Work : done = force × distance : moved
When lifting an object, the force is its weight, and the distance is the height.
The gain in energy equals mass × g × height, which is the equation for gravitational
potential energy GPE.
GPE = m × g × h
Speed, Velocity, and Acceleration
Speed and velocity are measured in m/s.
Velocity includes direction positive,negative,up,down,left,right.
Speed/Velocity =
Distance/TDimiseplacement
On a distance-time graph, the gradient gives the speed or velocity. Draw a tangent for curves.
A speed or velocity-time graph provides more information:
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The gradient gives acceleration changeinspeeddividedbytime.
Acceleration =
ChangTei:mine:Speed
The unit of acceleration is m/s².
Negative gradient indicates deceleration slowingdown.
In a velocity-time graph of an object thrown upwards, the velocity starts positive, decreases to
zero at the top, then becomes negative as it falls.
The acceleration is constant and negative 9.8m/s²duetogravity.
The area under a velocity-time graph gives the distance traveled.
Area under 0 m/s counts as negative displacement.
Equations of Motion SUVAT
SUVAT equations are used to predict an object's motion under constant acceleration.
s = displacement
u = initial velocity 0ifstartingatrest v = final
velocity 0ifdeceleratingtoastandstill a =
acceleration 9.8m/s²orgforfallingobjects t = time
To solve problems:
1. List the variables.
2. Put a question mark next to what you're trying to find.
3. Note the values of the other three variables.
4. Choose the equation with those four variables.
5. Rearrange if needed. 6. Plug in the numbers.
Newton's Laws of Motion
Newton's First Law: When there's no resultant force, an object's motion is constant
nochangeinvelocity.
Inertia: The tendency of an object's motion to stay constant unless acted on by a resultant
force.
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Newton's Second Law: When there is an unbalanced force aresultantforce, it equals mass
times acceleration.
F = ma
Only one of these laws can be true in any situation: either there's no resultant force, or there
is.
Newton's Second Law can be demonstrated using a trolley on a track pulled by weights hanging
over a pulley.
Use light gates to measure acceleration.
Change the weight on the string, ensuring that mass taken off the hanger is added to the
trolley.
Draw a graph of force against acceleration, which should be a straight line through the
origin.
Newton's Third Law
Newton's Third Law states that for every action force, there is an equal and opposite reaction force.
This law is always true but is often confused with balanced forces.
The key to understanding Newton's Third Law lies in perspective. The first two laws primarily consider
the object itself. For example, the force pulling downwards on a ball is its weight, even with air
resistance resulting in a net downward force.
However, zooming out to include the Earth in the system reveals that while the Earth pulls down on
the ball, the ball also pulls up on the Earth with an equal force. The Earth's massive size renders this
effect negligible, but the principle holds true.
Another example involves two ice skaters. If the male skater pushes the female skater, there's an equal
and opposite reaction force pushing back on him. This explains why both skaters move away from
their initial positions.
Stopping Distance
The overall stopping distance for a car comprises two components:
Thinking Distance: The distance traveled during the driver's reaction time upon seeing a hazard
e.g.,abunny.
Braking Distance: The distance traveled after the brakes are applied.
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If you double your speed, you double your thinking distance because you travel twice as far
during your reaction time.
However, doubling your speed quadruples your braking distance. The reason is that the car needs to
dissipate all of its kinetic energy KE, which is given by the formula:
KE = mv2
If you double the velocity (v), the kinetic energy increases by a factor of 4 since 22 = 4. Tripling the
speed increases the kinetic energy andthusbrakingdistance by a factor of 9 $32 = 9$.
Factors Affecting Stopping Distance
Momentum
The faster you go, the more momentum you have.
Momentum is a measure of how hard it is to stop an object.
It's calculated using the formula:
momentum = mass ⋅ velocity
The unit for momentum is kilogram meters per second kgm/s.
Momentum is a vector, meaning it has both magnitude and direction. Negative momentum indicates
negative velocity.
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Conservation of Momentum
In a collision, kinetic energy isn't always conserved, but total momentum always is. This means the
total momentum before the collision equals the total momentum after the collision.
Calculations can be tricky, so pay attention to signs. The total momentum before the collision can be
calculated using the formula:
m1u1 + m2u2
where m1 and m2 are the masses of the objects, and u1 and u2 are their initial velocities. If an object is
not moving initially, its initial velocity is zero, and its initial momentum is also zero.
After the collision, the total momentum is:
m1v1 + m2v2
where v1 and v2 are the final velocities of the objects.
If the objects couple together, the final momentum is:
mv
where m is the total mass of the combined objects and v is their combined velocity.
In cases like a cannon firing, the total momentum before is zero, meaning the total momentum after
must also be zero. The cannonball's forward momentum is counteracted by the cannon's recoil in the
opposite direction.
Force and Momentum
Newton's second law says that F = ma. However, we also know that acceleration $a$ is the change in
velocity $Δv$ over time $t$, so a = Δtv .
Therefore, force can also be expressed as:
F = Δmomtentum
This equation means that force is equal to the rate of change of momentum.
The shorter the time taken for momentum to change, the greater the force. This principle underlies
the use of safety features in cars:
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Seat belts Airbags
Crumple zones
These features extend the time over which momentum changes during a collision, reducing the force
experienced by the occupants and increasing the likelihood of survival.
Energy
Energy is not a tangible substance but rather a concept representing the capacity to cause
interactions and changes within a system.
Total energy in any interaction is always conserved. Energy cannot be created or destroyed, although it
can be converted into mass thisismainlyimportantfornuclearreactions.
Energy Stores TypesofEnergy
Energy exists in different stores, which change when objects interact. Here are the energy stores as
well as formulas for calculating the amount of energy in each store:
Kinetic Energy KE: Energy of motion.
KE = mv2 halftimesmassinkilogramstimesspeedsquared.
Gravitational Potential Energy GPE: Energy due to an object's height above a reference point.
GPE
=
mgh
masstimesgravitationalfieldstrengthtimesheight.
g
gravitationalfieldstrength is 10 or 9.8 N/kg. This value will be provided. Technically,
this equation gives the change in GPE, as h is the change in height.
Elastic Potential Energy EPE: Energy stored in a spring or elastic material when stretched or
compressed.
EPE = 12 ke2 halftimesspringconstanttimesextensionsquared. k is
the spring constant in N/m. e is the extension in meters
theamountthespringhasstretchedfromitsoriginallength.
Thermal Energy: Energy associated with the temperature of an object.
ΔE = mcΔT masstimesspecificheatcapacitytimeschangeintemperature.
c is the specific heat capacity SHC, which indicates how much energy is needed to
raise 1 kg of a substance by 1°C.
An increase in thermal energy results in particles moving faster, essentially
increasing their kinetic energy.
Chemical Potential Energy: Energy stored in chemical bonds e.g.,infoodorfuels.
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Energy Transfer
In a closed system, no energy is lost to or gained from the surroundings. This allows us to equate
different forms of energy.
For example, a roller coaster car at the top of a ride has gravitational potential energy GPE and
virtually no kinetic energy KE. As it descends, GPE is converted into KE. At the bottom, it has lost GPE
and gained KE.
Therefore, we can say:
GPElost = KEgained
If we know the GPE at the top, we know the KE at the bottom. We can then use the KE equation to
find the car's speed.
Rearranging Equations
To solve for speed $v$ in the KE equation, we rearrange as follows:
1. KE
2. 2KE = mv2 multiplybothsidesby2
3. 2KmE = v2 (divide both sides by m)
4. v = √ 2KmE takethesquarerootofbothsides
Energy Transformations
Kinetic and Potential Energy
To find the velocity $v$ using kinetic $KE$ and potential energy $PE$, you can use the formula:
v = √2 ∗ KE/m
where m is the mass.
Elastic potential energy can be equated to kinetic energy when a toy car is released from a spring. In
scenarios involving gravitational potential energy GPE, if you equate the equations, mass cancels out,
simplifying to:
v = √2gh
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where g is the acceleration due to gravity and h is the height. This shortcut allows you to find the
speed $v$ at the bottom of a fall knowing only the height.
Remember to rearrange the GPE equation by moving the terms from the right-hand side to the bottom
of the left-hand side, multiplying them together in brackets.
Closed vs. Open Systems
In a roller coaster example, if the gravitational potential energy at the top is greater than the kinetic
energy at the bottom, the remaining energy is lost to the surroundings, indicating an open system.
This loss can occur due to:
Work done against air resistance Friction
Work is another term for energy used.
Waves
Energy Transfer
Waves transfer energy without transferring matter. Oscillations or vibrations are passed along.
Longitudinal Waves
Longitudinal waves are waves in which the direction of the oscillations is parallel to the
direction of energy transfer.
Examples:
Sound waves
Seismic P-waves primarywaves
In longitudinal waves:
Compressions: Particles bunch up
Refractions: Particles spread out
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Transverse Waves
Transverse waves are waves in which the direction of oscillations is perpendicular to the
direction of energy transfer.
Examples:
Waves on the surface of water
Seismic S-waves secondarywaves
Light
All other electromagnetic waves
S-waves are slower than P-waves and produce earthquake aftershocks.
Wave Representation
A waveform represents any wave, including longitudinal waves.
Displacement: How far particles have oscillated from their original position y − axis X-axis: Can
represent either distance or time
Wave Properties
Amplitude: The peak of a wave, representing the maximum displacement from equilibrium.
Wavelength $λ$: The length of one complete wave ifthex − axisisdistance, measured in meters.
Time Period $T$: The time it takes for one complete wave to pass ifthex −
axisistime, measured in seconds.
Frequency $f$: How many waves pass a point every second, measured in Hertz.
Frequency and time period are reciprocals:
f = 1/T
Wave Equation
The wave equation is:
v = fλ
Where:
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v = wave speed f
= frequency λ =
wavelength
Measuring Wave Speed
Ripple Tank: Measure frequency, measure the distance between 10 peaks then divide by 10 to
get the wavelength, and use the wave equation to find speed.
Direct Timing: Time how long it takes for a ripple to travel the length of the tray 10 times, then
use speed = totaldistance/time.
Sound Waves: Use a microphone attached to an oscilloscope to measure the time it takes for a
sound to echo off a wall. Then, use speed = totaldistance/time.
Sound Waves and Hearing
Sound waves cause the eardrum to vibrate, converting vibrations into signals that travel to the brain.
Humans can hear frequencies between 20 Hz and 20 kHz. Frequencies above this range are called
ultrasound.
Transmission and Reflection
When sound reaches a boundary between two mediums, some is transmitted, and some is reflected.
Ultrasound is used to scan bodies by timing how long it takes for the waves to return off different
layers. Sonar uses sound waves in water to create images underwater.
Seismic Waves and Earth's Structure
Longitudinal P-waves can travel through liquids, but transverse S-waves cannot. The absence of
aftershocks on the opposite side of the Earth from an earthquake suggests a liquid center moltencore.
Reflection Types
Specular Reflection: Reflection off a smooth surface e.g.,amirror, where the angle of incidence
equals the angle of reflection.
Diffuse Reflection: Scattering of light off a rough surface.
All angles are measured from the normal, a line perpendicular to the surface.
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Electromagnetic Waves
EM Wave Basics
Electromagnetic EM waves do not need a medium to travel through and can travel through the
vacuum of space.
EM Spectrum
The EM spectrum includes inorderofincreasingfrequencyanddecreasingwavelength:
Radio waves
Microwaves
Infrared radiation
Visible light
Ultraviolet
X-rays
Gamma rays
EM waves are produced when electrons lose energy, emitting the energy as an EM wave. Gamma rays
are emitted by nuclei.
Energy and Absorption
The higher the frequency of an EM wave, the more energy it carries. EM waves are absorbed by
electrons. UV, X-rays, and gamma rays can cause ionization, which can be dangerous if absorbed by
DNA and cells.
Applications
All parts of the EM spectrum are used for various applications, including communications, cooking,
heating, imaging, and medical treatments.
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Refraction
Light and Mediums
When light waves move from one medium to another e.g.,airtoglass, they change speed. They slow
down, and the wavelength decreases.
Refraction
A change in medium results in a change in direction, called refraction, if the light is at an angle to the
normal. If light slows down, it moves closer to the normal, meaning the angle of refraction is smaller
than the angle of incidence.
Critical Angle and Total Internal Reflection
As the angle of incidence increases, the angle of refraction also increases. When the angle of
refraction reaches 90°, the angle of incidence is at the critical angle. If the angle of incidence exceeds
the critical angle, total internal reflection occurs, and all light is reflected back inside the medium.
Fiber Optics
Fiber optics work using total internal reflection. Light is sent down a thin glass fiber, and due to the
large angle of incidence, it is totally internally reflected, bouncing along the fiber without needing
mirrors.
Lenses
Lenses are curved blocks of glass that utilize refraction to manipulate light rays, either converging
meeting or diverging spreadingout them.
Convex Lenses
A convex lens converges light rays. Its symbol is a double-headed arrow pointing outwards at both
tips.
Rays entering parallel to the principal axis converge at the principal focus.
The focal length is the distance from the lens center to the principal focus. It remains constant
for a given lens.
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When an object is placed near a convex lens:
1. A ray goes straight through the center of the lens.
2. Another ray travels parallel into the lens, then through the principal focus.
The image forms where these rays meet, allowing projection onto a screen, retina, or camera sensor.
The image is:
Diminished smallerthantheobject
Inverted upsidedown
Object Very Close to the Lens
When an object is very close, rays don't meet. Extrapolating them back behind the lens reveals where
they meet. The image is:
Magnified larger
Upright
Virtual cannotbeprojected
This is how a magnifying glass works. Your eye interprets the diverging light to focus on the retina,
resulting in a magnified virtual image.
Concave Lenses
Concave lenses diverge light rays and always produce virtual images.
A ray parallel to the principal axis goes back through the other principal focus behind the lens.
The image forms where this ray meets another ray.
The image is:
Diminished
Upright
Magnification
Magnification: The ratio of image height to object height.
Magnification > 1: Image is bigger than the object.
Magnification < 1: Image is smaller than the object diminished.
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Color Perception
Color is perceived based on the wavelengths of light emitted or reflected by an object and absorbed
by cells in our retina.
Most objects absorb some wavelengths and reflect others.
Chlorophyll absorbs red wavelengths, reflecting green wavelengths hence,greenleaves.
A blue ball reflects blue wavelengths.
When shined with only red light, a blue ball appears black, as red light is absorbed.
Radiation
Radiation: Any particle or wave emitted by something.
The electromagnetic spectrum comprises all radiation emitted by electrons exceptgammaradiation.
Gamma Radiation
Emitted by the nucleus of an atom with excess energy.
High-energy electromagnetic waves.
Dangerous due to their ability to ionize atoms knockingoffelectrons, potentially causing cell
damage and cancer.
Alpha Radiation
Emitted when heavy nuclei decay e.g.,Americium − 241.
Consists of an alpha particle: two protons and two neutrons.
The nuclear decay equation represents this process:
$^{241}{95}Am \rightarrow ^{237}{93}Np + ^4_2He$
Americium Am decays into Neptunium Np and a Helium nucleus alphaparticle.
Beta Radiation
Emitted when lighter nuclei decay e.g.,Carbon − 14.
A neutron in the nucleus turns into a proton and an electron. The fast-moving electron is
ejected as beta radiation.
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The mass of an electron is negligible.
The nuclear decay equation represents this process:
$^{14}_6C \rightarrow ^{14}7N + ^0{-1}e$
Carbon C decays into Nitrogen N and an electron betaparticle.
Ionizing Ability and Penetrating Power
easily stopped.
Beta particles are less ionizing but more penetrating than alpha particles.
Gamma radiation is weakly ionizing but highly penetrating.
Background Radiation
Always present from natural and man-made sources:
Radon gas from concrete and rocks
Cosmic rays from space
Nuclear weapons testing
Corrected Count
To obtain an accurate radiation count, subtract the background count from the count with the source.
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Uses of Radiation
Alpha radiation: Smoke detectors
Beta radiation: Measuring thickness of thin materials
Gamma radiation: Radiotherapy, sterilizing medical equipment
Neutron Emission
Nuclei can emit neutrons under specific circumstances e.g.,nuclearfission.
Alpha, beta, and gamma radiation ionize atoms but don't make them radioactive. Neutrons can
make other atoms radioactive, requiring careful disposal of materials exposed to neutron
bombardment e.g.,concretearoundanuclearreactor.
Radioactivity and Half-Life
Radioactivity Activity: The rate of decay of a radioactive source.
Measured using a GM tube: radiation count divided by time in seconds. Units: counts
per second cps or Becquerel Bq.
As unstable nuclei decay, activity decreases over time.
Half-life: The time it takes for the activity ornumberofunstablenuclei,ormass of a
radioactive isotope to halve.
Half-Life Calculations
Half-life is the time it takes for half of a radioactive sample to decay. Half-lives can range from days to
millions of years.
Half-life: The time required for half of the atoms in a radioactive sample to decay.
If we graph the activity of a radioactive substance over time, we get a decay curve. To find the half-life
from the graph:
1. Find the initial activity.
2. Halve it.
3. Draw a line from the halved value to the curve.
4. Drop a line down to the time axis to see how long it took.
The time it takes to halve the activity again will be the same, regardless of the initial amount or when
timing starts.
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Example Calculation: A sample starts at 96 Bq activity and falls to 12 Bq after 1 year 12months.
1. How many half-lives occurred? Count how many times you halve 96 to reach 12:
1 half-life: 96 Bq → 48 Bq
2 half-lives: 48 Bq → 24 Bq
3 half-lives: 24 Bq → 12 Bq
2. It took three half-lives to decrease to 12 Bq.
3. If 12 months is three half-lives, then one half-life is 12 months / 3 = 4 months.
Nuclear Fission
Nuclear fission occurs when a neutron is fired at a nucleus like uranium-235. The neutron is absorbed,
making the nucleus unstable, causing it to split into two smaller nuclei.
Nuclear Fission: The splitting of a heavy nucleus into two lighter nuclei, accompanied by the
release of energy.
The total mass of the products is less than the initial mass. This missing mass is converted into energy
$E = mc2$. This energy is released as thermal or kinetic energy. The fission also releases up to three
more neutrons, leading to a chain reaction.
Chain Reaction: Neutrons released from one fission event trigger more fission events.
Uncontrolled Chain Reaction: Leads to a rapid, uncontrolled release of energy
atomicbomb.
Controlled Chain Reaction: Used in nuclear reactors to produce a consistent and safe
amount of energy to generate electricity.
Nuclear Fusion
Nuclear fusion is the process by which two light nuclei combine to form a heavier nucleus, releasing
energy.
Nuclear Fusion: The process in which two light nuclei combine to form a heavier nucleus,
releasing a large amount of energy.
An example is what happens in the Sun, where hydrogen nuclei fuse to form helium, releasing energy.
This requires a lot of kinetic energy to begin with. Scientists are trying to create fusion reactors but
haven't yet managed to harness enough energy to make them viable.
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Solar System and Stars
Solar System: Consists of the Sun, eight planets, asteroid belt betweenMarsandJupiter, and
dwarf planets likePluto.
Natural Satellites: Moons orbiting planets.
Galaxy: Our solar system is located in the Milky Way galaxy.
Star Formation and Life Cycle
Stars are formed from dust and gas particles in a nebula attracted to each other by gravity. The cloud
becomes hotter and denser until fusion starts.
Main Sequence: A star remains stable as long as the outward pressure from fusion balances the
inward force of gravity.
Red Giant/Super Red Giant: When a star dies, the outward pressure increases, causing it to
expand.
White Dwarf/Black Dwarf: A red giant collapses after all fuel for fusion runs out, leaving a white
dwarf, which then cools into a black dwarf.
Supernova: A super red giant explodes, leaving a dense neutron star or black hole. The outer
layers form new nebulae.
Satellites
Natural Satellites: Like the Moon, orbit the Earth naturally.
Artificial Satellites: Launched by humans e.g.,byElonMusk.
Geostationary Satellites: Orbit in a circle above the equator, used for GPS and communication.
They move at a constant speed, but their velocity is constantly changing because their direction is
changing. This means they are accelerating towards the Earth due to a centripetal force.
Centripetal Force: Any force that results in circular motion, always acting towards the center of
the orbit.
Elliptical Orbits: Satellites in elliptical orbits are used for reconnaissance and weather
monitoring. They move faster when closer to the Earth and slower when farther away.
Red Shift and the Big Bang Theory
When we look at distant stars and galaxies, the wavelengths of light appear longer, shifted towards the
red end of the spectrum (redshift). This indicates that galaxies are moving away from us. More distant
galaxies are more red-shifted, suggesting they are receding faster.
Big Bang Theory: The observation that galaxies are moving away from each other implies that
they originated from a single point in space in the past.
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Cosmic Microwave Background Radiation CMBR: Microwave radiation detected from all
directions, thought to be emitted as a result of matter cooling down after the Big Bang. It is
evidence for the Big Bang and the expansion of the universe.
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