Section 1 - Mechanics

The effect of a force on an objects velocity is dependent on its massThe resistance to a change in velocity is called inertia

Momentum

The momentum of an object is defined as its mass x velocity The units of momentum are Kgms-1

Newton’s laws of motion

Newton’s first law – Every object continues at rest with uniform velocity unless acted upon by a resultant force.

So an object moving through space will travel at a constant velocity without slowing down since there are no other forces such as friction and air resistance acting upon it.

Newton’s second law – The rate of change of momentum on an object is proportional to the resultant force acting upon it.

Rate of change of momentum for an object initially at rest is given by the change in momentum over time:

F= mv Since V = acceleration, F=ma T T

Newton’s third law – When two object interact, they exert equal and opposite forces upon each other

Impulse

Where forces only act briefly, an impulse is exerted by the force.

Impulse = Force x time. Its units are therefore Ns

Since Newton’s second law states that force = change in momentum/time, we can say that impulse must equal change in momentum.

Impulse = mv - mu

Conservation of momentum

For a system of interacting objects, provided that no external force is acting, the total momentum before any collisions is equal to the total momentum after any collisions. In an explosion, momentum is conserved. However since the momentum before an explosion is 0, the momentum after the explosion must also equal 0.

The reason for this is that momentum is a vector quantity and so since the objects are moving in different directions, they each have equal and opposite momentum and so when added together, they cancel out.

Deriving an equation for kinetic energy from momentum

Kinetic energy = ½ mv2

Momentum = mv

Therefore kinetic energy can also be expressed as follows:

K.E = (mv) 2 2m = p 2 2m

This formula allows us to calculate the kinetic energy of a particle providing that we know its momentum and its mass.

Momentum of particles in two dimensions

In order to solve momentum equations in two dimensions it is important to remember that momentum is always conserved in each dimension. So for example the total dimension in the x direction before a collision will equal the total momentum in the x direction after a collision.

There are two balls A and B of masses 10kg and 5 Kg respectively. Ball A has a horizontal velocity of 3ms-1

and collides with ball B which is initally stationary. After the collision, ball a moves away with a velocity of 2ms-1 at 300 to the horizontal. Calculate the velocity of B after the collision.

1. First we need to calculate the initial momentum in the x direction. Since B is not moving, Px = 3 x 10 = 30Ns. Therefore we know that the combined momentum of both balls after the collision in the x direction will equal 30.

2. Secondly we need to work out what the momenum is in the y direction. Seeing as ball a is moving horizontally with know vertcle component and ball b is stationary we can say that the momentum is 0. (Py = 0) and so the verticle momentum will also equal 0 after the collision.

3. Next we need to resolve the verticle and horizontal components of A’s momentum after the collion using trigonemtry. To do this we find its horizontal and verticle velocity and multiply by its mass. This gives as a horizontal momentum of 20cos30 Ns and a verticle momentum of 20sin30 Ns.

Momentum2

4. We have previously calculated the total momentum in the x direction will equal 30. Therefore:

20cos30 + 5x (momentum of B) = 30 5x = 30 – 20cos30 5x = 12.679 x = 2. 536 ms-1

5. In the y direction we know that: 5y = 20sin30

y = 2 ms-1

6. Using pythagorous’ theorum we can can find the magnitude of the X and Y components as well as its direction using trigonemtry

Elastic and inelastic collision

An elastic collision is a collision where no energy is lost during impact. These collisions do not happen in reality as some energy is often converted into other forms such as heat for example.Most collisions are inelastic, meaning energy is lost during impact.

Collisions can also be partially inelastic and totally inelastic.A partially inelastic collision could be one where the colliding objects move apart after impact.Whereas in a totally inelastic collision, the objects would couple together after colliding (this type of collision causes a greater energy loss)

Rotational dynamics

The time period of an object in uniform circular motion is the time taken for that object to complete one rotation. To complete one rotation, an object must move through 2π radians. Since the circumference of a circle is equal to 2πr. The length of any arc is given by:Arc length = Rθ (Where θ is the angle of the arc)The unit for angular velocity is rads-1

The frequency of a rotation is the number of rotations per second.F= 1/T (where T is the time period)

The angular velocity is given by 2π/T so the speed of an object in circular motion must equal the angular velocity multiplied by the radius. Angular velocity is normally denoted by ω.V= 2πR or V= ωR T

An object in uniform circular motion travels at a constant speed, however since its direction is constantly changing, its velocity is constantly changing, so it must be accelerating towards the centre of the circle. What causes the acceleration is a centripetal force.

Centripetal acceleration is given by:a = -v 2 R

Since V is equal to ωR, acceleration is also given by: A = - ω2R

Applying equations of circular motions to horizontal and vertical cicrcles

Horizontal

The diagram shows the forces at work on a revolving swing ride. The vertical components are in equilibrium. Therefore Tcosθ = mgHorizontally it is not in equilibrium. The resultant force acts toward the centre and is equal to Tsin θ

Tsin θ = mv 2 and Tcos θ = mg. RTsin θ = mv 2 /r = tan θ = V 2 Tcosθ mg gr

Vertical

For a car travelling over a hill, the support force S = mg – mv 2 R Therefore as velocity in crease mv2/r increases until S = 0. This is the critical velocity at which there is no support force.

For a car travelling through the bottom of a hill the support force is given by:S = mg + mv 2 R

For a person on the inside of a rollercoaster, the normal reaction force is given by:

R = mv 2 - mg R

For a car travelling around a banked track, V2 = gr tan θ

Section 2 – Electric, Magnetic and gravitational Fields

Summary

Gravitational fields

Any object of mass will have a force acting on it along the lines of force

The force is only attractive It acts over an infinite distance Cannot be blocked

Magnetic fields

Can be either attractive or repulsive Field lines point out of the north pole and

into the south pole

Electric fields

Can be either attractive or negative Lines of force go from positive to negative

Electric fields

A region where a charged body would experience a force The field is caused by charges

Electric fields are represented by diagrams which follow these rules:

Field lines show direction of the positive charge Spacing of lines shows the strength

Point source radial field

For a positive point charge For a negative point chargeElectric field strength for a point charge

To calculate the strength of an electric field we use the formula

E = F Q

E = field strength (NC-1)F= Force (N)Q = Charge (C)

Uniform electric field

Field lines go from positive to negativeField lines are equally spaced out

Field strength for a uniform electric field

To calculate the strength of a uniform field we use

E = V D

E = field strength (NC-1)V = potential difference (V)D = Distance between charged plates (m)

For both uniform and electric fields, the base units for the strength of the field are Kgms-3A-1

Equipotentials in a radial field

Potential energy of a particle is determined by its distance from the charged plate/pointEquipotentials are lines where the potential energy is the same throughoutIf a particle moves along an equipotential no work is done on the particle since its potential energy is the sameEquipotentials are closer towards the centre of a radial field

There is a 1V increase at each equipotential as you move into the centre of the fieldCoulombs law

The size of the force between two charged particles depends on the distance they are apart, their charges, and a constant K equal to 9x109 NM2C2

The formula used to calculate the size of the force between the two particles is:

F = KqQ K = 1 = 8.99 x 109 NM2C2

r2 4πεo

εo = is the permittivity of a vacuum = 8.85 x 10 -12 FM-1

If the force is positive there is repulsion, if the force is negative there is attraction

Coulombs law can also be used to find the potential or strength near a point charge.Consider a point charge q near a much bigger point charge Q. The force on the smaller charge is given by

F = qQ 4πεor2

Capacitors

Capacitance – The ability of a component to store chargeAny capacitor consists of two conductors isolated from eachotherTo calculate the capacitance of two parallel plates we use the formula:

Capacitance = charge displaced (C) (F) p.d across plates (V)

1 farad = 1 coulomb per volt F=CV-1

Graphs for capacitance

Charging

Discharging

Energy stored in a capacitor

When a capacitor is charged, it stores energy.

When a capacitor is discharged it releases energy.

At any stage in the charging process, the p.d across a capacitor is given by:

V=Q/C

Charge is proportional to P.D for a fixed capacitor

Energy stored = area under the graph = ½ QV

Since Q = VC and V=Q We can derive the C following formula:

W = 1/2 Q 2 and W = ½ V2 C C

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Capacitor Decay

The charge in a capacitor undergoes exponential decay. Meaning that the time it takes for the charge to reach a fraction of its original value is constant.

The formula for exponential decay of charge in a capacitor is given by:

Q = Qo e –t/RC

Q/Qo = Charge/original chargee = mathematical constant (2.718)t = timeR= resistanceC=capacitance

T = Rc Where T is a time constantTo use this information to determine what the value is for the time constant, consider the value of Q = Qo e –t/RC where t is equal to RC. This would mean that the power to which e is raised to would be -1. We can then rewrite the expression as follows Q = Qo e –1. From this we can deduce that the time taken for the original charge (Qo) to reach 37% (e-1) is equal to RC.

Since Q = Qo e –T/RC and Q=VC we can derive other expressions such as:

VC = VoC e –T/RC

V = Vo e –T/RC

I = Io e –T/RC

Magnets + Magnetic Fields

A magnetic field is a region where a particle of charge or with magnetic properties will experience a force.

There are 2 types of magnetic; permanent and electromagnet

Lines of force go from north to south

The closer the lines of flux, the strong the force

Force lines are called “lines of flux”

The strength of a magnetic field is called the flux density (B) and is measured in tesla (T) (vector quantity)

Magnetic flux density is greatest when a given area is perpendicular to the lines of flux. In other words when the angle between the lines of flux and the area is given by sin 90

Flux is measured in weber (Wb) and is given by ф = B x A (flux density x area)

Magnetic force effect of an electric current

Whenever a current flows, a circular magnetic field is produced around the wireA force is produced whenever a current flows with a magnetic force around it in the presence of a permanent magnetic

The force F on the wire depends on the lenth of the wire (L) and the current (I). The stronger the magnetic field, the larger the flux density (B) and therefore the large the force acting on the wire.

Flux density can be defined as the force per meter of wire per unit of current. F = BIL

If the field is at an angle, F = BILsin θ

Electricity from magnetism

When a conductor moves through a magnetic field, the electrons within it experience a force and therefore move producing a current. This is called an induced e.m.f

Faradays law – The induced e.m.f in a circuit is equal to the rate of change of flux

Lenz’s law – The direction of the induced emf is such that it opposes the change of flux. This explains why energy does not come from nowhere since work is done moving through the lines of flux

e.m.f = -d(n ф)/dt

D is the change in a propertyN is the number of turns in a solenoidФ is the fluxT is the time for the change

Section 3 – Particle Physics

The nucleus

Consists of protons and neutrons (nucleons)Makes up approximately 1/10000th of the atomThe size of the nucleus is determined by its mass number. (number of nucleons)The number of neutrons can vary depending on the isotopeProtons/Neutrons are made up of smaller particles called quarksQuarks as well as electrons are fundamental particles

Rutherford alpha scattering

A beam of alpha particles were fired at thin metal foils Alpha particles were scattered and detected using a small zinc sulphide screen Each alpha particle that hit the screen produced a beam of light The foil scattered the particles at all possible angles to the incidence beam At each angle, the number of alpha particles hitting the screen was measured Most of the alpha particles passed straight through

What this tells us about the structure of an atom

All of the atoms positive charge is concentrated in the centre Most of the atom’s mass is concentrated in the nucleus The electrons surround the nucleus at very large distances

Deep inelastic scattering

Deep inelastic scattering provides evidence for the existence of quarks Electrons are accelerated to

high energies and allowed to interact with protons

High energy electrons have wavelengths smaller that the size of a proton

The high energies disrupt the proton creating new particles called hadrons

It is inelastic because the target has been changed in the process

Jets of new particles are produced

Electrons as waves

Electrons can act as both particles and wavesDe broglie’s wavelength can relate the wavelength of a particle to its momentum

λ = h/p Where h is Planck’s constant (6.63x10-34Js), λ is the wavelength and P is momentum

Electron Gun

Metal filament is heated up Electrons are excited to high energy levels and leave the surface of the metal

(thermionic emission) Electrons can be affected by positively charged plates which will attract them A force is exerted on the electrons causing them to accelerate The electrons will travel through a series of tubes called a linear accelerator Once the electron has travelled through the tube, the charge of the tube behind

the electron becomes negative and the next tube in front becomes positive Since the electrons increase in velocity the length of the tube must increase so

that the time spent in each tube can remain constant A radio frequency controls the alternation of the P.D so that the tubes can

change from positive to negative at the correct times

Cyclotrons and synchrotrons

Linacs are very good at producing tightly collimated beams of electrons. However to excited electrons to such energy levels they must be several kilometres long.

The next step to producing high energy electron or particle beams is to make them move in a circle. This is done using magnetism and Fleming’s left hand law for motion

To calculate the force exerted on a charge particle moving through and electric field we use

F = BQV

Where F = force, B = magnetic field strength, Q = charge of particle and V = Velocity

Cyclotron

In a cyclotron, charged particles, such as protons, are produced at the centre

of the instrument. These particles move in a circular path because of the

confining magnetic field above and below. The particles are alternately pushed and pulled by the alternating

electric current thereby acquiring more and more energy. As they move faster, they spiral outward. After about one

hundred orbits they emerge from the instrument with great energy.

The motion of charges particles in a magnetic field is given by r=p/BQ

Detectors

Detectors fall into three main categories

Ionisation to produce a current Ionisation to produce condensation Excitation to produce a photon (scintillation)

Detector 1 – include spark counters, GM tubes, drift tubes and semi-conductor diodesDetector 2 – Cloud chambers, bubble chambersDetector 3 – Scintillation counters, photomultipliers

Cloud chambers

Used for charged particlesParticles passing through are detected by leaving trails where they have ionised and caused alcohol in the super saturated air to condense.

Bubble chambers

Particles cause tracks where they have ionised other particles to form sites where bubbles can form. Magnetic and electric fields are used to cause deflections in the particles that are detected so that it is possible to deduce their charge, and their momentum (which is given by p = BQr)

Producing new particles

Particles can decay during the high energy interactions that take place in particle acceleratorsThe energy from these collisions can be used to make new particlesMatter and antimatter collisions cause annihilation to occur. In other words, the particles are completely destroyed and are turned into energy according to the equations, E=mc2

Energy Units

The energy of particles is normally measured in electron-voltsAn electron volt is the energy released when an electron of charge 1.6 x 10-19As passes through a potential difference of 1V, thus the energy released is 1.6x10-19J according to the equation V=J/Q

So an electron volt = 1.6x10-19J

More non – si units

The relative atomic mass of a nucleon is 1/12 of that of an atom of Carbon 12

1u = 1.66x10-27

The mass of particle can also be measured using the unit eV/C2 which is a derivative of Einstein’s formula E=mc2

Particle families

Family 1 - lepton

Not subject to the string forceThey are fermions (have half integer spin_They are fundamental (can’t be broken down)They have integer charge

If a particle is a lepton, it has the lepton number, 1, if it is an anti-lepton, its lepton number is -1. If it is not a lepton, its lepton number is 0

Family 2 – Hadrons

Subject to the strong forceAre not fundamentalSome are fermions,Others are bosons (have integer spin)

Baryons (includes protons + neutrons) MesonsFermions BosonsHeavier than 1 nucleon Heavier than a electron, lighter than a

nucleonComposed of 3 quarks Composed of 2 quarks one regular the

other anti Can decay if not already a proton, into a proton (if that makes sense?)

Can decay into a lepton, photon, pions

Family 3 – Quarks

Quarks are fundamental particle and have half integer spin (fermions)

Quarks ChargeUp (u) +2/3 eDown (d) -1/3 eCharm (c) +2/3 eStrange (s) -1/3 eBottom (b) +2/3 eTop (t) -1/3 e

The up, charm and top quarks are heavier than their counter parts

Baryons – Proton (UUD) / Neutron (UDD) Mesons – π- - (anti-U D) / π+ (anti-D U) / πo (Anti-U U)