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Tue Mar 27, 2012 3:32 am
PiesAreSquared says...



Ok, so this is what I wrote out from my textbook (which I still am doing). I am trying to condense the book of 900+ pages into maybe less than thirty pages (For a quick review...). So far this is the first ten chapters. Have I missed out on anything? (Other than the units in which they are measured, that is) (So after awhile I got bored of bolding all of the stuff....)
Anyway, the units are:
Dynamics, Circular Motion, Simple Harmonic Motion, Forces in Equilibrium, Forces in fluid, Further topics in Mechanics and Fluids, Energy sources, Elasticity, Molecular forces, Solid Materials, The Gravitational Field, Electrostatics. The electric Field, Capacitors, Current electricity)
Level: A-levels.

Part 1: Mechanics and Solid Materials:
Conservation of Linear Motion:
If no external forces act on a system of colliding objects, the total momentum of the objects in a given direction before collision =total momentum in same direction after collision.
Newton’s Laws of motion:
First law:
The velocity of a body remains constant unless the body is acted upon by an external force.
Second law:
The acceleration a of a body is parallel and directly proportional to the net force F and inversely proportional to the mass m, i.e.,F = ma.
Third law:
The mutual forces of action and reaction between two bodies are equal, opposite and collinear.
Types of collisions:
Elastic-Collision where total K.E. is conserved
Inelastic-Collision where total K.E. is not conserved
Conditions for Equilibrium: 1. Concurrent Equilibrium the sum of vector forces through a point is zero.
2. Coplanar equilibrium, the sum of forces in a plane is zero and the sum of the torques around the axis of the plane is zero.
Archimedes’ Principle: Volume of object=volume of water displaced, Upthrust on object immersed in fluid=weight of fluid displaced by object
Classifications of solids: Crystalline--ordered structure, Amorphous--irregular structure, Glassy-disordered structure, Polymer-organic irregular structure.
Solid Structures: In crystalline solids such as metals, the atoms are grouped in a lattice structure with many planes rich in atoms. Crystals are imperfect. The most important defects are dislocations.
In elastic deformation, when Hooke’s law is obeyed and the atoms undergo small displacements, the energy stored is fully recovered when the load is removed.
[b/]Plastic deformation[/b] is due to the movement of dislocations. The energy in plastic deformation is converted to heat. Slip occurs due to movement of dislocations. Atomics bonds are broken one at a time. Due to this, tensile strength of a metal is much lower than the value calculated from the interatomic forces.
[b/]Stiffness[/b]=Young Modulus. Tensile Strength=Breaking point stress
Ductile materials show a large amount of plastic deformation. Brittle materials fracture at low strains close to their elastic limit.
In work hardening, moving dislocations are pinned or entangled and the metal may fracture as a brittle material. Annealing, heating to a suitable high temperature, restores the crystalline state and ductility.
Cracks near the surface produces high concentration at the tip. Rapid growth leads to brittle behavior. Toughness=ability to resist crack growth. Hardness=resistance to plastic deformation.
Composite materials improve mechanical properties. Composite reinforced materials prevent spreading, are strong in relation to their weight and may be stiffer than steel.
Polymers, widely used in plastics are made chemically with monomers. They consist of very long chains of carbon atoms bonded to other atoms. It can be linear, branched or cross-linked.
Thermosetting polymers have cross-links between chains of molecules. They are rigid and cannot be re-moulded by heating. Thermoplastic polymers have no cross-links. They become soft on reheating and can be re-moulded.
Rubber Molecules are coiled. Under increased tension the molecules uncoil and high strain (800%) can be produced. On removing the load a hysteresis loop is obtained. The “host” energy is converted to heat and is proportional to the area of the loop. Resilient materials have low hysteresis.
Metals and glass have high Young’s Modulus and low elastic strain, plastics have lower Young's Modulus, large elongation, and do not obey Hooke’s law like metals do.
Hooke’s Law: The extension is proportional to the force or tension in a wire if the proportional limits are not exceeded.
Kepler's Laws:
The orbit of every planet is an ellipse with the Sun at one of the two foci.
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.


Velocity: (Dynamics) s/t , v=u+at, v2=u2 + 2as, Gradient of (s-t) graph, terminal velocity = maximum velocity of an object, at such a velocity, resultant force=0,
(Circular Motion, Simple harmonic Motion) v=rw, v=w (r2-y2)^(½), Max. v=wr,
(Rotational Dynamics, Fluid in Motion) w=w0+at, w=v/t, v2=grE2/R, w=angle/t, angle=wt, w=2pi/t =2pi*frequency

Acceleration (vector): (Dynamics) v/t, Gradient of (v-t) graph,
(Circular Motion, Simple harmonic Motion) acceleration towards center=v2/r = r2w2/r = rw2 , a=v2/r, a=-w2y (y is the displacement. y=r sin wt ), Max. a = -w2r (where r is the amplitude),

Distance: (Dynamics) s=ut+ ½ at2 , Range=(u2 sin 2 angle)/g , Area between (v-t) graph and time-axis

Angles: (Circular Motion, Simple harmonic Motion) T sin angle=mv2/r , T cos=mg, tan angle=v2/rg, 2pi = 360o

Frequency: (Circular Motion, Simple harmonic Motion) f=1/t =w/2pi, f=½(Iw2)/n1,

Time: (Dynamics) (2u sin angle)/g, Force * time = mv-mu=momentum change,
(Circular Motion, Simple harmonic Motion) Time to describe a circle= 2pi/w=2pi(e/g)^(½),
(The Gravitational Field) t2 directly proportional to r3, T2=4pi2R3/grE2, (R is distance from center of the earth)

Components: (Dynamics) A resolved P of a force F=F cos angle, F sin (90-angle)

Momentum: (Dynamics) Mass * velocity, momentum change=mv-mu, Total momentum is always conserved,
(Rotational Dynamics, Fluid in Motion) Total angular momentum of body=Iw

Moment: (Forces in Equilibrium and Fluids) F*shortest s from axis, Torque=one of a couple of force*shortest s, Torque = Force applied x lever,

(Rotational Dynamics, Fluid in Motion)Torque=Moment of inertia(I)*angle=F.r, T*angle=work done=k.e. change, T*t=angular momentum change=Iw1-Iw2, Moment of inertia=Mgh=½Mr2w2+½Iw2(1+n/n1),


Energy: (Dynamics) Total kinetic energy is shared inversely as the ratio of the mass of the exploding stationary original mass, Change in gravitational potential energy = mgh, Total energy in a closed system is always constant, k.e. = ½ * m * speed2 ,
(Forces in Equilibrium and Fluids) Max. k.e. at center (½*m*r2*w2) For springs,
(Rotational Dynamics, Fluid in Motion) Rotational k.e.=½Iw2, Total rotational k.e.=½Mv2+½Iw2 (M=mass),
(Elasticity, Molecular forces, Solid Materials) energy W=½EAe2/l, Energy per unit volume=½ stress*strain (If Hooke's law is obeyed), energy in stretched wire=area between F against e graph and axis.
Potential energy=a/rp-b/rq (p and q are powers of r, a and b are constants)
(The Gravitational Field) Normal k.e. =½mv2 =GmM/2r0, P.e.=½kx2, (in orbit= -GmM/r0),

Force: (Dynamics) F=ma= -mg sin angle (use to find if m is constant), Force moving an object along incline = force due to engine - frictional force - component of weight down incline, Frictional force =kv, v is velocity F= (mv-mu)/t, F=m/t * (mv-mu) (use if mass varies), F=mv2/r,
(Elasticity, Molecular forces, Solid Materials) F=EAa*angle, (a=linear expansivity of magnitude), F=EAe/l, Average separation=3/(22.4x10-3)/(6x1023)=33x10-10,
(The Gravitational Field) Force of attraction=GmM/r2, (M and m are masses, G is the gravitational constant)

Work: (Dynamics) Force * distance moved in direction of force, F cos angle * s,
(Rotational Dynamics, Fluid in Motion)Torque*angle of rotation,
(Elasticity, Molecular forces, Solid Materials) Average force*extension=½Fe

Power: (Dynamics)Work done/time taken, Power of engine=F * v

Tension: Resolved through (horizontal) (mv2/r) (Vertically) (mg)

Pressure: (Forces in Equilibrium and Fluids) Force/area, p=hpg, p+½pv2+pgh=constant,

Weight: (Dynamics) Weight is mass times gravity, i.e. W=mg,
(The Gravitational Field) g=GM/rE2, (M refers to mass of earth and r to radius of earth), g’=GM/r2, M=grE2/G, gravitational potential=-GM/rE,

Simple harmonic Motion: (Circular Motion, Simple harmonic Motion) Motion of a particle whose acceleration is always (i) directed towards a fixed point (ii) directly proportional to its distance from that point. a=-w2x (x is distance from center of oscillation)

Tensile stress, strain, Young's Modulus: (Elasticity, Molecular forces, Solid Materials) Stress=pressure=force per unit area= F/A, Strain=extension per unit length=e/l, Modulus=(F/A)/(e/l)=F/e*l/A=E, Field strength potential gradient numerically.



Part 2: Electricity:
Magnetic Poles: Like charges repel. Unlike charges attract.
Surface density increases with the curvature of the body.
Electric Potential is a scalar.
Potential is the work done per coulomb in bringing a positive charge from infinity to a point.
Capacitors: All capacitors consist of two metal plates separated by an insulator.
When two capacitors are connected, the p.d. V across both capacitors is the same after connection. The total charge before connection=total charge after connection.
CR=time constant, C in farads, R is resistance, in ohms.
Heating increases the amplitude of the atom vibrations, causing the temperature of the metal to rise.
Ohm’s Law: Under constant physical conditions, the resistance V/I is a constant independent of V or I and their directions.
Kirchhoff's Laws: The algebraic sum of currents in a network of conductors meeting at a point is zero. The sum of the emfs in any closed loop is equivalent to the sum of the potential drops in that loop.
The e.m.f (E) of a battery or any other generator as the total energy per coulomb it delivers round a circuit joined to it.
Resistance. A a balance, Wheatstone bridge gives ratio relation P/Q=R/X, where X is the fourth resistance in the bridge.
A meter (slide-wire) bridge can measure (a) an unknown resistance X using a known resistance, (b) resistivity, © temperature coefficient of resistance.
Potentiometer. Advantage: At a balance, no current taken from p.d. source. So, unlike using a moving-coil voltmeter, p.d. is undisturbed.
p.d. is proportional to the balance-length on potentiometer wire.

(Misc formulas) L=length=R.A/resistivity,

Force: (Electrostatics. The electric Field) 1/4pi*e0xQQ’/r2, F=EQ’,
(Current Electricity) E=P/I, EI=I2R+I2r, E=a(angle)+b(angle)2, E varies as a parabola with the angle,

Permittivity: (Electrostatics. The electric Field) e0=QQ’/4pi*Fr2, e0 usually = 8.854x10-12. er=e/e0,
(Capacitors) er=Cd/Cv, Cd is the capacitance with a dielectric filling the space, Cv is the capacitance with vacuum. e0=Id/50VA farad meter-1, 50 being the frequency. er=Gsubstance/Cair=I/I0,

Electric Field: (Electrostatics. The electric Field) E= strength of the field, E=F/Q’, E x area=Charge inside sphere/permittivity, E.A=sigmaA/e, therefore, E=sigma/e, E=-dV/dx, E=Q/4pi*e0r2, E=V/d=Potential gradient,
(Capacitors) E=IR+Ir, E=V+Ir, (The r is for internal resistance in these cases), E=e.m.f=electromotive force
(Current Electricity) (Drift Velocity of electrons) I=nAve (v=electrons drift velocity, Av is volume, n=number of electrons per m3, it contains nAv electrons, and a charge nAve, Total potential difference=Sum of individual potential difference), Current density=J=I/A=nev, E=Vdistance, E/E0=L/L0, E=(L/L0)*E0, E=a(temperature)+b(temperature)2, (a and b are denoting different temperatures),

Potential difference: (Electrostatics. The electric Field) VAB=Work per coulomb in moving charge from B to A, 1 Volt=1 Joule per coulomb (1V =1 J/C), V=Q/4pi*e0r,
(Capacitors) V=Q/C, V=IR, V=p.d.=power dissipation,
(Current Electricity) V=IR, V1=IR1=R1/R1+R2xV0, V=E-Ir, VDistance=E=Ir, VAC is directly proportional to the length of the wire, A=end of wire, C=any distance on it.

Power: (Current Electricity) P=W/t=IVAB, P=I2R=V2AB/R, Total electrical power generated=P=W/t=EI, Power output to R is a maximum when R=r, internal resistance,

Energy: (Electrostatics. The electric Field) W=QV, W=½CV2=½Q2/C=½QV
(Current Electricity) W=QVAB, W=IVABt, Total electrical energy liberated=W=QE=IEt

Electrical charges: (Capacitors) I=Charge per second=fQ, Q=CV, Q=Q0e-t/CR, Q=Q0(1-e-t/CR), (Current Electricity) I=V/R, I=E/R+r, E=e.m.f., (R+r) is the total resistance, r=internal resistance, I1=I2+I3,

Capacitance: (Capacitors) C=Q/V, C=e0A/d, C=4pi*e0r, C=4pi*e0ab/b-a, (b-a is the distance, and b and a are the points), C=C1+C2+C3, 1/C=1/C1+1/C2+1/C3, Ratio of Capacitors to currant is: C1/C2=I1/I2,

Resistance:(Current Electricity) R=V/I (Individual potential difference directly proportional to individual resistances), R=VDistance/I=R1+R2+R3, Current is same through all resistors in series, I/VAB=1/R=1/R1+1/R2+1/R3, Potential difference same across each resistor in parallel, R=resistivity*l/A, r=(E-V)/l, r=r1+r2+r3, 1/r=1/r1+1/r1, P/Q=R/X (P, Q R, and X are resistors) X=QR/P, X/R=l1/l2(l=length of wire), R temperatures=R0(1+a* temperatures) (a=temeperature coefficient of resistance), Inner resistance r=(E/V-1)R=(L0/L-1)R,
Resistivity: (Current Electricity) resistivity=Rl/A, Conductivity=1/resistivity,G=I/V G=conductivity*A/l, G=conductance,

Heat: (Current Electricity) H=IVt=I2Rt=V2t/R,
The moment you say that one set of moral ideas can be better than another, you are, in fact, measuring them both by a standard, saying that one of them conforms to that standard more nearly than the other. C. S. Lewis

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