Finnish national matriculation exam in physics:

additions to IB Physics HL

Mechanics

Rotational mechanics

(this section is now also included in the IB Mechanics chapter of the Compendium, although it is not required in the IB)

For rotational motion a set of mechanics formulas similar to those for linear mechanics (objects moving in a straight line) can be developed. Instead of the distance or displacement s we can study the angle turned, or the angular displacement q. In radians we have by definition

q = s/r

where r = the radius of the circle and s = the distance covered along its circumference. In a similar way we can define an angular velocity w = q/t (the angle turned per time; unit: radians per second, rads^-1) and an angular acceleration a = w/t (the change in angular velocity per time; unit rads^-2).

The results on a rotational motion of a force depend on how far from the center of rotation it is applied, so force will be replaced by torque (Sw. kraftmoment eller vridmoment, Fi. vääntömomentti), t = Fr. (the torque is a quantity familiar from the IB programme)

Without proof we will notice that mass also will be replaced by "moment of inertia", symbolised I (in many English books) or J (in Finland) where J = mr² if all the mass is at the same distance r from the center or axis of rotation. If not, then it can be shown that J follows certain formulas like J = 2/5*mr² for a sphere, J = (1/3)ml² for a bar of length l rotating around one end (like a baseball bat) or J = (1/12)ml² for the bar rotating around its center (like a propeller). Time is the same for linear (translational) and rotational motion. Summary:

TRANSLATIONAL => ROTATIONAL

s => q = s/r

v => w = v/r

a => a = a/r

F => t = Fr

m => J = mr² (or other)

Using the "word list" above we can "translate" the known translational formulas into the corresponding rotational ones, for example:

v = u + at => w_final = w_initial + at

s = ut + ½at² => q = w_initialt + ½at²

F = ma => t = Ja

E_k = ½mv² => E_rotational = ½Jw²

p = mv => p_rot = Jw

The rotational or angular momentum p_rot is often symbolised L and is also relevant in Atomic physics in the IB. We may notice that:

L = Jw = mr²(v/r) = mvr

for an electron in a circular orbit around the nucleus of an atom.

Fluid mechanics

The pressure at a depth h in a liquid = hydrostatic pressure p = rgh or

p = p₀ + rgh

where g = 9.81 ms^-2 and r ("rho") = the density = m/V in kgm^-3 of the liquid. We may include the atmospheric pressure p₀ acting on the surface of the liquid.

· Pascal's principle: pressure applied to a fluid is the same everywhere at the same depth in it, and "acts" in all directions (is a scalar quantity).

An application of this is the hydraulic lift (used in car brakes) where force is applied to a liquid (oil) on a large area and then spreads through the liquid where it is allowed to act on a much smaller area attached to the brake mechanism or other. Then with F = P/A => P = FA from the IB programme

P_in = P_out gives F_inA_in = F_outA_out and F_out = F_inA_in/A_out

which means that a small force in causes a larger force out (but to keep the volume of liquid constant, the F_in must move a piston or other a longer distance than F_out; therefore the work done is the same.

Archimede's law: the upwards force buoyancy (lyftkraft, nostovoima) on a submerged object is

F = rVg

= the force of gravity on the mass of the amount of water displaced by the object. r = density of the liquid the body is immersed in, V = its volume, g = 9.81 ms^-2.

Electricity and magnetism

Capacitors

Capacitors (Sw. kondensator, Fi. kondensaattori) are devices where electric charge can be stored on plates or sheets of metal (often wrapped to a small cylinder) with a thin layer or air or insulating material in between. The amount of stored charge is much smaller than in a battery, but the advantage is that it can be taken out of there very quickly, in a small fraction of a second, which makes them useful in alternating current (AC) circuits where the direction of the current may change many times per second.

For the capacitor, the capacitance (kapacitans, kapasitanssi) is

C = Q / U

in the unit 1 farad = 1 F = 1CV^-1, if we use the symbol U instead of V for "voltage" = potential difference and as usual Q for the amount of electric charge stored in the capacitor (the charge is of opposite signs on opposing plates, but there is equally much positive and negative).

If the area of the plate (only one is counted, not both the positively and the negatively charged one) is A and the distance between the foils d then

C = e_re₀A/d

where e₀ = the electric permittivity in vacuum (same as in F = 1/4pe₀)(q₁q₂/r²). e_r = the relative permittivity = a number without unit which gives the correction for the vacuum value. Found in MAOL's tables for various substances.

The energy stored in a capacitor (in the unit J) is

E = ½QU

(which using C =Q/U can be written E = Q²/2C or E = ½CU² ) where the voltage U is in volts and the charge Q in coulombs.

Capacitors in series and parallel

Capacitors can be connected in series or in parallel like resistors. Then we have:

series => same charge, parallel = > same voltage

Compare to resistors:

series => same current, parallel => same voltage

For the total capacitance we have the formulas

series 1/C_tot = 1/C₁ + 1/C₂ + .... parallel C_tot = C₁ + C₂ + ...

(Note that is opposite to the formulas for resistors! There it was

series R_tot = R₁ + R₂ + ..., parallel 1/R_tot = 1/R₁ + 1/R₂ + ...

Self-inductance

Change in flux through a loop or a solenoid gives induced emf. If we connect or disconnect a solenoid to a voltage source, the current I through it will increase or decrease quickly, causing a "self-inductance" which opposes the change. For this:

emf_self = - LDI/Dt

where L = inductance (induktans, induktanssi), unit 1 henry = 1 H = 1 VsA^-1 = 1 Ws. For a solenoid with the cross-section area A, the length l and the number of turns we have when m₀ = the magnetic permeability

L = m₀N²A/l

The energy stored in a coil or solenoid when the current I runs through it is:

E = ½LI²

AC circuits and impedance = "AC resistance"

When a circuit is connected to an AC power source the total resistance is affected not only by resistors but also by capacitors and solenoids. The total

resistance for AC is called

· impedance Z = U/I (compare R = U/I), unit 1 ohm

Capacitors cause

· capacitive reactance X_C = 1/wC where w = 2pf and f = the frequency in Hz.

This "reactance" has the same unit as resistance and impedance and only means the resistance caused by the capacitors in the circuit. (Note: Direct current can not go through a capacitor since the metal foils in it are separated by a layer of insulating material. High frequency AC can, since the charges that cannot go through the capacitor are stored on one of the plates or foils, but soon the current changes direction and they move back through the circuit and are stored on the opposite foil etc.)

·        Solenoids cause inductive reactance X_L = wL

The total impedance is (can be shown with complex number mathematics)

Z = Ö(R² + (X_L - X_C)²)

If there is no capacitor in the circuit Z = Ö(R² + X_L²); inserting C = 0 would in principle give infinitely high X_C and therefore also Z but that would represent a circuit which is broken (there is a place with insulating material that the current would have to go through) but where this break in the circuit is an infinitely bad capacitor, with so small area or high d that C = e_re₀A/d » zero). For an unbroken circuit with no capacitor in it, the term for capacitive reactance is just left out.

If there is a resistor and a capacitor then L = 0 gives X_L = 0 and Z = Ö(R² + (-X_C)²) = Ö(R² + X_C²) and if there is only a resistor then Z = Ö(R²) = R. (But L is never quite = 0 since any closed circuit contains at least one "loop" of wire).

Phase shift

When there is L and C in the circuit the current I and voltage U as a function of time will not be in phase; the phase shift j in radians (2p means one whole "wavelength" of the sine curve ) is given by

tan j = (X_L - X_C) / R

The AC circuit is in resonance when (X_L - X_C) = 0 so that Z = R which happens when X_L = X_C or wL = 1/wC which gives w = 1/ÖLC or

f = 1/2pÖ(LC)

This can be used radio tuning where the C-value of a capacitor is changed either by turning plates so that the area opposing a plate with opposite charge is changed, or by changing the distance between plates); thereby the resonance frequency is changed and difference stations can be tuned in.

Atomic physics

(if the Biomedical physics option is not taken)

Radiation dose

The dose of ionising radiation absorbed by living organisms is the absorbed dose D = E/m = energy absorbed per mass, unit 1 gray = 1 Gy = 1 Jkg^-1 .The equivalent dose H = QD where Q = a "quality factor" which says how this type of radiation affects living organisms; unit 1 sievert = 1 Sv (Q has no unit).

Optics

More or less the entire Optics option is included in the Finnish national programme and should be self-studied if that option is not chosen by the IB group.