NSSC Physics Advanced Subsidiary Level Study Notes

Essential Physical Constants and Formulae Reference

In the study of Physics at the Advanced Subsidiary level, specific fundamental constants are utilized across various calculations. The speed of light in free space, denoted as $c$, is defined as $3.00 \times 10^8\,\text{m}\,\text{s}^{-1}$. The elementary charge, represented by $e$, is $1.60 \times 10^{-19}\,\text{C}$. Mass is quantified using the unified atomic mass unit, where $1\,u = 1.66 \times 10^{-27}\,\text{kg}$. Particle masses are specifically noted, with the rest mass of an electron $m_e$ being $9.11 \times 10^{-31}\,\text{kg}$ and the rest mass of a proton $m_p$ being $1.67 \times 10^{-27}\,\text{kg}$. The standard acceleration of free fall, $g$, is taken as $9.81\,\text{m}\,\text{s}^{-2}$.

Mathematical relationships for uniformly accelerated motion include the equations $s = ut + \frac{1}{2}at^2$ and $v^2 = u^2 + 2as$. For thermodynamics and fluids, work done on or by a gas is expressed as $W = p\Delta V$, and hydrostatic pressure change is determined by $\Delta p = \rho g\Delta h$. Wave phenomena involve the Doppler effect formula $f^o = f_s \frac{v}{v \pm v_s}$. Electrical principles are guided by the equation for electric current $I = Anvq$, and the combination of resistors. For series circuits, total resistance is $R = R_1 + R_2 + \dots$, while for parallel circuits, it is calculated using $\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$.

Physical Quantities, Units, and Measurements

Physical quantities are categorized as either base or derived, and must be expressed in SI base units. For example, mass has the SI base unit of the kilogram, symbolized as $\text{kg}$, while velocity is a derived quantity with the base units $\text{m}\,\text{s}^{-1}$. It is crucial to distinguish between scalar quantities, which possess only magnitude, and vector quantities, which possess both magnitude and direction. Mass and distance are defined as scalars, whereas moment of force is categorized as a vector. When selecting measuring instruments, precision is a primary factor; a micrometer is the appropriate tool for measuring the diameter of a thin copper wire when a precision of $0.01\,\text{mm}$ is required.

Measurement analysis involves dealing with absolute and percentage uncertainties. For a rectangular tank with dimensions $L = (2.14 \pm 0.04)\,\text{m}$, $w = (1.98 \pm 0.05)\,\text{m}$, and $h = (1.50 \pm 0.04)\,\text{m}$, the volume is found by multiplying the three dimensions, resulting in $6.36\,\text{m}^3$. The uncertainty in a product is determined by summing the fractional uncertainties: $\frac{\Delta V}{V} = \frac{0.04}{2.14} + \frac{0.05}{1.98} + \frac{0.04}{1.50}$. Converting this to a percentage yields an uncertainty of approximately $7\%$. Therefore, the volume of water needed to fill the tank is summarized as $6.36\,\text{m}^3 \pm 7\%$.

Kinematics and Dynamics of Motion

Motion can be analyzed through velocity-time graphs, where the area under the curve represents the total distance covered. For a cyclist accelerating from rest to a velocity of $12\,\text{m}\,\text{s}^{-1}$ in a certain period and continuing for a total of $25\,\text{s}$, the displacement or distance is the integral of the velocity over time. In a specific scenario where a cyclist's velocity increases and then sustains, the distance covered in $25\,\text{s}$ may be calculated as $150\,\text{m}$. Vertical motion under gravity requires consideration of time and acceleration. If a ball is thrown vertically upwards and takes $4.10\,\text{s}$ to return to its starting point with negligible air resistance, the total time of flight is divided by two to find the time to reach maximum height ($2.05\,\text{s}$). Using $v = u + at$ where $v = 0$ at the peak and $a = -9.81\,\text{m}\,\text{s}^{-2}$, the initial velocity $u$ is calculated as approximately $20.1\,\text{m}\,\text{s}^{-1}$.

Force application on Inclined planes involves decomposing weight into components. For a $50\,\text{kg}$ trolley on an $8.0^{\circ}$ frictionless slope pulled by a rope with a tension $T = 150\,\text{N}$, the resultant force is the difference between the tension and the component of weight acting down the slope ($mg\sin(8.0^{\circ})$). The acceleration $a$ is determined by the equation $F_{net} = ma$, leading to a value of $1.6\,\text{m}\,\text{s}^{-2}$. Momentum is conserved in collisions. When two identical gliders, $Q$ and $S$, travel toward each other at $70\,\text{cm}\,\text{s}^{-1}$ and $40\,\text{cm}\,\text{s}^{-1}$ respectively and stick together upon collision, the final velocity is found using $m_1u_1 + m_2u_2 = (m_1 + m_2)v$. Calculating with direction ($70 - 40$) for identical masses gives a combined speed of $15\,\text{cm}\,\text{s}^{-1}$.

Equilibrium, Moments, and Fluid Mechanics

Statics and the principle of moments are used to maintain system balance. A metal rod of total length $1.5\,m$ pivoted at $0.75\,m$ from one end (the center) requires the sum of clockwise moments to equal the sum of anticlockwise moments. If a force $F$ is applied at $0.25\,m$ from the pivot and a $5.0\,N$ force is applied at $0.75\,m$ on the opposite side, the balance is achieved when $(F \times 0.25) = (5.0 \times 0.75)$, meaning $F = 15.0\,N$. For a couple, which consists of two equal and opposite forces not acting along the same line, the torque is the product of one of the forces and the perpendicular distance between them. For a $150.0\,cm$ ($1.5\,m$) bar with forces of $45.0\,N$ at each end, the torque is $45.0 \times 1.5 = 67.5\,N\,m$. An object in equilibrium must satisfy two conditions: there is no resultant force acting on it, and there is no resultant torque.

Fluid properties like density are additive when mixing substances without chemical reactions. If oils $M$, $N$, and $O$ are mixed, with mass and density values of ($670\,kg$, $650\,kg\,m^{-3}$), ($1200\,kg$, $840\,kg\,m^{-3}$), and ($2000\,kg$, $950\,kg\,m^{-3}$) respectively, the mixture density is calculated as $\rho_{mix} = \frac{\sum m}{\sum \frac{m}{\rho}}$. Total mass is $3870\,kg$ and total volume is the sum of individual volumes ($1.03 + 1.43 + 2.11 \approx 4.57\,m^3$), resulting in a mixture density of $850\,kg\,m^{-3}$. In thermodynamics, if a gas at atmospheric pressure ($101\,kPa$) in a syringe expands from $3.0\,cm^3$ to $9.0\,cm^3$ upon heating, the work done by the gas is $W = p\Delta V = 101 \times 10^3 \times (9.0 - 3.0) \times 10^{-6}$, which equals $606\,mJ$.

Work, Energy, Power, and Material Properties

Energy conversion and power in systems like hydroelectricity involve gravity and efficiency. For water falling $300\,m$ at a rate of $2500\,kg$ per minute ($41.67\,kg/s$), the input power is $P_{in} = \frac{m}{t}gh = 41.67 \times 9.81 \times 300 = 122610.3\,W$. With an efficiency of $75\%$, the average output power is $0.75 \times 122.6\,kW \approx 92\,kW$. Kinetic energy change occurs when velocity varies; for a $900\,kg$ car where speed doubles from $6.00\,\text{m}\,\text{s}^{-1}$ to $12.0\,\text{m}\,\text{s}^{-1}$, the change in kinetic energy is $\Delta E_k = \frac{1}{2}m(v^2 - u^2) = \frac{1}{2}(900)(144 - 36) = 48600\,J$. Speed can also be determined from power and force: if an engine does $36000\,J$ of work in $1.3\,s$ to move a car against a driving force of $960\,N$, the power is $P = \frac{36000}{1.3}$. Since $P = Fv$, the speed $v$ is calculated as approximately $29\,\text{m}\,\text{s}^{-1}$.

Material deformation is described by the Young modulus $E$. For a wire of length $l$ and diameter $d$ extending by $e$ under a mass $m$, the formula for the Young modulus is $E = \frac{\text{stress}}{\text{strain}} = \frac{\frac{mg}{A}}{\frac{e}{l}}$. Since the cross-sectional area $A = \frac{\pi d^2}{4}$, substituting gives $E = \frac{4mgl}{\pi d^2 e}$, which rearranges to solve for extension: $e = \frac{4mgl}{\pi d^2 E}$. The work done on a spring is represented by the area under its force-extension graph. If a spring reaches an extension of $39.0\,cm$ with a force of $4.0\,N$, the work done is $\frac{1}{2}Fx = 0.5 \times 4.0 \times 0.39 = 0.78\,J$.

Waves, Sound, and the Electromagnetic Spectrum

Wave mechanics are visualized through displacement-time or displacement-distance graphs. On a displacement-time graph, the maximum displacement from the equilibrium position represents the amplitude, while the time for one complete cycle represents the period. A fundamental relationship exists between a wave's power and its amplitude, where power is proportional to the square of the amplitude ($P \propto A^2$), thus $A \propto \sqrt{P}$. Sound wave frequency can be determined using a cathode-ray oscilloscope (c.r.o). If a trace shows one full cycle across $2\,cm$ and the time-base is set to $0.20\,ms\,cm^{-1}$, the period $T$ is $0.40\,ms$. The frequency is then $f = \frac{1}{T} = \frac{1}{0.0004} = 2500\,Hz$.

The Doppler effect occurs when a source and observer are in relative motion. For a train sounding a $500\,Hz$ horn while moving at $15.0\,\text{m}\,\text{s}^{-1}$ away from a stationary observer (speed of sound $336\,\text{m}\,\text{s}^{-1}$), the observed frequency is $f^o = 500 \times \frac{336}{336 + 15.0} = 479\,Hz$. Stationary waves in a tube closed at one end are longitudinal and stationary in nature. The electromagnetic spectrum is ordered by decreasing frequency (increasing wavelength) as follows: gamma rays, X-rays, ultraviolet, visible light, infrared, microwaves, and radio waves. Light behavior includes diffraction, which is the spreading of a wave as it passes through a narrow gap. For double-slit interference, fringe spacing is determined by the wavelength, distance to screen $D$ ($e.g. 50\,cm$), and slit separation $a$ ($e.g. 0.50\,mm$). Diffraction gratings produce patterns where the maximum order $n$ is limited by $n \le \frac{d}{\lambda}$. For $750\,lines\,mm^{-1}$, $d = \frac{1 \times 10^{-3}}{750}\,m$. For a wavelength of $590\,nm$, the maximum order is $2$.

Electricity, Resistance, and Circuits

Electric charge and fields exert forces on particles. A charge of $3.0\,μ C$ between plates separated by $30\,cm$ with a potential difference of $6.0\,V$ experiences a force $F = qE = q\frac{V}{d} = 3.0 \times 10^{-6} \times \frac{6.0}{0.30} = 6.0 \times 10^{-5}\,N$. In a metal wire, current $I$ is related to drift velocity $v$ by $I = Anvq$. For a current of $2.0\,A$, drift velocity of $0.4\,mm\,s^{-1}$, and $2.0 \times 10^{28}$ electrons per unit volume, the cross-sectional area is $A = \frac{2.0}{(2.0 \times 10^{28} \times 0.0004 \times 1.6 \times 10^{-19})} = 1.6 \times 10^{-6}\,m^2$. Electrical power dissipation in parallel circuits involves four bulbs of $960\,\Omega$ each connected to $240\,V$. The power for one bulb is $P = \frac{V^2}{R} = \frac{240^2}{960} = 60\,W = 0.060\,kW$. The property of resistivity has SI units of $\Omega\,m$.

In circuit theory, electromotive force (e.m.f.) and internal resistance influence potential difference ($p.d.$). When a switch is closed in a circuit with internal resistance, the potential difference across the battery terminals decreases because work must be done against the internal resistance. Kirchhoff's first law, based on the conservation of charge, dictates that the sum of currents entering a junction equals the sum of currents leaving it ($e.g. I_1 = I_2 + I_3$, so $I_3 = I_1 - I_2$). Potential dividers allow for specific voltage outputs; in a circuit with a $9.0\,V$ battery and resistors of $3.0\,k\Omega$ and $2.0\,k\Omega$ in series, a voltmeter across the $2.0\,k\Omega$ resistor will read $V_{out} = 9.0 \times \frac{2.0}{3.0 + 2.0} = 3.6\,V$.

Particle and Nuclear Physics

Radioactive decay transforms nuclei through types of radiation. Thorium-232 ($^{232}_{90}Th$) emitting an alpha particle ($^4_{2}He$) forms a daughter nucleus $X$ with a nucleon number of $228$ and proton number of $88$ ($X = ^{228}_{88}Ra$). If $X$ then undergoes beta-minus decay ($^0_{-1}e$), the granddaughter nucleus $Y$ maintains a nucleon number of $228$ but the proton number increases to $89$ ($Y = ^{228}_{89}Ac$). Alpha radiation is characterized by a short range (a few millimeters) in air and deflection by electric fields. Isotopes of an element are defined by having the same proton number but different neutron numbers. Fundamental particles are those not composed of smaller components, such as electrons, neutrinos, and quarks; neutrons and protons are not fundamental as they are composed of quarks.

Quark composition determines the charge of composite particles. Up quarks ($u$) have a charge of $+\frac{2}{3}e$ and down quarks ($d$) have a charge of $-\frac{1}{3}e$. To form a particle with a $2+e$ charge, a combination of four up quarks and one down quark is required calculation: $4(+\frac{2}{3}) + 1(-\frac{1}{3}) = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$ (adjustment: $4u$ and $2d$ gives $4(2/3) + 2(-1/3) = 6/3 = +2e$). Neutrinos are fundamental particles classified as leptons, possess no charge (neutral), and are emitted during processes such as beta-minus decay alongside an electron.