Physics Formulas

$$\text{Density } \rho = \frac{\text{mass}}{\text{volume}} = \frac{m}{V} \qquad m = \rho V \qquad V = \frac{m}{\rho}$$

• $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$.
• Volume: cuboid $= lwh$, cylinder $= \pi r^2 h$, sphere $= \tfrac{4}{3}\pi r^3$. Area of a circle $= \pi r^2$.
• $1\ \text{m}^3 = 10^6\ \text{cm}^3 = 1000$ litres; $1\ \text{cm}^3 = 1$ ml.
• Relative density $= \dfrac{\text{density of the substance}}{\text{density of water}}$.
• Thickness of one sheet (or diameter of one wire turn) $= \dfrac{\text{total thickness}}{\text{number of sheets}}$.
• Area of an oil film: thickness (molecule diameter) $d = \dfrac{V}{A}$.

$$F = ma \qquad W = mg \quad (g \approx 10\ \text{N/kg or } 9.8\ \text{N/kg})$$

• Resultant of two forces at right angles: $R = \sqrt{F_1^2 + F_2^2}$, direction $\tan\theta = \dfrac{F_2}{F_1}$.
• Weight on another body: $W = mg_{\text{body}}$; mass never changes.

Momentum and impulse

$$p = mv \qquad \text{Impulse} = Ft = \Delta p = mv - mu \qquad F = \frac{mv - mu}{t}$$

Conservation of linear momentum

$$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \quad\text{(total momentum before = after)}$$

• Elastic collision: kinetic energy is also conserved.
• Bodies that join (inelastic): $m_1u_1 + m_2u_2 = (m_1 + m_2)v$.

Friction

$$F_{\text{friction}} = \mu R \qquad \mu = \frac{F}{R}\ (R = \text{normal reaction})$$

$$\text{Speed} = \frac{\text{distance}}{\text{time}} \qquad \text{Velocity} = \frac{\text{displacement}}{\text{time}} \qquad a = \frac{v - u}{t}$$

Equations of uniformly accelerated motion

$$v = u + at \qquad s = ut + \tfrac{1}{2}at^2 \qquad v^2 = u^2 + 2as \qquad s = \frac{u + v}{2}\,t$$ $u$ = initial velocity, $v$ = final velocity, $a$ = acceleration, $t$ = time, $s$ = displacement.

Free fall (take $a = g$)

$$v = gt \qquad h = \tfrac{1}{2}gt^2 \qquad v^2 = 2gh$$ Going up: $a = -g$; at the highest point $v = 0$. Time to the top $= \dfrac{u}{g}$, maximum height $= \dfrac{u^2}{2g}$.

Graphs

• Displacement-time graph: gradient $=$ velocity.
• Velocity-time graph: gradient $=$ acceleration; area under the graph $=$ displacement.

Projectiles (launched at speed $u$ and angle $\theta$)

$$\text{Time of flight} = \frac{2u\sin\theta}{g} \qquad \text{Range} = \frac{u^2\sin 2\theta}{g} \qquad \text{Max height} = \frac{u^2\sin^2\theta}{2g}$$

$$\text{Work } W = F\times d \qquad \text{Power } P = \frac{W}{t} = Fv \qquad \text{Energy is measured in joules (J)}$$ $$\text{Kinetic energy } E_k = \tfrac{1}{2}mv^2 \qquad \text{Gravitational potential energy } E_p = mgh$$

• Conservation of energy: $mgh = \tfrac{1}{2}mv^2$ (falling object, no friction).
• Work done by a force at an angle: $W = Fd\cos\theta$.
• 1 kilowatt-hour $= 3\,600\,000$ J.

Efficiency

$$\text{Efficiency} = \frac{\text{useful output energy (or power)}}{\text{total input energy (or power)}}\times 100\%$$

Machines

$$\text{MA} = \frac{\text{load}}{\text{effort}} \qquad \text{VR} = \frac{\text{distance moved by effort}}{\text{distance moved by load}} \qquad \text{Efficiency} = \frac{\text{MA}}{\text{VR}}\times 100\%$$

• Inclined plane: $\text{VR} = \dfrac{l}{h}$. Pulley system: $\text{VR} =$ number of rope sections supporting the load.
• Screw jack: $\text{VR} = \dfrac{2\pi r}{p}$ ($r$ = handle length, $p$ = pitch). Gear wheels: $\text{VR} = \dfrac{\text{teeth on driven wheel}}{\text{teeth on driving wheel}}$.

$$\text{Moment} = \text{force}\times\text{perpendicular distance from the pivot} \quad (\text{N m})$$ Principle of moments: at equilibrium $$\text{sum of clockwise moments} = \text{sum of anticlockwise moments}$$ $$F_1 d_1 = F_2 d_2$$

• For a body in equilibrium: total upward force $=$ total downward force.
• For a uniform rod, the weight acts at its midpoint (the centre of gravity).
• Couple: moment $= F \times$ (perpendicular distance between the two forces).

$$P = \frac{F}{A} \quad (\text{Pa} = \text{N/m}^2) \qquad \text{Liquid pressure } P = \rho g h$$

• Total pressure at depth $h$: $P_{\text{total}} = P_{\text{atm}} + \rho g h$. Atmospheric pressure $\approx 1.01\times10^5$ Pa ($760$ mm Hg).
• Pressure in a liquid acts equally in all directions; it depends only on depth and density.

Hydraulic machine (Pascal's principle)

$$\frac{F_1}{A_1} = \frac{F_2}{A_2} \qquad \text{MA} = \frac{F_2}{F_1} = \frac{A_2}{A_1}$$

Upthrust (Archimedes' principle)

$$\text{Upthrust} = \text{weight of fluid displaced} = \rho_{\text{fluid}}\, g\, V_{\text{displaced}}$$ $$\text{Apparent weight} = \text{real weight} - \text{upthrust}$$

• Floating: upthrust $=$ weight of the body. Fraction submerged $= \dfrac{\rho_{\text{body}}}{\rho_{\text{liquid}}}$.

Gas laws (temperature in kelvin)

$$\text{Boyle (constant } T\text{): } P_1V_1 = P_2V_2$$ $$\text{Charles (constant } P\text{): } \frac{V_1}{T_1} = \frac{V_2}{T_2} \qquad \text{Pressure law (constant } V\text{): } \frac{P_1}{T_1} = \frac{P_2}{T_2}$$ $$\text{General gas law: } \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$ $$T(\text{K}) = \theta(^\circ\text{C}) + 273$$

$$F = ke \qquad k = \frac{F}{e}\ \ (\text{N/m})$$ $F$ = force, $e$ = extension, $k$ = spring constant. Valid up to the elastic limit.

• Springs in series: $\dfrac{1}{k} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$. Springs in parallel: $k = k_1 + k_2$.
• Energy stored in a stretched spring: $E = \tfrac{1}{2}Fe = \tfrac{1}{2}ke^2$ (area under the force-extension graph).

$$v = \omega r \qquad \omega = \frac{2\pi}{T} = 2\pi f \qquad T = \frac{1}{f}$$ $$\text{Centripetal acceleration } a = \frac{v^2}{r} = \omega^2 r \qquad \text{Centripetal force } F = \frac{mv^2}{r} = m\omega^2 r$$

• $\omega$ is measured in rad/s; $1$ revolution $= 2\pi$ rad $= 360^\circ$.
• The centripetal force points towards the centre of the circle.
• Vertical circle: at the top, $T + mg = \dfrac{mv^2}{r}$; at the bottom, $T - mg = \dfrac{mv^2}{r}$.

$$Q = mc\Delta\theta \qquad \text{Heat capacity } C = mc = \frac{Q}{\Delta\theta}$$ $c$ = specific heat capacity (J/kg K), $\Delta\theta$ = temperature change.

Latent heat (change of state, no temperature change)

$$Q = mL \qquad L_f = \text{specific latent heat of fusion}, \quad L_v = \text{specific latent heat of vaporisation}$$

Heat exchange

Heat lost by the hot body $=$ heat gained by the cold body: $\ m_1c_1(\theta_1 - \theta) = m_2c_2(\theta - \theta_2)$.

Electrical heating

$$\text{Electrical energy} = Pt = VIt = mc\Delta\theta \ (\text{no losses}) \qquad \text{with a latent change: } Pt = mL$$

Expansion

$$\text{Linear: } \Delta L = \alpha L_0 \Delta\theta \qquad \text{Area: } \Delta A = \beta A_0\Delta\theta \qquad \text{Volume: } \Delta V = \gamma V_0\Delta\theta$$

Thermometers

$$\text{Temperature} = \frac{X_\theta - X_0}{X_{100} - X_0}\times 100^\circ\text{C}$$

Waves

$$v = f\lambda \qquad f = \frac{1}{T}$$ $v$ = speed, $f$ = frequency, $\lambda$ = wavelength, $T$ = period.

Sound

$$\text{Speed of sound} = \frac{2d}{t} \ \text{(echo)} \qquad \text{Speed of sound in air} \approx 330\text{ - }340\ \text{m/s}$$

• Closed pipe (one end closed): $\lambda = 4L$ (fundamental). Open pipe: $\lambda = 2L$.

Reflection and refraction

Angle of incidence $=$ angle of reflection. $$n = \frac{\sin i}{\sin r} \qquad n = \frac{\text{real depth}}{\text{apparent depth}} \qquad n = \frac{1}{\sin C}\ (C = \text{critical angle})$$ $$n = \frac{c_{\text{air}}}{v_{\text{medium}}} = \frac{\lambda_{\text{air}}}{\lambda_{\text{medium}}}$$

Mirrors and lenses

$$\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \qquad \text{Magnification } m = \frac{v}{u} = \frac{\text{image height}}{\text{object height}}$$

• Mirrors: $f = \dfrac{R}{2}$ ($R$ = radius of curvature). Lens power $P = \dfrac{1}{f}$ (dioptres, $f$ in metres).
• Real-is-positive convention for a convex lens: $u$ and $v$ positive for real objects and images.
• Two thin lenses in contact: $P = P_1 + P_2$.

Optical instruments

• Magnifying glass (image at the near point): $M = 1 + \dfrac{D}{f}$; (image at infinity) $M = \dfrac{D}{f}$, $D = 25$ cm.
• Astronomical telescope in normal adjustment: $M = \dfrac{f_o}{f_e}$.

$$Q = It \qquad V = IR \qquad P = VI = I^2R = \frac{V^2}{R} \qquad E = Pt = VIt$$ $Q$ in coulombs, $I$ in amperes, $V$ in volts, $R$ in ohms, $E$ in joules.

Resistors

$$\text{Series: } R = R_1 + R_2 + R_3 \qquad \text{Parallel: } \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$$ For two resistors in parallel: $R = \dfrac{R_1R_2}{R_1 + R_2}$.

• Series: same current, voltages add. Parallel: same voltage, currents add.

Resistivity

$$R = \frac{\rho L}{A} \qquad \text{(resistance increases with length, decreases with cross-sectional area)}$$

Cells

$$E = V + Ir = I(R + r) \qquad V = E - Ir$$ $E$ = e.m.f., $r$ = internal resistance, $V$ = terminal p.d., $R$ = external resistance.

• Cells in series: e.m.f.s add. Identical cells in parallel: e.m.f. is the same as one cell.

Cost of electricity

$$\text{Energy (kWh)} = \text{power (kW)}\times\text{time (h)} \qquad \text{Cost} = \text{units used}\times\text{cost per unit}$$

Fuses

Fuse rating should be slightly above the working current $I = \dfrac{P}{V}$.

Capacitors

$$C = \frac{Q}{V} \qquad E = \tfrac{1}{2}CV^2 \qquad \text{Series: } \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \qquad \text{Parallel: } C = C_1 + C_2$$

Force on a current-carrying conductor in a magnetic field

$$F = BIl \qquad (\text{conductor perpendicular to the field})$$

Electromagnetic induction

$$\text{Induced e.m.f.} = -N\frac{\Delta\Phi}{\Delta t}\ (\text{Faraday's law; Lenz: opposes the change}) \qquad E = Blv \ \text{(straight conductor moving across a field)}$$

Transformer

$$\frac{V_s}{V_p} = \frac{N_s}{N_p} \qquad \text{Ideal: } V_pI_p = V_sI_s$$ $$\text{Efficiency} = \frac{V_sI_s}{V_pI_p}\times 100\%$$

• Step-up: $N_s > N_p$ (voltage increases, current decreases). Step-down: $N_s < N_p$.

Power transmission

$$\text{Power loss in cables} = I^2R \qquad I = \frac{P}{V}$$ Transmitting at a high voltage reduces the current and therefore the power loss.

A.C.

$$V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \qquad I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \qquad f = \frac{1}{T}$$

Atomic structure

$$A = Z + N \qquad {}^{A}_{Z}X$$ $A$ = mass number, $Z$ = atomic number (protons), $N$ = neutrons.

Radioactive decay

• $\alpha$-decay: $\ {}^{A}_{Z}X \to {}^{A-4}_{Z-2}Y + {}^{4}_{2}\text{He}$. $\quad\beta$-decay: $\ {}^{A}_{Z}X \to {}^{A}_{Z+1}Y + {}^{0}_{-1}e$. $$N = N_0\left(\frac{1}{2}\right)^{n} \qquad n = \frac{t}{T_{1/2}} \qquad \text{Fraction remaining} = \left(\tfrac{1}{2}\right)^{t/T_{1/2}}$$ $$\text{Activity } A = \lambda N \qquad T_{1/2} = \frac{0.693}{\lambda}$$

Photons and the photoelectric effect

$$E = hf = \frac{hc}{\lambda} \qquad hf = \phi + \tfrac{1}{2}mv_{\max}^2 \qquad \phi = hf_0$$ $h = 6.63\times10^{-34}$ J s, $c = 3.0\times10^{8}$ m/s, $1\ \text{eV} = 1.6\times10^{-19}$ J.

X-rays

$$eV = \tfrac{1}{2}mv^2 \quad\text{(energy of electrons accelerated through a p.d. } V\text{)}$$ $$\text{Maximum photon energy} = eV = \frac{hc}{\lambda_{\min}}$$