FSc Part 1 Physics formulas

Measurements

SI units, prefixes and significant figures

• Base units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
• Supplementary units: radian (rad) for plane angle, steradian (sr) for solid angle.
• newton: N = kg m s⁻²; joule: J = kg m² s⁻²; pascal: Pa = kg m⁻¹ s⁻².
• Prefixes: pico 10⁻¹², nano 10⁻⁹, micro 10⁻⁶, milli 10⁻³, kilo 10³, mega 10⁶, giga 10⁹.
• Zeros on the left of the first non-zero digit are not significant; zeros between digits, and zeros at the end after a decimal point, are significant.
• Multiplying or dividing: keep as many significant figures as the factor with the fewest.
• Adding or subtracting: keep as many decimal places as the term with the fewest.

Errors and uncertainties

• Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100.
• Addition and subtraction: add the absolute uncertainties.
• Multiplication and division: add the percentage uncertainties.
• Power: multiply the percentage uncertainty by the power, e.g. for r³ it is 3 × (% uncertainty in r).
• Timing experiment: uncertainty in one period = least count of the stopwatch ÷ number of oscillations timed.
• Corrected reading = observed reading − zero error (a positive zero error is subtracted).

Dimensions of physical quantities

• Velocity [LT⁻¹]; acceleration [LT⁻²]; force [MLT⁻²].
• Work, energy and torque [ML²T⁻²]; power [ML²T⁻³]; momentum and impulse [MLT⁻¹].
• Pressure [ML⁻¹T⁻²]; frequency and angular frequency [T⁻¹].
• Homogeneity: dimensions on the left side = dimensions of every term on the right side.
• Dimensional analysis cannot find a numerical constant such as 2π or ½.
• A dimensionally correct equation may still be wrong; a dimensionally wrong equation is certainly wrong.

Vectors and equilibrium

Vectors: representation, addition and rectangular components

• A_x = A cos θ and A_y = A sin θ (θ measured from the positive x-axis).
• Magnitude: A = √(A_x² + A_y²).
• Direction: φ = tan⁻¹(|A_y| ÷ |A_x|), then use the signs of the components to pick the quadrant.
• Quadrants: 1st θ = φ; 2nd θ = 180° − φ; 3rd θ = 180° + φ; 4th θ = 360° − φ.
• Resultant: R_x = A_x + B_x and R_y = A_y + B_y.
• Unit vector in the direction of A = A ÷ (magnitude of A).
• The resultant of two vectors A and B lies between (A − B) and (A + B) in magnitude.

Product of two vectors

• A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z.
• A · B = B · A; i · i = j · j = k · k = 1; i · j = j · k = k · i = 0.
• Angle between vectors: cos θ = (A · B) ÷ (AB).
• Magnitude of A × B = AB sin θ = area of the parallelogram with A and B as sides.
• A × B = −(B × A); i × j = k, j × k = i, k × i = j; i × i = j × j = k × k = 0.
• Work W = F · d; power P = F · v; torque τ = r × F.

Torque and equilibrium

• Torque = r × F; magnitude τ = rF sin θ, where θ is the angle between r and F. Unit: N m.
• Torque = force × moment arm, where the moment arm is r sin θ.
• First condition: ΣF = 0, which means ΣF_x = 0 and ΣF_y = 0.
• Second condition: Στ = 0, so total clockwise torque = total anticlockwise torque.
• The weight of a uniform body acts at its centre.
• Take torques about a point where an unknown force acts, so that force drops out.

Motion and force

Displacement, velocity, acceleration and motion graphs

• Average velocity = displacement ÷ time (v_av = d/t).
• Acceleration = change in velocity ÷ time taken (a = (v_f − v_i)/t).
• v_f = v_i + at.
• S = v_it + ½at².
• 2aS = v_f² − v_i².
• Distance with uniform acceleration = average velocity × time = ½(v_i + v_f) × t.
• Free fall: taking upward as positive, a = −9.8 m s⁻² on the way up and on the way down.

Newton's laws, momentum and collisions

• F = ma; momentum p = mv (unit kg m s⁻¹ or N s).
• Force = change in momentum ÷ time (F = (mv_f − mv_i)/t); impulse = F × t = mv_f − mv_i.
• Conservation: m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′.
• Elastic, head-on, target at rest: v₁′ = (m₁ − m₂)v₁/(m₁ + m₂) and v₂′ = 2m₁v₁/(m₁ + m₂).
• Equal masses, elastic: the bodies exchange velocities. Light body hitting a massive body at rest: it bounces back with almost the same speed.
• Force due to a flowing fluid = mass per second × change in velocity.
• Rocket: acceleration = (mass of gas ejected per second × speed of gas relative to rocket) ÷ mass of rocket (a = mv/M).

Projectile motion

• Components of the launch velocity: v_ix = v_i cos θ and v_iy = v_i sin θ.
• Time of flight: t = 2v_i sin θ ÷ g.
• Maximum height: h = v_i² sin²θ ÷ (2g).
• Horizontal range: R = v_i² sin 2θ ÷ g.
• Range is greatest at θ = 45°, where R_max = v_i² ÷ g.
• Complementary angles such as 30° and 60° give the same range at the same speed.
• Thrown horizontally from height h: time to fall t = √(2h ÷ g); distance = v_x × t.

Work and energy

Work, power and conservative field

• Work = force × displacement × cos θ (W = F · d = Fd cos θ). Unit: joule, 1 J = 1 N m.
• Variable force: work = area under the force-displacement graph.
• Work against gravity = mgh, for any path between the same two levels.
• Conservative forces: gravitational, elastic spring and electric forces. Non-conservative: friction, air resistance, tension in a string, the push of a rocket.
• Average power = work done ÷ time taken (P = W/t). Unit: watt, 1 W = 1 J s⁻¹.
• Instantaneous power = F · v = Fv cos θ.
• 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J (a unit of energy, not power).

Work-energy principle and conservation of energy

• kinetic energy = ½ × mass × speed² (K.E. = ½mv²)
• gravitational potential energy near the Earth's surface = mass × g × height (P.E. = mgh), with g = 9.8 m s⁻²
• work-energy principle: work done = change in kinetic energy (Fd = ½mv_f² − ½mv_i²)
• free fall with no friction: loss of P.E. = gain in K.E., so mgh = ½mv² and v = √(2gh)
• with friction: loss of P.E. = gain in K.E. + work done against friction (mgh = ½mv² + fh)
• for the same momentum p, K.E. = p²/2m, so the lighter body has more kinetic energy
• the total energy of an isolated system stays constant

Absolute potential energy, escape velocity and energy sources

• absolute potential energy at distance r from the Earth's centre: U = −GMm/r (at the surface, U = −GMm/R)
• escape velocity: ½mv_esc² = GMm/R, so v_esc = √(2GM/R) = √(2gR)
• for the Earth, v_esc is about 11.2 km s⁻¹, and it is the same for every mass
• tidal energy: the gravitational pull of the Moon (and Sun) raises and lowers sea water, which can drive turbines
• solar energy: sunlight is used directly for heating or turned into electricity by solar cells
• geothermal energy: heat from hot rocks and molten material inside the Earth
• biomass: plant and animal waste; decay of dung in a digester gives biogas, which is mainly methane

Circular motion

Angular motion and centripetal force

• angle in radians = arc length ÷ radius (θ = s/r); 1 revolution = 2π rad = 360°, so 1 rad = 57.3°
• angular velocity = angular displacement ÷ time (ω = Δθ/Δt), in rad s⁻¹; for n revolutions per second, ω = 2πn
• angular acceleration = change in angular velocity ÷ time (α = Δω/Δt), in rad s⁻²
• linear and angular: s = rθ, v = rω, a_t = rα (θ must be in radians)
• centripetal acceleration = v²/r = rω², directed towards the centre
• centripetal force = mv²/r = mrω²
• least speed at the top of a vertical circle: weight alone gives the centripetal force, so v = √(gr)

Moment of inertia, angular momentum and rotational kinetic energy

• moment of inertia of a point mass = mass × (distance from axis)² (I = mr²), unit kg m²
• hoop: I = mr²; disc: I = ½mr²; solid sphere: I = (2/5)mr²
• angular momentum = moment of inertia × angular velocity (L = Iω), unit kg m² s⁻¹ (J s)
• conservation of angular momentum: with no external torque, I₁ω₁ = I₂ω₂
• rotational kinetic energy = ½Iω²; for a disc it is ¼mv², for a hoop ½mv²
• rolling from rest down a height h: disc v = √(4gh/3), hoop v = √(gh); the disc is faster and arrives first
• these speeds do not depend on the mass or radius of the body

Artificial satellites and weightlessness

• orbital velocity: GMm/r² = mv²/r, so v = √(GM/r); close to the Earth v = √(gR), about 7.9 km s⁻¹
• lift accelerating upwards: apparent weight = m(g + a); accelerating downwards: m(g − a); falling freely: zero
• artificial gravity in a spaceship of radius R: Rω² = g, so frequency of rotation f = (1/2π)√(g/R)
• geostationary orbit: period 24 hours, above the equator, moving in the direction of the Earth's rotation
• geostationary orbit radius is about 4.23 × 10⁴ km from the Earth's centre (about 3.6 × 10⁴ km above the surface)
• three geostationary satellites can cover the whole Earth except the polar regions

Fluid dynamics

Viscous drag, Stokes' law and terminal velocity

• Stokes' law: drag force on a sphere = 6π × viscosity × radius × speed (F = 6πηrv)
• the unit of viscosity η is N s m⁻², the same as Pa s
• at terminal velocity: drag = weight, so 6πηrv_t = mg and the acceleration is zero
• with m = (4/3)πr³ρ: terminal velocity v_t = 2gr²ρ/9η
• v_t is proportional to r²: twice the radius gives four times the terminal velocity
• streamlines never cross; turbulent flow is irregular and unsteady

Equation of continuity and Bernoulli's equation

• equation of continuity: area × speed = constant (A₁v₁ = A₂v₂); it comes from conservation of mass
• Bernoulli's equation: P + ½ρv² + ρgh = constant
• horizontal pipe: P₁ − P₂ = ½ρ(v₂² − v₁²)
• Torricelli's theorem: speed of efflux from a hole at depth h below the surface, v = √(2gh)
• venturi relation (speed in the wide part negligible): P₁ − P₂ = ½ρv₂²
• lift: air moves faster over the top of the wing, so the pressure above is lower than below
• blood pressure is measured on the upper arm, at heart level; normal values are about 120 torr (systolic) and 80 torr (diastolic)

Oscillations

Simple harmonic motion and the reference circle

• condition for SHM: a = −ω²x (compare a given equation with this to read off ω²)
• time period T = 2π/ω; frequency f = 1/T; angular frequency ω = 2πf
• displacement (starting from the mean position): x = x₀ sin ωt
• speed at displacement x: v = ω√(x₀² − x²)
• maximum speed (at the mean position): v₀ = ωx₀
• maximum acceleration (at the extreme positions): ω²x₀
• a phase difference of π rad (180°) means the two bodies are always at opposite points of their motion

Mass-spring system, simple pendulum and energy in SHM

• mass-spring system: T = 2π√(m/k), where k is the spring constant in N m⁻¹; it does not depend on g
• simple pendulum: T = 2π√(l/g); it does not depend on the mass of the bob
• a second's pendulum has a period of 2.0 s (length about 0.99 m where g = 9.8 m s⁻²)
• potential energy at displacement x: P.E. = ½kx²
• kinetic energy at displacement x: K.E. = ½k(x₀² − x²)
• total energy = ½kx₀², constant at every point
• K.E. = P.E. when x = x₀/√2

Free and forced oscillations, resonance and damping

• free oscillation: no outside periodic force; frequency = natural frequency
• forced oscillation: frequency = frequency of the driving force
• resonance: driving frequency = natural frequency, giving the largest amplitude
• examples of resonance: pushing a swing in time with its motion, tuning a radio, a microwave oven, soldiers marching in step on a bridge
• damping: energy is dissipated by friction, so the amplitude decreases with time
• light damping gives a tall, sharp resonance curve; heavy damping gives a low, broad one
• shock absorbers in a car use damping to stop the body oscillating for long after a bump

Waves

Progressive waves and the speed of sound

• wave speed = frequency × wavelength (v = fλ)
• Newton's formula: speed = √(pressure ÷ density), v = √(P/ρ), about 280 m s⁻¹ at STP (about 16% too low)
• Laplace's correction: v = √(γP/ρ), with γ = 1.4 for air, about 333 m s⁻¹
• Pressure: no effect at constant temperature, because density rises in the same ratio and P/ρ stays the same
• Density: v is proportional to 1/√ρ at the same pressure, so sound is faster in a less dense gas (moist air, hydrogen)
• Temperature: v is proportional to √T, with T in kelvin
• Near room temperature: v_t = v₀ + 0.61t, with t in °C (speed rises about 0.61 m s⁻¹ for each 1 °C)

Superposition, interference, beats and Doppler effect

• Constructive interference: path difference = nλ (n = 0, 1, 2, ...)
• Destructive interference: path difference = (n + ½)λ
• beat frequency = difference of the two frequencies (f₁ − f₂)
• Denser medium: crest returns as trough (180° change). Rarer medium: crest returns as crest; for sound, a compression returns from an open end as a rarefaction
• Observer moving, source at rest: f′ = f(v + u_o)/v towards, f′ = f(v − u_o)/v away
• Source moving, observer at rest: f′ = fv/(v − u_s) towards, f′ = fv/(v + u_s) away
• Uses: radar speed checks, sonar, red shift of light from stars and galaxies moving away, blue shift for those approaching

Stationary waves in strings and air columns

• node to node = λ/2; node to next antinode = λ/4
• speed of a wave on a string = √(tension ÷ mass per unit length), v = √(F/m)
• String fixed at both ends: f₁ = v/(2l), and f_n = nf₁ with n = 1, 2, 3, ... (all harmonics)
• Frequency of a string is proportional to √F, and to 1/l
• Pipe open at both ends: f_n = nv/(2l), n = 1, 2, 3, ... (all harmonics)
• Pipe closed at one end: f_n = nv/(4l), n = 1, 3, 5, ... (odd harmonics only)
• For the same length, the open pipe's fundamental is twice that of the closed pipe

Physical optics

Wavefronts and interference of light

• Conditions for interference: the beams must be monochromatic and coherent
• Bright fringe: d sin θ = mλ. Dark fringe: d sin θ = (m + ½)λ, with m = 0, 1, 2, ...
• fringe spacing = wavelength × slit-to-screen distance ÷ slit separation (Δy = λL/d)
• Bright and dark fringes are equally spaced; in water λ becomes smaller, so the fringes move closer together
• Thin films (oil on water, soap bubble): light reflected from the top and bottom surfaces interferes, giving colours
• Newton's rings: circular fringes from the air film between a plano-convex lens and a glass plate; the centre is dark in reflected light because of the 180° phase change at the denser medium
• Michelson's interferometer: moving the mirror by λ/2 shifts one fringe, so distance moved L = mλ/2

Diffraction and polarization

• Single slit of width a: first minimum where a sin θ = λ
• Grating: d sin θ = nλ, with n = 0, 1, 2, ... the order of the maximum (written m in some questions)
• grating element d = 1 ÷ (number of lines per unit length); 5000 lines per cm gives d = 2.0 × 10⁻⁶ m
• Highest order: sin θ cannot be more than 1, so n must be a whole number not greater than d/λ
• Bragg's equation: 2d sin θ = nλ, where d is the spacing of the crystal planes and θ is the glancing angle between the beam and the planes
• Two crossed Polaroids (axes at 90°) let no light through; parallel axes let the polarized light through

Optical instruments

Magnifying glass, microscopes and telescopes

• Simple microscope (image at near point): M = 1 + d/f, with d = 25 cm
• Compound microscope: M = (q/p)(1 + d/f_e) for the objective's image and object distances, or about (L/f_o)(1 + d/f_e), where L is the length of the microscope
• Astronomical telescope in normal adjustment: M = f_o/f_e
• Length of the telescope in normal adjustment = f_o + f_e; final image at infinity and inverted
• Telescope: long-focus, wide objective and short-focus eyepiece. Microscope: both lenses of short focal length
• Smallest angle resolved: α_min = 1.22λ/D; resolving power R = 1/α_min, so a wider aperture D and a shorter wavelength resolve better
• Resolving power of a grating: R = λ/Δλ = N × m (N lines, order m)

Spectrometer, speed of light and optical fibres

• Collimator: slit at the focus of a convex lens, gives a parallel beam. Turntable: holds the prism or grating, with a circular scale. Telescope: turns about the same axis, with a vernier to read the angle
• Michelson's method: c = 16fd, where f is the number of revolutions per second and d is the distance to the far mirror; c is about 3.0 × 10⁸ m s⁻¹
• Critical angle: sin θ_c = n₂/n₁ (rarer ÷ denser); into air, sin θ_c = 1/n
• Total internal reflection needs: light going from denser to rarer, and angle of incidence greater than θ_c
• Single mode step index fibre: very thin core, about 5 μm in diameter, one path
• Multimode step index: core about 50 μm in diameter, many paths. Multimode graded index: core index falls gradually from the centre outwards, light bends by continuous refraction, less pulse spreading
• Signal losses: absorption and scattering by impurities in the glass, and dispersion (spreading of the pulse)

Heat and thermodynamics

Kinetic theory of gases

• P = ⅔N₀⟨½mv²⟩, where N₀ is the number of molecules per unit volume; the same result is P = ⅓ρ⟨v²⟩
• T = (2/3k)⟨½mv²⟩, so average translational kinetic energy = (3/2)kT, with k = 1.38 × 10⁻²³ J K⁻¹
• rms speed is proportional to √T: doubling T (in kelvin) multiplies it by √2
• Boyle's law: PV = constant at constant temperature (P₁V₁ = P₂V₂)
• Charles's law: V/T = constant at constant pressure, T in kelvin
• Ideal gas equation: PV = nRT, with R = 8.31 J mol⁻¹ K⁻¹; also PV = NkT for N molecules
• T (K) = t (°C) + 273

Internal energy and the first law of thermodynamics

• work done by a gas at constant pressure = pressure × change in volume (W = PΔV)
• First law: Q = ΔU + W. Q is positive for heat supplied to the system; W is positive for work done by the system and negative for work done on it
• Isothermal (ideal gas): ΔU = 0, so Q = W; PV = constant
• Adiabatic: Q = 0, so W = −ΔU; PV^γ = constant. Expansion cools the gas, compression heats it
• Constant volume: W = 0, so Q = ΔU
• Molar specific heat: heat needed to raise the temperature of 1 mole by 1 K, at constant volume (C_v) or at constant pressure (C_p); Q = nC_vΔT or Q = nC_pΔT
• C_p − C_v = R; C_p is larger because at constant pressure the gas also does work as it expands

Second law, heat engines and entropy

• Kelvin statement: no engine working in a cycle can convert all the heat taken from a source into work; a sink is necessary
• Clausius statement: heat cannot flow by itself from a colder body to a hotter body; work must be done to make it happen
• Carnot cycle order: isothermal expansion, adiabatic expansion, isothermal compression, adiabatic compression
• efficiency = work output ÷ heat input = 1 − Q₂/Q₁; for a Carnot engine η = 1 − T₂/T₁, with temperatures in kelvin
• Petrol engine: four strokes (intake, compression, power, exhaust), petrol-air mixture lit by a spark plug, efficiency about 25 to 30%
• Diesel engine: no spark plug; air alone is compressed until very hot, then diesel is sprayed in and ignites; efficiency about 35 to 40%
• change in entropy = heat transferred ÷ absolute temperature (ΔS = ΔQ/T), unit J K⁻¹; positive when heat is added