“Add the absolutes, add the relatives”
Which error rule to use
For a sum or difference, add the absolute errors. For a product, quotient or power, add the relative errors. Two sentences cover every error numerical in the paper.
Gujarat Board, CBSE, GUJCET and JEE · free
Class 11 Physics formula book
Every chapter on one page, to read straight through or print. 13 chapters, 214 formulas with every symbol defined, 44 memory hooks, and the mistakes that cost the most marks.
1Units and Measurements
Units and measurements is the chapter that decides whether the rest of your physics answers score. It is the smallest chapter by formula count and the largest by marks lost, because significant figures and error propagation are applied in every numerical you write for two years.
Errors in a measurement
| What | Formula | Symbols and notes |
|---|---|---|
| Absolute error | Δa = amean − ai | the difference of one reading from the mean; always taken as positive |
| Mean absolute error | Δamean = (|Δa₁| + |Δa₂| + … + |Δan|)/n | average of the absolute errors of n readings |
| Relative error | relative error = Δamean / amean | a pure number, no unit |
| Percentage error | percentage error = (Δamean / amean) × 100% | what the question usually asks for |
How errors combine
| What | Formula | Symbols and notes |
|---|---|---|
| Sum or difference | Z = A + B or A − B, then ΔZ = ΔA + ΔB | absolute errors add, even when the quantities subtract |
| Product or quotient | Z = AB or A/B, then ΔZ/Z = ΔA/A + ΔB/B | relative errors add |
| Power | Z = ApBq/Cr, then ΔZ/Z = p(ΔA/A) + q(ΔB/B) + r(ΔC/C) | a squared term doubles its share of the error |
Dimensions worth knowing
| What | Formula | Symbols and notes |
|---|---|---|
| Force | [F] = M L T−2 | also the dimension of weight and tension |
| Work, energy, heat | [W] = M L2 T−2 | all three share one dimension, so they share one unit |
| Power | [P] = M L2 T−3 | energy per unit time |
| Pressure, stress | [p] = M L−1 T−2 | force per unit area |
| Gravitational constant | [G] = M−1 L3 T−2 | read it off F = Gm₁m₂/r² |
| Planck constant | [h] = M L2 T−1 | same dimension as angular momentum |
How to remember this chapter
“Powers multiply the blame”
Error in a squared or cubed term
A quantity raised to the power p contributes p times its relative error. If radius is measured to 2% and you need volume, the error is 6%, not 2%.
“Last doubtful digit”
Significant figures in an answer
An answer carries as many significant figures as the least precise measurement used. Rounding early is the most common way to lose the mark.
The mistakes that cost the most
- Subtracting the errors when the quantities subtract. Errors always add, whichever way the quantities go.
- Reporting more significant figures than the data allows, for example writing 2.3456 m when the metre scale reads to 0.1 cm.
- Treating a dimensional check as a proof. A dimensionally correct equation can still be wrong by a numerical factor; dimensions cannot see ½ or 2π.
- Forgetting that angles, relative error and refractive index are dimensionless, so they must never be given a unit.
2Kinematics
Kinematics describes motion without asking what caused it. The three equations of motion and the projectile results here reappear in laws of motion, in circular motion, and in every entrance paper you will sit for the next two years, so they are worth over-learning.
Equations of motion, uniform acceleration
| What | Formula | Symbols and notes |
|---|---|---|
| First equation | v = u + at | u initial velocity, v final velocity, a acceleration, t time |
| Second equation | s = ut + ½at² | s is displacement, not distance travelled |
| Third equation | v² = u² + 2as | use it when time is not given and not asked |
| Displacement in the nth second | sn = u + (a/2)(2n − 1) | the displacement during the nth second alone, not the first n seconds |
| Average velocity | vav = (u + v)/2 | valid only for uniform acceleration |
Motion under gravity
| What | Formula | Symbols and notes |
|---|---|---|
| Taking up as positive | v = u − gt, h = ut − ½gt², v² = u² − 2gh | g = 9.8 m s−2 downwards |
| Maximum height | H = u²/2g | at the top v = 0, but a is still g |
| Time to the top | t = u/g | time up equals time down for the same level |
| Total time of flight | T = 2u/g | for a body thrown up and returning to the same point |
Projectile motion
| What | Formula | Symbols and notes |
|---|---|---|
| Time of flight | T = 2u sinθ/g | θ is the angle of projection from the horizontal |
| Maximum height | H = u² sin²θ/2g | vertical velocity is zero at the top, horizontal velocity is not |
| Horizontal range | R = u² sin2θ/g | maximum when θ = 45° |
| Equal ranges | θ and (90° − θ) give the same range | for example 30° and 60° |
| Path equation | y = x tanθ − gx²/(2u² cos²θ) | a parabola |
| Horizontal projection from a height h | T = √(2h/g), R = u√(2h/g) | the time of fall does not depend on the horizontal speed |
Relative velocity
| What | Formula | Symbols and notes |
|---|---|---|
| Relative velocity | vAB = vA − vB | velocity of A as seen by B |
| River crossing, shortest time | t = d/vb, drift = vrd/vb | head straight across; d is river width |
| River crossing, shortest path | sinθ = vr/vb, t = d/√(vb² − vr²) | aim upstream at angle θ |
How to remember this chapter
“SUVAT”
What the letters in the equations of motion mean
s displacement, u initial velocity, v final velocity, a acceleration, t time. Every problem gives you three of them and asks for a fourth; pick the equation that leaves out the one you neither have nor want.
“Horizontal is lazy, vertical does the work”
Projectile motion
Horizontal velocity never changes because there is no horizontal force. Everything interesting, the rise, the fall, the time, comes from the vertical part.
“30 and 60 land together”
Equal ranges
Two angles adding to 90° give the same range. The larger angle simply spends longer in the air and less time moving forward.
The mistakes that cost the most
- Using s for distance travelled. In the equations of motion s is displacement, so a ball thrown up and caught again has s = 0.
- Thinking acceleration is zero at the highest point of a projectile. Velocity is momentarily zero vertically; acceleration is still g, downwards.
- Using the equations of motion when acceleration is not uniform. If a changes, they do not apply at all.
- Reading sn as the distance in n seconds. It is the distance during the nth second only.
- Forgetting the sign convention. Choose up as positive once, then keep it for the whole question.
3Laws of Motion and Friction
Laws of motion turns a described situation into an equation. Almost every mark in this chapter comes from one skill: drawing the free-body diagram correctly before writing anything. Friction, which sits inside this chapter, is the single most searched topic of Class 11 mechanics.
Newton's laws
| What | Formula | Symbols and notes |
|---|---|---|
| Second law | F = ma = dp/dt | p = mv is linear momentum; the vector form is what the law actually says |
| Impulse | J = FΔt = Δp | a large force for a short time, or a small force for long, can give the same change |
| Conservation of momentum | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | holds when the net external force is zero |
| Recoil of a gun | vgun = −(mbullet/mgun)vbullet | the minus sign is the recoil direction |
Friction
| What | Formula | Symbols and notes |
|---|---|---|
| Static friction | fs ≤ μsN | static friction is whatever it needs to be, up to this maximum |
| Limiting friction | fmax = μsN | the value just before sliding starts |
| Kinetic friction | fk = μkN | constant once sliding, and μk < μs |
| Angle of friction | tanλ = μs | λ is the angle the total contact force makes with the normal |
| Angle of repose | tanα = μs | the incline angle at which a body just begins to slide; numerically equal to the angle of friction |
| Body on a rough incline | a = g(sinθ − μcosθ) | when it slides down; it stays put while tanθ ≤ μ |
| Stopping distance | d = v²/(2μg) | independent of mass, which is why a loaded truck does not stop later for this reason |
Circular motion
| What | Formula | Symbols and notes |
|---|---|---|
| Centripetal force | F = mv²/r = mω²r | directed towards the centre; it is a requirement, not a new force |
| Flat road, maximum speed | vmax = √(μrg) | friction alone supplies the centripetal force |
| Banked road, no friction | tanθ = v²/rg | the ideal speed needs no friction at all |
| Banked road with friction | vmax = √(rg(μ + tanθ)/(1 − μtanθ)) | the full result used in entrance papers |
| Vertical circle, top | vtop ≥ √(gr) | the minimum speed to keep the string taut |
| Vertical circle, bottom | vbottom ≥ √(5gr) | follows from energy conservation between top and bottom |
Connected bodies
| What | Formula | Symbols and notes |
|---|---|---|
| Two masses over a pulley | a = (m₁ − m₂)g/(m₁ + m₂) | for an ideal pulley and a light inextensible string |
| Tension in that string | T = 2m₁m₂g/(m₁ + m₂) | the same throughout a light string over a frictionless pulley |
| Lift going up with acceleration a | apparent weight = m(g + a) | reading of a weighing machine inside the lift |
| Lift going down with acceleration a | apparent weight = m(g − a) | zero in free fall, which is weightlessness |
How to remember this chapter
“Draw before you write”
Every laws of motion question
Free-body diagram first: weight down, normal perpendicular to the surface, friction along the surface opposing relative motion, then tension. Most lost marks in this chapter are a missing force, not a wrong formula.
“Static is lazy, kinetic is fixed”
Telling the two frictions apart
Static friction only supplies as much as is needed to prevent sliding, up to μsN. Kinetic friction is a fixed μkN once sliding has begun.
“Repose equals friction”
Angle of repose and angle of friction
Both have tangent equal to μs, so the two angles are numerically the same. One is measured from the incline, the other from the normal.
“Centripetal is a job, not a force”
What supplies the centripetal force
Nothing is labelled centripetal on a free-body diagram. Friction, tension, gravity or the normal does that job. Write which one before using mv²/r.
The mistakes that cost the most
- Adding a centrifugal force in an inertial frame. In the ground frame there is no outward force on the body.
- Using f = μN when the body has not started sliding. Below the limit, friction equals the applied force, not μN.
- Assuming friction always opposes motion. It opposes relative sliding, which is why friction pushes a car forward.
- Using the mass of only one block for a connected system. Acceleration is common; write the equation for each block, or for the whole system.
- Forgetting that the string tension is the same throughout only if the string is light and the pulley frictionless.
4Work, Energy and Power
Work, energy and power gives you a second way to solve mechanics problems, and it is usually the shorter one. When a question gives speeds and heights but no time, the energy method will beat the equations of motion every time.
Work
| What | Formula | Symbols and notes |
|---|---|---|
| Work by a constant force | W = Fs cosθ | θ is the angle between force and displacement |
| Work by a variable force | W = ∫F dx | the area under a force-displacement graph |
| Work by a spring | W = ½kx² | the work done against the spring in stretching it by x |
| Work by gravity | W = −mgh going up, +mgh coming down | depends only on the height difference, not the path |
Energy
| What | Formula | Symbols and notes |
|---|---|---|
| Kinetic energy | KE = ½mv² = p²/2m | always positive; p is momentum |
| Gravitational potential energy | PE = mgh | near the Earth's surface, measured from a chosen level |
| Elastic potential energy | U = ½kx² | k is the spring constant, x the extension or compression |
| Work-energy theorem | Wnet = ΔKE = ½mv² − ½mu² | net work equals the change in kinetic energy, always |
| Conservation of mechanical energy | KE + PE = constant | only when no friction or other non-conservative force acts |
Power
| What | Formula | Symbols and notes |
|---|---|---|
| Average power | Pav = W/t | one watt is one joule per second |
| Instantaneous power | P = F·v = Fv cosθ | the dot product of force and velocity |
| Power of a pump | P = mgh/t = ρVgh/t | lifting a volume V of liquid of density ρ through height h |
Collisions
| What | Formula | Symbols and notes |
|---|---|---|
| Coefficient of restitution | e = (v₂ − v₁)/(u₁ − u₂) | relative speed after, over relative speed before |
| Perfectly elastic collision | e = 1, KE conserved | for equal masses the velocities are simply exchanged |
| Perfectly inelastic collision | e = 0, bodies move together | v = (m₁u₁ + m₂u₂)/(m₁ + m₂) |
| Elastic collision, unequal masses | v₁ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂) | momentum and kinetic energy both conserved |
| Height after n bounces | hn = e2nh | for a ball dropped from height h |
How to remember this chapter
“No time, use energy”
Choosing between kinematics and energy
If the question mentions speed and height but never time, the energy method is shorter. If it asks about time, go back to the equations of motion.
“Momentum always, energy sometimes”
Collisions
Momentum is conserved in every collision. Kinetic energy is conserved only if the collision is elastic. Write the momentum equation first, always.
“Perpendicular means no work”
When work is zero
A force at 90° to the displacement does no work. That is why the normal reaction, the tension in a circular path and the magnetic force on a moving charge all do zero work.
The mistakes that cost the most
- Calling work a vector. Work is a scalar, and it can be negative, as friction always shows.
- Using conservation of mechanical energy on a rough surface. With friction present, the loss equals the work done against friction.
- Assuming kinetic energy is conserved in every collision. It is not, unless the collision is stated to be elastic.
- Writing potential energy without saying from where. PE is measured from a reference level you must choose and state.
- Mixing up average and instantaneous power in a question that gives a velocity at one instant.
5Rotational Motion
Rotational motion is linear motion with every symbol swapped: mass becomes moment of inertia, force becomes torque, velocity becomes angular velocity. Learn the translations and half the chapter is already done, because every equation you learned last month has a rotational twin.
The translations
| What | Formula | Symbols and notes |
|---|---|---|
| Angular velocity | ω = dθ/dt = 2πf | measured in radian per second |
| Angular acceleration | α = dω/dt | the rotational twin of a |
| Equations of rotation | ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ | identical in shape to the SUVAT equations |
| Torque | τ = r × F = rF sinθ = Iα | the rotational twin of F = ma |
| Angular momentum | L = Iω = r × p | conserved when the net external torque is zero |
| Rotational kinetic energy | KE = ½Iω² | add it to ½mv² for a rolling body |
Moments of inertia
| What | Formula | Symbols and notes |
|---|---|---|
| Thin ring, about its axis | I = MR² | all the mass sits at distance R |
| Solid disc or cylinder | I = ½MR² | about the central axis |
| Solid sphere | I = ⅖MR² | about a diameter |
| Hollow sphere | I = ⅔MR² | about a diameter |
| Rod about its centre | I = ML²/12 | about a perpendicular axis |
| Rod about one end | I = ML²/3 | four times the value about the centre |
| Parallel axis theorem | I = Icm + Md² | d is the distance between the two parallel axes |
| Perpendicular axis theorem | Iz = Ix + Iy | for a flat lamina only |
Rolling
| What | Formula | Symbols and notes |
|---|---|---|
| Rolling without slipping | v = ωR | the contact point is momentarily at rest |
| Total kinetic energy | KE = ½mv²(1 + k²/R²) | k is the radius of gyration, I = mk² |
| Down an incline | a = g sinθ/(1 + k²/R²) | a ring is slowest, a solid sphere is fastest |
| Radius of gyration | k = √(I/m) | the distance at which all the mass could be placed |
Centre of mass
| What | Formula | Symbols and notes |
|---|---|---|
| Two particles | xcm = (m₁x₁ + m₂x₂)/(m₁ + m₂) | it lies nearer the heavier mass |
| Velocity of the centre of mass | vcm = (m₁v₁ + m₂v₂)/(m₁ + m₂) | unchanged by internal forces |
How to remember this chapter
“Same equations, new alphabet”
Rotational versions of the laws of motion
m becomes I, F becomes τ, v becomes ω, a becomes α, p becomes L. Every linear equation has an identical rotational partner.
“Ring, disc, sphere: 1, ½, ⅖”
The three moments of inertia asked most
Ring MR², disc ½MR², solid sphere ⅖MR². The further the mass sits from the axis, the larger the value, which is why the ring is largest.
“Skater pulls arms in”
Conservation of angular momentum
Pulling the arms in reduces I, so ω rises to keep Iω constant. Kinetic energy rises too, paid for by the work the skater does.
“Fat rings lose races”
Rolling down an incline
The larger the k²/R², the slower the body. Order down a slope: solid sphere, then disc, then ring, and mass and radius make no difference at all.
The mistakes that cost the most
- Using I = MR² for every round body. The shape decides the number in front, and that number decides who wins a rolling race.
- Applying the perpendicular axis theorem to a solid body. It works for flat laminas only.
- Forgetting the rotational kinetic energy for a rolling body. Its total energy is ½mv² plus ½Iω², not ½mv² alone.
- Assuming angular momentum is conserved when an external torque acts. Check the torque first, then conserve.
- Writing v = ωR when the body is slipping. That relation holds only for rolling without slipping.
6Gravitation
Gravitation is where the mechanics of Class 11 meets the sky. It supplies the satellite results asked in every GUJCET and JEE paper, and it is the first place where potential energy is negative, which is the idea most students never fully accept.
The law and g
| What | Formula | Symbols and notes |
|---|---|---|
| Newton's law of gravitation | F = Gm₁m₂/r² | G = 6.674 × 10−11 N m² kg−2 |
| Acceleration due to gravity | g = GM/R² | independent of the falling body's mass |
| At a height h | gh = g(1 − 2h/R) | valid only when h is much smaller than R |
| At a depth d | gd = g(1 − d/R) | g is zero at the centre of the Earth |
| Effect of rotation | g' = g − ω²R cos²λ | λ is the latitude, so g is least at the equator |
Satellites
| What | Formula | Symbols and notes |
|---|---|---|
| Orbital velocity | vo = √(GM/r) = R√(g/r) | for a circular orbit of radius r from the centre |
| Time period | T = 2π√(r³/GM) | this is Kepler's third law written out |
| Escape velocity | ve = √(2GM/R) = √(2gR) ≈ 11.2 km s−1 | does not depend on the escaping body's mass |
| Escape and orbital | ve = √2 × vo | about 41% faster is all it takes to leave |
| Geostationary satellite | T = 24 h, height ≈ 36,000 km | equatorial orbit, moving west to east |
Energy in orbit
| What | Formula | Symbols and notes |
|---|---|---|
| Gravitational potential energy | U = −GMm/r | negative, and zero only at infinity |
| Kinetic energy in orbit | KE = GMm/2r | half the magnitude of the potential energy |
| Total energy in orbit | E = −GMm/2r | negative, which is what bound means |
| Binding energy | BE = GMm/2r | the energy needed to free the satellite |
| Potential | V = −GM/r | potential energy per unit mass |
Kepler's laws
| What | Formula | Symbols and notes |
|---|---|---|
| First law | Orbits are ellipses with the Sun at one focus | not circles |
| Second law | Equal areas are swept in equal times | a planet moves fastest when nearest the Sun |
| Third law | T² ∝ r³ | r is the semi-major axis |
How to remember this chapter
“Negative means trapped”
Why gravitational energy is negative
Zero energy is defined at infinity, where the body is free. Anything bound has less than that, so its total energy is negative. Reaching zero is exactly what escaping means.
“Root two to be free”
Escape velocity from orbital velocity
Escape velocity is √2 times orbital velocity at the same radius. Roughly 8 km per second to orbit, 11.2 to leave.
“Height halves twice as fast as depth”
Variation of g
Going up, g falls as (1 − 2h/R). Going down, it falls as (1 − d/R). The factor 2 appears only in the height formula.
The mistakes that cost the most
- Using g = GM/R² with r measured from the Earth's surface. Distances in gravitation are measured from the centre.
- Writing potential energy as mgh for a satellite. mgh works only near the surface where g is roughly constant.
- Dropping the minus sign in U = −GMm/r, which then makes every energy answer wrong in sign.
- Thinking astronauts float because there is no gravity. They are in free fall, and g at that height is still about 8.7 m s−2.
- Forgetting that escape velocity does not depend on the direction of projection or on the mass thrown.
7Mechanical Properties of Solids
This short chapter is mostly one idea, stress over strain, wearing three different names. It is worth learning properly because the stress-strain graph is asked as a theory question almost every year, and the numerical work is straightforward marks.
Stress and strain
| What | Formula | Symbols and notes |
|---|---|---|
| Stress | σ = F/A | unit N m−2 or pascal, same unit as pressure |
| Longitudinal strain | ε = ΔL/L | a ratio, so no unit |
| Volume strain | ΔV/V | used for the bulk modulus |
| Shear strain | θ = Δx/L | the angle through which a face is tilted |
The three moduli
| What | Formula | Symbols and notes |
|---|---|---|
| Young's modulus | Y = stress/strain = FL/(AΔL) | for stretching a wire |
| Bulk modulus | B = −pV/ΔV | for uniform compression; the minus keeps B positive |
| Shear modulus | G = F/(Aθ) | also called the modulus of rigidity |
| Compressibility | K = 1/B | how easily a material is squeezed |
| Poisson's ratio | ν = (Δd/d)/(ΔL/L) | lateral strain over longitudinal strain, between 0 and 0.5 |
Energy stored
| What | Formula | Symbols and notes |
|---|---|---|
| Elastic potential energy | U = ½ × stress × strain × volume | the area under the stress-strain curve |
| In terms of extension | U = ½ FΔL = ½ (YA/L)(ΔL)² | a stretched wire behaves like a spring of constant YA/L |
| Energy density | u = ½ × stress × strain | energy stored per unit volume |
How to remember this chapter
“Y stretches, B squeezes, G slides”
Which modulus goes with which deformation
Young's for a pulled wire, bulk for pressure from all sides, shear for a face pushed sideways. Pick the modulus by what is being done to the body.
“Steel is stiffer, rubber is stretchier”
What a large Y means
A large Young's modulus means hard to stretch, not strong. Steel has a Y about 100,000 times rubber's, which is why bridges are not built of rubber.
“Elastic limit, then no return”
Reading the stress-strain graph
Up to the elastic limit the wire returns to its original length. Past the yield point it stays deformed, and at the breaking point it snaps.
The mistakes that cost the most
- Calling stress the same as pressure. They share a unit, but stress can be shear or tensile, and pressure cannot.
- Giving strain a unit. It is a ratio of two lengths, so it has none.
- Forgetting the factor of ½ in the stored energy. The force grows from zero, so the average force does the work.
- Using Young's modulus where the body is squeezed from all sides. That is the bulk modulus.
- Thinking a larger Young's modulus means a stronger material. It means a stiffer one.
8Mechanical Properties of Fluids
Fluids is three separate chapters joined together: pressure, flow, and surface tension with viscosity. Each part has its own small set of formulas, and questions almost never mix them, so it is easier than its length suggests.
Pressure and floating
| What | Formula | Symbols and notes |
|---|---|---|
| Pressure in a liquid | p = ρgh | depends on depth, never on the container's shape |
| Absolute pressure | p = p₀ + ρgh | p₀ is atmospheric pressure |
| Pascal's law | pressure applied is transmitted undiminished | the basis of the hydraulic lift |
| Hydraulic lift | F₂ = F₁(A₂/A₁) | a small force on a small piston lifts a large load |
| Buoyant force | FB = ρliquidVg | V is the volume submerged, not the whole body |
| Floating condition | ρbody/ρliquid = fraction submerged | ice floats with about 90% under water |
Flow
| What | Formula | Symbols and notes |
|---|---|---|
| Equation of continuity | A₁v₁ = A₂v₂ | a narrower pipe means faster flow |
| Bernoulli's theorem | p + ½ρv² + ρgh = constant | for steady, non-viscous, incompressible flow |
| Speed of efflux | v = √(2gh) | Torricelli's result, the same as a free fall through h |
| Venturi meter | v₁ = A₂√(2(p₁ − p₂)/ρ(A₁² − A₂²)) | used to measure flow speed |
Viscosity
| What | Formula | Symbols and notes |
|---|---|---|
| Newton's law of viscosity | F = −ηA(dv/dx) | η is the coefficient of viscosity, unit Pa s |
| Stokes' law | F = 6πηrv | drag on a small sphere moving slowly |
| Terminal velocity | vt = 2r²(ρ − σ)g/9η | ρ sphere density, σ fluid density |
| Reynolds number | Re = ρvD/η | below about 2000 the flow is streamlined, above 3000 turbulent |
| Poiseuille's equation | Q = πpr⁴/8ηL | the r to the fourth is why narrow arteries matter so much |
Surface tension
| What | Formula | Symbols and notes |
|---|---|---|
| Surface tension | T = F/L | force per unit length, unit N m−1 |
| Excess pressure in a drop | Δp = 2T/r | one surface |
| Excess pressure in a bubble | Δp = 4T/r | two surfaces, so twice the drop |
| Capillary rise | h = 2T cosθ/(rρg) | θ is the angle of contact; narrow tube, higher rise |
| Work to make a drop | W = T × ΔA | the increase in surface area times the surface tension |
How to remember this chapter
“Bubble has two skins”
Why the bubble formula has a 4
A soap bubble has an inner and an outer surface, so its excess pressure 4T/r is twice a liquid drop's 2T/r.
“Narrow means fast means low pressure”
Continuity plus Bernoulli
A narrower pipe forces the liquid faster, and faster flow means lower pressure. That one chain explains aeroplane lift, the spin on a cricket ball and the atomiser.
“Submerged volume, not total”
Buoyancy
The upthrust uses the volume of liquid displaced. For a floating body that is only the part under the surface.
“Terminal when drag equals weight”
Terminal velocity
A falling drop speeds up until viscous drag plus upthrust balances weight. After that the speed is constant and the acceleration is zero.
The mistakes that cost the most
- Using the whole volume in the buoyancy formula for a floating body. Only the submerged part counts.
- Thinking pressure depends on the shape or width of the vessel. It depends only on depth, density and g.
- Using 2T/r for a soap bubble. A bubble has two surfaces, so it is 4T/r.
- Applying Bernoulli's theorem to a viscous or turbulent flow. It assumes neither.
- Forgetting that terminal velocity is proportional to r², so a drop twice as wide falls four times as fast.
9Thermal Properties of Matter
Heat and temperature are not the same thing, and this chapter exists to make that difference precise. Expansion, calorimetry and the three modes of heat transfer each bring a small formula set, and the radiation laws at the end carry marks in both the board and the entrance papers.
Thermal expansion
| What | Formula | Symbols and notes |
|---|---|---|
| Linear expansion | ΔL = LαΔT | α is the coefficient of linear expansion, per °C |
| Area expansion | ΔA = AβΔT, β = 2α | for a flat sheet |
| Volume expansion | ΔV = VγΔT, γ = 3α | for solids; liquids have their own γ |
| Relation | α : β : γ = 1 : 2 : 3 | the single most asked line of this chapter |
| Thermal stress | σ = YαΔT | when a rod is held so that it cannot expand |
Calorimetry
| What | Formula | Symbols and notes |
|---|---|---|
| Heat and temperature change | Q = mcΔT | c is specific heat capacity, J kg−1 K−1 |
| Latent heat | Q = mL | during a change of state the temperature does not move |
| Latent heat of fusion of ice | Lf = 3.34 × 10⁵ J kg−1 | melting ice at 0 °C |
| Latent heat of vaporisation of water | Lv = 22.6 × 10⁵ J kg−1 | boiling water at 100 °C |
| Principle of calorimetry | heat lost = heat gained | in an insulated mixture |
Heat transfer
| What | Formula | Symbols and notes |
|---|---|---|
| Conduction | Q/t = kA(T₁ − T₂)/L | k is thermal conductivity; needs matter and no bulk motion |
| Thermal resistance | R = L/kA | resistances in series add, as in an electric circuit |
| Stefan-Boltzmann law | E = σeAT⁴ | σ = 5.67 × 10−8 W m−2 K−4, e is emissivity |
| Net radiation | Enet = σeA(T⁴ − T₀⁴) | T₀ is the surroundings' temperature |
| Wien's displacement law | λmT = 2.898 × 10−3 m K | hotter bodies glow at shorter wavelengths |
| Newton's law of cooling | dT/dt ∝ (T − T₀) | valid for a small temperature difference |
How to remember this chapter
“One, two, three”
Linear, area and volume expansion
α : β : γ = 1 : 2 : 3. One dimension, two dimensions, three dimensions, in that order.
“Latent means hidden”
Why temperature stops during melting
During a change of state the heat goes into breaking bonds, not into raising temperature, so the thermometer does not move while ice melts.
“Fourth power punishes heat”
Stefan's law
Radiated power goes as T⁴, so doubling the absolute temperature multiplies the radiation by sixteen.
The mistakes that cost the most
- Using °C in Stefan's law or Wien's law. Both need kelvin, because both use absolute temperature.
- Forgetting the latent heat step when ice at 0 °C becomes water at 0 °C. Q = mcΔT gives zero there, and mL is the whole answer.
- Mixing up specific heat capacity and heat capacity. Specific heat is per kilogram; heat capacity is for the whole body.
- Treating β as 3α. Area uses 2α; only volume uses 3α.
- Applying Newton's law of cooling to a large temperature difference, where it stops being accurate.
10Thermodynamics
Thermodynamics is bookkeeping for energy: heat in, work out, and what is left inside. The sign convention is the whole battle. Get ΔQ = ΔU + ΔW right and the four processes fall out of it almost automatically.
The first law
| What | Formula | Symbols and notes |
|---|---|---|
| First law | ΔQ = ΔU + ΔW | heat supplied to the gas is positive, work done by the gas is positive |
| Work done by a gas | W = ∫p dV | the area under the p-V curve |
| Internal energy of an ideal gas | U = (f/2)nRT | f is the degrees of freedom; U depends on temperature alone |
The four processes
| What | Formula | Symbols and notes |
|---|---|---|
| Isothermal | T constant, ΔU = 0, W = nRT ln(V₂/V₁) | slow process, in contact with a reservoir |
| Adiabatic | Q = 0, ΔU = −W, pVγ = constant | fast process, insulated from the surroundings |
| Adiabatic work | W = (p₁V₁ − p₂V₂)/(γ − 1) | also nR(T₁ − T₂)/(γ − 1) |
| Isobaric | p constant, W = pΔV = nRΔT | Q = nCpΔT |
| Isochoric | V constant, W = 0, ΔQ = ΔU | all the heat raises the temperature |
Heat capacities
| What | Formula | Symbols and notes |
|---|---|---|
| Mayer's relation | Cp − Cv = R | per mole, for an ideal gas |
| Ratio of heat capacities | γ = Cp/Cv | 1.67 monatomic, 1.40 diatomic, 1.33 triatomic |
| Monatomic gas | Cv = 3R/2, Cp = 5R/2 | three translational degrees of freedom |
| Diatomic gas | Cv = 5R/2, Cp = 7R/2 | three translational plus two rotational |
Engines and refrigerators
| What | Formula | Symbols and notes |
|---|---|---|
| Efficiency of a heat engine | η = 1 − Q₂/Q₁ | work out over heat in |
| Carnot efficiency | η = 1 − T₂/T₁ | kelvin only; the highest possible between two temperatures |
| Coefficient of performance | β = Q₂/W = T₂/(T₁ − T₂) | for a refrigerator, and it can exceed 1 |
How to remember this chapter
“Heat in, work out”
The first law sign convention
Heat supplied to the gas is positive, work done by the gas is positive. ΔQ = ΔU + ΔW then reads: what you put in either stays inside or comes out as work.
“Isothermal is slow, adiabatic is sudden”
Telling the two apart
Isothermal needs time for heat to flow, so it is slow. Adiabatic gives heat no time at all, so it is sudden, like a bursting tyre.
“Gamma: 5 over 3, 7 over 5”
Values of γ
Monatomic 5/3 = 1.67, diatomic 7/5 = 1.40. More ways to store energy means a smaller γ.
“Carnot is the ceiling”
Maximum possible efficiency
No engine between the same two temperatures can beat 1 − T₂/T₁, and reaching it would need infinitely slow, reversible operation.
The mistakes that cost the most
- Using °C in the Carnot efficiency. Both temperatures must be in kelvin.
- Saying ΔU = 0 for an adiabatic process. That is the isothermal one; in an adiabatic process Q = 0 instead.
- Forgetting that internal energy of an ideal gas depends only on temperature, so any process returning to the same T has ΔU = 0.
- Using W = pΔV when the pressure is not constant. Then the work is the area under the curve, not a product.
- Expecting a coefficient of performance below 1. For a refrigerator it is normally greater than 1, and that is not a violation of anything.
11Kinetic Theory of Gases
Kinetic theory explains pressure and temperature by counting molecular collisions. It is the bridge between mechanics and thermodynamics, and its three speeds, with the order between them, are asked nearly every year.
The gas laws
| What | Formula | Symbols and notes |
|---|---|---|
| Ideal gas equation | pV = nRT = NkT | R = 8.314 J mol−1 K−1, k = 1.38 × 10−23 J K−1 |
| Boltzmann constant | k = R/NA | the gas constant per molecule |
| Pressure from kinetic theory | p = ⅓ρvrms² = ⅓(mN/V)vrms² | ρ is the gas density |
Molecular speeds
| What | Formula | Symbols and notes |
|---|---|---|
| Root mean square speed | vrms = √(3RT/M) = √(3kT/m) | M is the molar mass in kg per mole |
| Average speed | vav = √(8RT/πM) | slightly smaller than vrms |
| Most probable speed | vmp = √(2RT/M) | the peak of the Maxwell distribution |
| Order of the three | vrms > vav > vmp | in the ratio √3 : √(8/π) : √2 |
Energy and degrees of freedom
| What | Formula | Symbols and notes |
|---|---|---|
| Kinetic energy per molecule | KE = (3/2)kT | depends only on temperature, not on the gas |
| Kinetic energy per mole | KE = (3/2)RT | for a monatomic ideal gas |
| Law of equipartition | each degree of freedom gets ½kT per molecule | ½RT per mole |
| Degrees of freedom | monatomic 3, diatomic 5, triatomic 6 | at ordinary temperatures |
| Mean free path | λ = 1/(√2 nπd²) | n is molecules per unit volume, d the molecular diameter |
How to remember this chapter
“Root three, root eight by pi, root two”
The three molecular speeds
vrms uses 3, average uses 8/π, most probable uses 2, all over M. The numbers fall in that order, and so do the speeds.
“Temperature is just kinetic energy”
What temperature means
Average translational kinetic energy is (3/2)kT for every gas. At the same temperature hydrogen and oxygen molecules carry the same energy, so the lighter one moves faster.
“Half kT per freedom”
Equipartition
Every degree of freedom, translational or rotational, gets ½kT per molecule. Count the freedoms and you have the internal energy.
The mistakes that cost the most
- Using grams per mole for M in the speed formulas. They need kilograms per mole, so 32 g per mole is 0.032.
- Thinking a heavier gas is hotter because it has more energy. At the same temperature every gas has the same average kinetic energy per molecule.
- Getting the speed order backwards. Root mean square is the largest of the three, most probable the smallest.
- Using pV = nRT with temperature in °C. Every gas law needs kelvin.
- Forgetting that an ideal gas has no intermolecular force, so its internal energy is entirely kinetic.
12Oscillations
Simple harmonic motion is one equation wearing many costumes: a spring, a pendulum, a floating block, a vibrating column. Once you can spot that the restoring force is proportional to displacement, the time period follows in one line.
Describing SHM
| What | Formula | Symbols and notes |
|---|---|---|
| Displacement | x = A sin(ωt + φ) | A amplitude, ω angular frequency, φ phase constant |
| Velocity | v = Aω cos(ωt + φ) = ω√(A² − x²) | maximum at the mean position, zero at the ends |
| Acceleration | a = −ω²x | always directed towards the mean position; this defines SHM |
| Angular frequency | ω = 2π/T = 2πf | T is the time period |
| Maximum values | vmax = Aω, amax = Aω² | at x = 0 and x = A respectively |
Time periods
| What | Formula | Symbols and notes |
|---|---|---|
| Spring-mass system | T = 2π√(m/k) | independent of gravity, so it works the same on the Moon |
| Springs in series | 1/k = 1/k₁ + 1/k₂ | softer than either alone |
| Springs in parallel | k = k₁ + k₂ | stiffer than either alone |
| Simple pendulum | T = 2π√(l/g) | independent of the mass of the bob |
| Pendulum in a lift | T = 2π√(l/(g ± a)) | g + a going up, g − a going down |
| Floating cylinder | T = 2π√(hρ/ρlg) | h is the submerged height |
Energy in SHM
| What | Formula | Symbols and notes |
|---|---|---|
| Kinetic energy | KE = ½mω²(A² − x²) | greatest at the mean position |
| Potential energy | PE = ½mω²x² | greatest at the extremes |
| Total energy | E = ½mω²A² = ½kA² | constant, and proportional to the square of the amplitude |
| Frequency of energy | energy varies at twice the frequency of displacement | because both KE and PE peak twice per cycle |
How to remember this chapter
“Acceleration towards the middle, size follows distance”
The test for SHM
If a = −ω²x, it is simple harmonic motion. Restoring, proportional, opposite in sign: all three must hold.
“Spring ignores gravity, pendulum lives on it”
Which time period contains g
T = 2π√(m/k) has no g at all, so a spring keeps its period on the Moon. T = 2π√(l/g) does, so a pendulum runs slower there.
“Fast in the middle, still at the ends”
Where velocity and acceleration peak
Velocity is largest at the mean position where acceleration is zero; acceleration is largest at the extremes where velocity is zero.
The mistakes that cost the most
- Thinking the time period depends on the amplitude. For SHM it does not, which is what makes pendulum clocks possible.
- Adding springs in series like resistors in parallel. Series springs add their reciprocals; parallel springs add directly.
- Forgetting the minus sign in a = −ω²x, which is the part that makes the motion oscillate at all.
- Using the length of the string alone for a physical pendulum. Only a simple pendulum uses T = 2π√(l/g).
- Saying total energy is proportional to the amplitude. It goes as the amplitude squared.
13Waves
Waves carries the most formulas of any Class 11 physics chapter, and the most marks. Sound, strings, pipes, beats and Doppler all sit here, and the Doppler sign convention is the single most common place students lose an easy three marks.
Wave basics
| What | Formula | Symbols and notes |
|---|---|---|
| Wave equation | v = fλ | speed, frequency, wavelength, for every wave there is |
| Progressive wave | y = A sin(ωt − kx) | moving in the positive x direction |
| Wave number | k = 2π/λ | also called the propagation constant |
| Speed on a string | v = √(T/μ) | T tension, μ mass per unit length |
| Speed of sound in a gas | v = √(γp/ρ) | Laplace's correction; Newton's version used p/ρ and was wrong |
| Speed of sound and temperature | v = 331 + 0.61t m s−1 | t in °C, for air |
Stationary waves
| What | Formula | Symbols and notes |
|---|---|---|
| Stationary wave | y = 2A sin(kx) cos(ωt) | nodes stay still, antinodes move most |
| Distance between nodes | λ/2 | and node to next antinode is λ/4 |
| String fixed at both ends | fn = n·v/2L | all harmonics present, n = 1, 2, 3, … |
| Open organ pipe | fn = n·v/2L | all harmonics present |
| Closed organ pipe | fn = (2n − 1)v/4L | odd harmonics only, and the fundamental is half that of an open pipe |
| End correction | closed pipe: L + 0.6r | r is the radius of the pipe |
Beats and Doppler
| What | Formula | Symbols and notes |
|---|---|---|
| Beat frequency | fbeat = |f₁ − f₂| | audible when the difference is under about 10 Hz |
| Doppler, general | f' = f(v ± vo)/(v ∓ vs) | upper signs for approach, lower for recession |
| Source approaching, observer still | f' = f·v/(v − vs) | pitch rises |
| Observer approaching, source still | f' = f(v + vo)/v | pitch rises, but by a different amount |
| Intensity | I ∝ A² | and I ∝ 1/r² for a point source |
How to remember this chapter
“Towards means higher”
Every Doppler question
Any approach raises the pitch, any recession lowers it. Work out the direction first, then choose the signs to match that answer. If your formula disagrees with the direction, the signs are wrong.
“Closed pipes are odd”
Harmonics in pipes
A closed pipe gives odd harmonics only, and its fundamental is half that of an open pipe of the same length.
“Node to node is half”
Spacing in a stationary wave
Node to node is λ/2, node to antinode is λ/4. Almost every stationary wave numerical comes down to this.
“Tighter and lighter is faster”
Speed on a string
v = √(T/μ), so more tension speeds a wave up and more mass per unit length slows it down. That is what tuning a guitar does.
The mistakes that cost the most
- Mixing up the source and observer speeds in the Doppler formula. The observer's speed changes the top, the source's changes the bottom.
- Treating beats as a sum of frequencies. Beat frequency is the difference, and only the difference.
- Using all harmonics for a closed pipe. Only the odd ones exist there.
- Forgetting that sound cannot travel through vacuum while light can, because sound needs a medium.
- Thinking the speed of sound depends on its frequency or loudness. It depends on the medium and its temperature.
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