Formula Sheet for Physics Mechanics

S h e e t f or Ph y si cs | pg. 1 Physics formulas from Mechanics, Waves, Optics, Heat and Thermodynamics, Electricity and Magnetism and Modern Physics. Also includes the value of Physical Constants. Helps in quick revision for CBSE, NEET, JEE Mains, and Advanced. c h hc G k R NA e µ0 Gravitation constant Boltzmann constant Molar gas constant Avogadro’s number Charge of electron Permeability of vacuum Permitivity of vacuum 0 1 Coulomb constant 4π0 Faraday constant F Mass of electron me Mass of proton mp Mass of neutron mn Atomic mass unit u u Atomic mass unit σ Stefan-Boltzmann constant Rydberg constant R∞ Bohr magneton µB Bohr radius a0 Standard atmosphere atm Wien displacement b constant s = ut + 21 at2 , v = u + at, 3 × 108 m/s 6.63 × 10−34 J s 1242 eV-nm 6.67 × 10−11 m3 kg−1 s−2 1.38 × 10−23 J/K 8.314 J/(mol K) 6.023 × 1023 mol−1 1.602 × 10−19 C 4π × 10−7 N/A2 u Projectile Motion: u cos θ 1 2 gt 2 x = ut cos θ, y = ut sin θ − g x2 2u2 cos2 θ u2 sin 2θ 2u sin θ , T = , R= g g y = x tan θ − H= u2 sin2 θ 2g 1.3: Newton’s Laws and Friction Linear momentum: p~ = m~v Newton’s first law: inertial frame. Newton’s second law: F~ = −1 1.097 × 10 m 9.27 × 10−24 J/T 0.529 × 10−10 m 1.01325 × 105 Pa 2.9 × 10−3 m K ~ = m~a F d~ p , dt Newton’s third law: F~AB = −F~BA Frictional force: fstatic, max = µs N, Banking angle: v2 rg = tan θ, v2 rg = mv2 , r fkinetic = µk N µ+tan θ 1−µ tan θ ac = Pseudo force: F~pseudo = −m~a0 , MECHANICS v2 r Fcentrifugal = − mv r 2 Minimum speed to complete vertical circle: p p vmin, bottom = 5gl, vmin, top = gl 1.1: Vectors Notation: ~a = ax ı̂ + ay ̂ + az k̂ q Magnitude: a = |~a| = a2x + a2y + az2 l Conical pendulum: T = 2π Dot product: ~a · ~b = ax bx + ay by + az bz = ab cos θ Cross product: x θ R 8.85 × 10−12 F/m 9 × 109 N m2 /C2 96485 C/mol 9.1 × 10−31 kg 1.6726 × 10−27 kg 1.6749 × 10−27 kg 1.66 × 10−27 kg 931.49 MeV/c2 5.67 × 10−8 W/(m2 K4 ) 7 y H O Centripetal force: Fc = 1 v2 − u2 = 2as Relative Velocity: ~vA/B = ~vA − ~vB 0.1: Physical Constants Speed of light Planck constant Motion in a straight line with constant a: u sin θ F or m u l ae ~ a ×~ b l cos θ g ~ a k̂ mg ̂ ~a ×~b = (ay bz − az by )ı̂ + (az bx − ax bz )̂ + (ax by − ay bx )k̂ |~a × ~b| = ab sin θ θ θ T ı̂ ~ b θ q 1.4: Work, Power and Energy ~ = F S cos θ, Work: W = F~ · S W = Kinetic energy: K = 21 mv2 = p2 2m R ~ · dS~ F Potential energy: F = −∂U/∂x for conservative forces. 1.2: Kinematics Average and Instantaneous Vel. and Accel.: ~vav = ∆~r/∆t, ~vinst = d~r/dt ~aav = ∆~v /∆t ~ainst = d~v /dt Ugravitational = mgh, Uspring = 12 kx2 Work done by conservative forces is path independent and depends only on initial and final points: H F~conservative · d~r = 0. Work-energy theorem: W = ∆K F or m u l ae S h e e t f or Ph y si cs | pg. 2 Mechanical energy: E = U + K. Conserved if forces are conservative in nature. Power Pav = ∆W ∆t , Pinst = F~ · ~v P x m Pi i, mi ω = ω0 + αt, θ = ωt + 21 αt2 , Moment of Inertia: I = 1.5: Centre of Mass and Collision Centre of mass: xcm = Rotation about an axis with constant α: xcm = R R xdm dm P m i ri 2 , i ω2 − ω0 2 = 2αθ I= R r 2 dm mr2 1 mr2 2 2 mr2 3 2 mr2 5 1 ml2 12 mr2 1 mr2 m(a2 +b2 ) 2 12 ring disk shell sphere rod hollow solid rectangle b a CM of few useful configurations: m1 m2 r 1. m1 , m2 separated by r: C m2 r m1 +m2 2. Triangle (CM ≡ Centroid) yc = 3. Semicircular ring: yc = 2r π 4. Semicircular disc: yc = 4r 3π 6. Solid Hemisphere: yc = h 3 C Ik Theorem of Parallel Axes: Ik = Icm + md2 h 3 C r 3r 8 z y x 2r π C p I/m 4r 3π Angular Momentum: L~ = ~r × p~, C r 2 Torque: ~τ = ~r × F~ , C 3r 8 r cm Theorem of Perp. Axes: Iz = Ix + Iy Radius of Gyration: k = r Ic d h r r 2 5. Hemispherical shell: yc = m1 r m1 +m2 7. Cone: the height of CM from the base is h/4 for the solid cone and h/3 for the hollow cone. ~τ = ~ dL , dt ~L = I~ ω y τ = Iα O P θ ~ F ~ r x ~ ~τext = 0 =⇒ ~L = const. Conservation of L: P~ P Equilibrium condition: F = ~0, ~τ = ~0 Kinetic Energy: Krot = 21 Iω 2 P Motion of the CM: M = mi P mi~vi , p~cm = M~vcm , ~vcm = M Impulse: J~ = R Dynamics: ~τcm = Icmα ~, F~ext M ~acm = F~ext = m~acm , ~pcm = m~vcm ~L = Icm ~ω + ~rcm × m~vcm K = 21 mvcm 2 + 21Icm ω 2 , F~ dt = ∆~p 1.7: Gravitation Before collision After collision Collision: m1 m2 v1 m1 m2 v10 v2 v20 Momentum conservation: m1 v1 +m2 v2 = m1 v 01 +m2 v20 2 2 Elastic Collision: 12 m1 v1 2+ 12 m2 v2 2 = 21m1 v 01 +21 m2 v 02 Coefficient of restitution:  −(v 01 − v20 ) 1, completely elastic e= = 0, completely in-elastic v1 − v2 If v2 = 0 and m1  m2 then v 01 = −v1 . If v2 = 0 and m1  m2 then v 02 = 2v1 . Elastic collision with m1 = m2 : v 01 = v2 and v20 = v1 . Angular Accel.: αav = ∆θ , ∆t ∆ω , ∆t F F r Potential energy: U = − GMr m Gravitational acceleration: g = GM R2   Variation of g with depth: ginside ≈ g 1 − 2h R   Variation of g with height: goutside ≈ g 1 − hR Effect of non-spherical earth shape on g : gat pole > gat equator (∵ Re − Rp ≈ 21 km) Effect of earth rotation on apparent weight: 1.6: Rigid Body Dynamics Angular velocity: ωav = m1 2 Gravitational force: F = G mr1 m 2 ω= α= dθ , dt dω dt , ~v = ~ω × ~r ~a = α ~ × ~r m2 F or m u l ae S h e e t f or Ph y si cs | pg. 3 ω ~ ~ A 2 mω R cos θ mg mg 0θ = mg − mω2 R cos2 θ Superposition of two SHM’s: x1 = A1 sin ωt, Escape velocity: ve = q q GM R 2GM R vo 1.9: Properties of Matter a Modulus of rigidity: Y = First: Elliptical orbit with sun at one of the focus. Second: Areal velocity is constant. (∵ d ~L/dt = 0). 4π 2 3 Third: T 2 ∝ a3 . In circular orbit T 2 = GM a . 1.8: Simple Harmonic Motion Time period: T = 2π ω k x = −ω 2 x = −m pm = 2π k Velocity: v = Aω cos(ωt + φ) = ±ω A2 lateral strain longitudinal strain Elastic energy: U = 1 2 − stress × strain × volume ∆pair = 2S/R, 0 1 mv2 2 2S cos θ rρg Hydrostatic pressure: p = ρgh K q 0 x A Buoyant force: FB = ρV g = Weight of displaced liquid Equation of continuity: A1 v1 = A2 v2 l dv Viscous force: F = −ηA dx F I mgl q Stoke’s law: F = 6πηrv v Poiseuilli’s equation: I k Volume flow time Terminal velocity: vt = 1 k1 + v2 v1 Bernoulli’s equation: p + 21 ρv2 + ρgh = constant √ Torricelli’s theorem: vefflux = 2gh l g q Torsional Pendulum T = 2π = ∆psoap = 4S/R x A Total energy: E = U + K = 21 mω2 A2 1 keq ∆D/D ∆l/l U −A Springs in series: = Excess pressure in bubble: x2 −A Physical Pendulum: T = 2π F Aθ , η= dV dP Poisson’s ratio: σ = Capillary rise: h = Simple pendulum: T = 2π ∆P ∆V Surface energy: U = SA Potential energy: U = 12 kx2 Kinetic energy K = B = −V Surface tension: S = F/l Displacement: x = A sin(ωt + φ) √ F/A , ∆l/l Compressibility: K = B1 = − V1 Hooke’s law: F = −kx (for small elongation x.) d2 x dt2 x2 = A2 sin(ωt + δ ) x = x1 + x2 = A sin(ωt + ) q A = A1 2 + A2 2 + 2A1 A2 cos δ A2 sin δ tan  = A1 + A2 cos δ Kepler’s laws: Acceleration: a = δ ~1 A θ R Orbital velocity of satellite: vo = ~2 A  1 k2 Springs in parallel: keq = k1 + k2 k1 k2 k2 k1 = 2r 2 (ρ−σ)g 9η πpr4 8ηl r l