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Position, velocity and acceleration
∆x/∆t = vav ∆v/∆t = aav dx/dt = v(t) dv/dt = a(t) ∫ a(t)dt = ∆v
v(t)dt = ∆x
Uniformly accelerated motion
v = v 0 + at x = x 0 + v 0 t + 12 at^2
v^2 = v^20 + 2a(x − x 0 ) x = x 0 +
v 0 + v 2
t
Projectile motion 2D (uniform field g=const.) x′(0) = ˙x(0) = vx(0) = v(0) cos θ = v 0 cos θ y′(0) = ˙y(0) = vy (0) = v(0) sin θ = v 0 sin θ
y′′^ = −g = const. x′′^ = 0 = const. y′(t) = y′(0) − gt x′(t) = x′(0) = const. y(t) = y(0) + y′(0)t − 12 gt^2 x(t) = x(0) + x′(0)t
Alternative form
ay = −g = const. ax = 0 = const. vy (t) = vy (0) − gt vx(t) = vx(0) = const. y(t) = y(0) + vy (0)t − 12 gt^2 x(t) = x(0) + vx(0)t
Trajectory equation for x(0) = 0
y(x) =
[
vy (0) vx(0)
]
x −
[
g v x^2 (0)
]
x^2 + y(0)
y(x) = x tan θ −
[
g v^2 (0) cos^2 θ
]
x^2 + y(0)
Velocity-position equations
v^2 (y) = v^2 (0) − 2 gy v^2 y (y) = v^2 y (0) − 2 gy
Special points: Range R, height h, flight time T
h = v^2 y (0)/ 2 g h = v^2 (0) sin^2 θ/ 2 g R = 2vx(0)vy (0)/g R = v^2 (0) sin 2θ/g T = 2vy (0)/g T = 2v(0) sin θ/g
R = 4h cot θ
Circular motion
Centripetal acceleration ar = v^2 /r = rω^2
Arc length s = rθ
Tangential speed v = rω = 2πr/T
Tangetial acceleration at = rα
Angular frequency ω = 2πf = 2π/T
Frequency and time period f = 1/T
Uniform circular motion at = 0, α = 0
Forces and Momentum
Newton’s second law (general) F~ = d~p/dt
Potential energy and force (1D) F = −dU/dx Potential energy and force (3D) F~ = −∇U Linear Momentum ~p = m~v Friction (static) fs ≤ fs,max = μsN Friction (kinetic) fk = μkN
Grav. fields due to point or spherical sources
Force between masses F = Gmm′/r^2 Gravity field of mass m g = Gm/r^2 G.P.E. two masses U = −Gmm′/r Grav. potential of m V = −Gm/r
Orbital motion
Kepler’s 2nd^ Law T 2 = (4π^2 /GM )r^3 Orbit (circular) v^2 = GM/r Escape velocity v^2 = 2GM/r
Constants related to gravity
Universal const. of gravitation G = 6. 67 × 10 −^11 N · m^2 /kg^2 Earth surface gravity g = 9.81 m/s^2 Earth mass & G GME = 3. 98 × 1014 m^3 /s^2 Solar mass & G GM = 1. 33 × 1020 m^3 /s^2 Moon mass & G GM$ = 4. 91 × 1012 m^3 /s^2
Work and energy
Kinetic energy K = 12 mv^2
Work W =
F^ ~ · d~r
Power P = dE/dt = dW/dt Average Power Pav = ∆E/∆t = W/∆t Instantaneous Power P = F~ · ~v = F‖v Work-energy theorem Wnet = Wc + Wnc = ∆K Work done by con. forces Wc = −∆U Mechanical energy Emech = K + U Conservation of mech. energy Ki + Ui + Wnc = Kf + Uf Work done by non-con. forces Wnc = −∆Emech GPE uniform field U (y) = mgy + U (0) Potential uniform grav. field V (y) = gy + V (0) GPE uniform field ∆Ugrav. = mg∆y = mgh Mechanical energy Emech. = K + Utotal
Centre of mass
R^ ~cm = 1 M
mi~ri R~cm =
M
~rdm =
M
~rρdV
Theorems for variable forces
Impulse-momentum J~ = ∆~p = F~av∆t =
F^ ~net(t)dt
Work-energy Wnet =
F^ ~net · d~r = ∆K
Types of collision
- totally elastic: No loss of K.E. , e = 1
- inelastic: Some loss of K.E., 0 < e < 1
- completely inelastic: v 1 = v 2 = v, e = 0 Max K.E. loss
Collision conservation laws (1D & 2D)
Momentum m 1 ~u 1 + m 2 ~u 2 = m 1 ~v 1 + m 2 ~v 2 K.E. (elastic only) 12 m 1 u^21 + 12 m 2 u^22 = 12 m 1 v 12 + 12 m 2 v^22
Newton’s collision law (1D only)
Newton’s collision law (1D) (v 2 − v 1 ) = −e(u 2 − u 1 )
Collisions 1D Elastic (derived from the above)
v 1 = m 1 − m 2 m 1 + m 2
u 1 + 2 m 2 m 1 + m 2
u 2
v 2 =
2 m 1 m 1 + m 2
u 1 +
m 2 − m 1 m 1 + m 2
u 2
Collisions 1D Inelastic (derived from the above)
v 1 =
m 1 − em 2 m 1 + m 2
u 1 +
(1 + e)m 2 m 1 + m 2
u 2
v 2 =
(1 + e)m 1 m 1 + m 2 u 1 +
m 2 − em 1 m 1 + m 2 u 2
Rotational motion
Anguar velocity, acceleration ω = dθ/dt, α = dω/dt
Angular displacement ∆θ =
ωdt
Linear and angular connection vt = Rω, at = Rα Torque ~Γ = ~r × F~ Magnitude of torque Γ = rF sin ϕ = rF⊥ Angular momentum (particle) L~ = ~r × ~p Angular momentum (solid) L~ = I~ω Moment of inertia (particles) I = Σ mr^2 axis
Moment of inertia (solid) I =
r^2 axisdm
N2 for rotation (general form) ~Γ = d~L/dt N2 for rotation (I=const.) ~Γ = I~α Rotational K.E. Kr = 12 Iω^2
Work done by a torque W =
Γ · dθ = ∆Kr
Work done by const. or av. torque W = Γ · ∆θ = ∆Kr Rotational power P = Γω Conservation of ~L Iiωi = If ωf Rolling without slipping vcm = Rω, acm = Rα Parallel axis theorem I = Icm + M D^2
Total kinetic energy I = 12 Icmω^2 + 12 M v cm^2
Rotational motion with (α = const.)
ω = ω 0 + αt ∆θ = ω 0 t + 12 αt^2
ω^2 = ω^20 + 2α∆θ ∆θ =
ω 0 + ω 2 t
Substitutions for rotational dynamics
s =⇒ ∆θ F~ =⇒ ~Γ u =⇒ ω 0 m =⇒ I v =⇒ ω K = 12 mv^2 =⇒ Kr = 12 Iω^2 a =⇒ α ~p = m~v =⇒ ~L = I~ω
Moments of inertia
Moment Object Axis I = M R^2 Uniform ring/tube Through C.M. I = 12 M R^2 Uniform disk/cylinder Through C.M. I = 121 M L^2 Uniform rod Through C.M. I = 13 M L^2 Uniform rod Through end I = 25 M R^2 Uniform sphere Through C.M. I = 23 M R^2 Hollow sphere Through C.M. I = 13 M a^2 Slab width a Along edge (door)
Simple harmonic motion (SHM)
Hooke’s Law F (x) = −kx acceleration a(x) = −ω^2 x = −n^2 x Velocity v(x) = ±ω
A^2 − x^2 SPE or EPE for a spring U (x) = 12 kx^2 Total energy E = 12 kA^2 = 12 mω^2 A^2 Position x(t) x(t) = A cos(ωt + ϕ) Velocity v(t) v(t) = −Aω sin(ωt + ϕ) Acceleration a(t) a(t) = −Aω^2 cos(ωt + ϕ)
Period, mass-spring T =
f
2 π ω
= 2π
m k
Period, simple pendulum T =
f
2 π n
= 2π
l g
Period, physical pendulum T =
f
2 π n
= 2π
I
mgr
Period, torsional pendulum T =
f
2 π n
= 2π
I
κ
Trigonometry
cos(±π/6) = sin π/3 = sin(2π/3) =
cos(±π/3) = sin π/6 = sin(5π/6) = 1/ 2 cos(±π/4) = sin π/4 = sin(3π/4) = 1/
cos(± 5 π/6) = sin(−π/3) = sin(− 2 π/3) = −
cos(± 2 π/3) = sin(−π/6) = sin(− 5 π/6) = − 1 / 2 cos(± 3 π/4) = sin(−π/4) = sin(− 3 π/4) = − 1 /
a^2 = b^2 + c − 2 bc cos A Law of cosines a sin A
b sin B
c sin C
Law of sines
sin (θ ± φ) = sin θ cos φ ± cos θ sin φ cos (θ ± φ) = cos θ cos φ ∓ sin θ sin φ
sin(π ± θ) = ∓ sin θ sin(π/ 2 ± θ) = cos θ cos(π ± θ) = − cos θ cos(π/ 2 ± θ) = ∓ sin θ sin(θ ± π) = − sin θ sin(θ ± π/2) = ± cos θ cos(θ ± π) = − cos θ cos(θ ± π/2) = ∓ sin θ
sin(ωt ± π) = − sin ωt sin(ωt ± π/2) = ± cos ωt cos(ωt ± π) = − cos ωt cos(ωt ± π/2) = ∓ sin ωt
sin^2 θ + cos^2 θ = 1 sin 2θ = 2 sin θ cos θ cos 2θ = cos^2 θ − sin^2 θ
Small angle formulae for small θ 1 (in radians)
sin θ ≈ θ cos θ ≈ 1 − θ^2 / 2 tan θ ≈ θ
Inverse trig functions where α = principal value
cos θ = cos α =⇒ θ = ±α + 2nπ sin θ = sin α =⇒ θ = (−1)nα + nπ tan θ = tan α =⇒ θ = α + nπ
Binomial formulae
(1 + x)n^ = 1 + nx +
n(n − 1)x^2 + ... if |x| 1
(a + b)n^ =
∑^ n
r=
nCr an−r (^) br (^) integer n
Combinatorics
nCr = n! r!(n − r)!
nPr = n! (n − r)!
Quadratic equation y = ax^2 + bx + c
Roots at x = − b 2 a
b^2 − 4 ac 2 a
max, min at x = −b/ 2 a
Linear Equation y = mx + b
Given m, (x 1 , y 1 ) y − y 1 = m(x − x 1 )
Given (x 1 , y 1 ), (x 2 , y 2 ) y − y 1 =
y 2 − y 1 x 2 − x 1
(x − x 1 )
Dot and cross product
A^ ~ · B~ = AxBx + Ay By + Az Bz = | A~|| B~| cos θ | A~ × B~| = | A~|| B~| sin θ
Differential equations
dy dt
- c 1 y = c 2 y(0) = 0 y(t) = c 2 c 1
1 − e−c^1 t
dy dt
- c 1 y = 0 y(0) > 0 y(t) = y(0)e−c^1 t
Exponential behaviour
y(t) = y(0)e−t/τ^ = y(0)e−λt^ Exponential decay y(t) = y(0)2−t/Thalf^ Exponential decay Thalf = τ ln 2 Half life
y(t) = ymax
1 − e−t/τ^
Exponential growth
Percent difference between quantities A, B
% diff (A, B) =
|A − B|
av(A, B)
× 100 =
|A − B|
A + B
× 200
Percent error
% error =
|measured − true| true
× 100
Possibly useful integrals ∫ dx (x^2 ± a^2 )^3 /^2
±x a^2
x^2 ± a^2
∫ (^) π
0
sin^3 θdθ =
xdx (x^2 ± a^2 )^3 /^2
∫ x^2 ±^ a^2 dx √ x^2 ± a^2
= ln
[√
x^2 ± a^2 + x
]
xdx √ x^2 + a^2
x^2 + a^2 ∫ dx √ a^2 − x^2
= arcsin(x/a)
Taylor series x − a = h, |x − a| = |h| < 1
f (x) = f (a) + (x − a)f ′(a) +
f ′′(a)(x − a)^2 + ...
f (a + h) = f (a) + hf ′(a) +
h^2 2!
f ′′(a) + ...
Mathematical constants
e = 2. 71828 ... 1 o^ = 1. 745 × 10 −^2 rad π = 3. 14159 ... 1 ′^ = 2. 9089 × 10 −^4 rad log 10 e = 0. 434 ... 1 ′′^ = 4. 8481 × 10 −^6 rad ln 10 = 2. 3025 ... 1 rad = 57. 296 o ln 2 = 0. 693 ... π/6 rad = 30o e−^1 = 0. 368 ... π/3 rad = 60o (1 − e−^1 ) = 0. 632 ... π/4 rad = 45o √ 3 /2 = 0. 866 ... 1 rpm = 0.1047 rad/s 1 /
2 = 0. 707 ... 1 rad/s = 9.549 rpm
Greek alphabet
Letter Upper case Lower case Alpha A α Beta B β Gamma Γ γ Delta ∆ δ Epsilon E , ε Zeta Z ζ Eta H η Theta Θ θ Iota I ι Kappa K κ Lambda Λ λ Mu M μ Nu N ν Xi Ξ ξ Omicron O o Pi Π π Rho P ρ Sigma Σ σ Tau T τ Upsilon Y υ Phi Φ φ, ϕ Chi X χ Psi Ψ ψ Omega Ω ω
SI units and derived units
Quantity Symbol Unit Name Basic Units Mass m kg kilogram kg Length l m meter m Time t s second s Force F N Newton kg ms−^2 Energy E J Joule kg m^2 s−^2 Power P W = Js−^1 Watt kg m^2 s−^3 Pressure p Pa = N.m^2 Pascal kg/ms^2
Abbreviations used: atm.=atmosphere (pressure) con. = conservative (force) AC = Alternating Current BVP = Boundary Value Problem CM = Centre of Mass DC = Direct Current (or Detective Comics) EM or E&M = ElectroMagnetism EMF = ElectroMotive Force (voltage) EPE = Elastic Potential Energy GR = General Relativity GPE = Gravitational Potential Energy G.T. = Galilean Transformation IC = Initial Condition IVP = Initial Value Problem ODE = Ordinary Differential Equation PD = Potential Difference PDE = Partial Differential Equation PE = Potential Energy L.T. = Lorentz Transformation SHM = Simple Harmonic Motion SHO = Simple Harmonic Oscillator SPE = Strain/Spring Potential Energy SR = Special Relativity STP = Standard Temperature and Pressure (20o^ C, 1 atm) TIR = Total Internal Reflection N1,N2,N3= Newton’s laws of motion T0,T1,T2,T3= the laws of thermal physics K1,K2,K3= Kepler’s laws of planetary motion
Metric Prefixes
exa E 1018 peta P 1015 tera T 1012 giga G 109 mega M 106 kilo k 103 hecto h 102 deci d 10 −^1 centi c 10 −^2 milli m 10 −^3 micro μ 10 −^6 nano n 10 −^9 pico p 10 −^12 femto f 10 −^15 atto a 10 −^18
Symbols used in mechanics:
A Amplitude for SHM A, A 1 , A 2 Cross sectional area of pipe a Acceleration at Tangential component of acceleration ar Radial component of acceleration e Coefficient of resitution E Total energy F , Fav Force, average force f Frequency (rev/second or cycles/second) f Friction (force) G Universal gravitation constant g Gravitational field strength h depth or height I Moment of inertia J^ ~ Impulse (change in momentum J~ = ∆~p) K Kinetic energy Kr Rotational kinetic energy k Spring constant k wavenumber 2π/λ L~ Angular momentum l Length M , m Mass n Normal force P Power Pav Average power p Momentum r radius s Displacement T Time period/ time of flight T tension U Potential energy u velocity at time t = 0 v velocity at time t W Work Wc Work done by a con. force(s) Wnc Work done by non-con. force(s) Wnet Work done by net force Y Young’s modulus
α Angular acceleration (rad/s^2 ) ∆ change in... μk Coefficient of kinetic friction μs Coefficient of static friction ω Angular speed at time t (rad/s) ω 0 Angular speed at time t = 0 (rad/s) ∆θ angular displacement ∆θ = θ − θ 0 θ 0 Angular position at time t = 0 Γ Torque ρ density (mass/volume)
