Motion in a straight line with constant a:
v = u + at, s = ut +12at2, v2−u2= 2as
Relative Velocity: ⃗vA/B = ⃗vA −⃗vB
0.1: Physical Constants
Speed of light c 3 × 108m/s
u
Planck constant h 6.63 × 10−34J s y
hc 1242 eV-nm H
x
Gravitation constant G 6.67×10−11m3kg−1s−2 Projectile Motion: θ
Boltzmann constant k O
1.38 × 10−23J/K u cos θ
Molar gas constant R 8.314 J/(mol K) R
Avogadro’s number NA 6.023 × 1023mol−1
Charge of electron e 1.602 × 10−19C x = ut cos θ, y = ut sin θ −12gt2
Permeability of vac- µ0 4π × 10−7N/A2 y = x tan θ − 2u2cos2θx2
uum
Permitivity of vacuum ϵ0 8.85 × 10−12F/m T = 2u sin θ R =u2 sin 2θ H =u2 sin2 θ
Coulomb constant 1
9 × 109N m2/C2 , , 2g
4πϵ0 g g
Faraday constant F 96485 C/mol
Mass of electron me 9.1 × 10−31kg 1.3: Newton’s Laws and Friction
Mass of proton mp 1.6726 × 10−27kg
Mass of neutron mn 1.6749 × 10−27kg Linear momentum: ⃗p = m⃗v
Atomic mass unit u 1.66 × 10−27kg
Atomic mass unit u 931.49 MeV/c2 Newton’s first law: inertial frame.
Stefan-Boltzmann σ 5.67×10−8W/(m2K4) Newton’s second law:⃗F =d⃗p ⃗F = m⃗a
constant dt,
Rydberg constant R∞ 1.097 × 107m−1 Newton’s third law:⃗FAB = −⃗FBA
Bohr magneton µB 9.27 × 10−24J/T
Bohr radius a0 0.529 × 10−10m Frictional force: fstatic,max = µsN, fkinetic = µkN
Standard atmosphere atm 1.01325 × 105Pa
2.9 × 10−3m K µ+tan θ
Wien displacement b Banking angle: rg= tan θ, v2 rg= 1−µ tan θ
constant
Centripetal force: Fc =mv2 ac =v2
r, r
Pseudo force:⃗Fpseudo = −m⃗a0, Fcentrifugal = −mv2
1 MECHANICS
Minimum speed to complete vertical circle:
1.1: Vectors
vmin,bottom = 5gl, vmin,top = gl
Notation: ⃗a = ax ˆı + ay ˆ+ azˆk
Magnitude: a = |⃗a| = a2x + a2y + a2z l
θ
l cos θ
Conical pendulum: T = 2π θ T
Dot product: ⃗a ·⃗b = axbx + ayby + azbz = ab cos θ g
ˆı mg
⃗a ×⃗b ⃗b
Cross product:
θ ˆk ˆ
⃗a
1.4: Work, Power and Energy
⃗a×⃗b = (aybz −azby)ˆı+(azbx −axbz)ˆ+(axby −aybx)ˆk Work: W =⃗F ·⃗S = FS cos θ, W = ⃗F · d⃗S
|⃗a ×⃗b| = ab sin θ Kinetic energy: K =1 2mv2 = p2
Potential energy: F = −∂U/∂x for conservative forces.
1.2: Kinematics
Ugravitational = mgh, Uspring =1 2kx2
Average and Instantaneous Vel. and Accel.:
⃗vav = ∆⃗r/∆t, ⃗
⃗vinst = dr/dt Work done by conservative forces is path indepen-
dent and depends only on initial and final points:
⃗aav = ∆⃗v/∆t ⃗
⃗ainst = dv/dt ⃗Fconservative · d⃗r = 0.
Work-energy theorem: W = ∆K
, Mechanical energy: E = U +K. Conserved if forces are Rotation about an axis with constant α:
conservative in nature.
ω = ω0 + αt, θ = ωt +12αt2, ω2−ω0 2 = 2αθ
Power Pav =∆W∆t, Pinst =⃗F ·⃗v
Moment of Inertia: I = imiri2, I= r2dm
1.5: Centre of Mass and Collision
ximi xdm 2mr2
Centre of mass: xcm = mi , xcm = dm
mr2 2mr2 3mr2 5mr2 12ml2
mr2 m(a2+b2)
12
b
a
CM of few useful configurations: ring disk shell sphere rod hollow solid rectangle
m1 r m2
1. m1, m2 separated by r: C
m2 r m 1r I∥ Ic
m1+m2 m 1+m2
Theorem of Parallel Axes: I∥= Icm + md2 d
cm
2. Triangle (CM ≡Centroid) yc =h h
C
h
3 z y
Theorem of Perp. Axes: Iz = Ix + Iy
x
3. Semicircular ring: yc =2r C
2r
r π
Radius of Gyration: k = I/m
4. Semicircular disc: yc =4r C
r
4r
3π Angular Momentum:⃗L = ⃗r × ⃗p, ⃗
⃗L = Iω
5. Hemispherical shell: yc =r C r y
P θ ⃗F
r 2
Torque: ⃗τ = ⃗r ×⃗F, ⃗τ =d⃗L τ = Iα
dt, ⃗r
O x
6. Solid Hemisphere: yc =3r C 3r
r 8
Conservation of⃗L: ⃗τext = 0 =⇒⃗L = const.
7. Cone: the height of CM from the base is h/4 for Equilibrium condition:⃗F =⃗0, ⃗τ = 0
⃗
the solid cone and h/3 for the hollow cone.
Kinetic Energy: Krot =1 2Iω2
Motion of the CM: M =mi Dynamics:
⃗Fext ⃗τcm = Icm⃗α, ⃗Fext = m⃗acm, ⃗pcm = m⃗vcm
miv⃗ i
⃗vcm = ⃗ cm, ⃗acm =
⃗pcm = Mv
M , M K =1 2mvcm 2 + 12Icmω2, ⃗L = Icm⃗ω + ⃗rcm × m⃗vcm
Impulse:⃗J = ⃗F dt = ∆⃗p
1.7: Gravitation
Before collision After collision
Collision: m1 m2 m1 m2 m1 F F m2
Gravitational force: F = Gm1m2r2
v1 v2 v1′ v2′
r
Momentum conservation: m1v1+m2v2 = m1v′ 1+m2v′
Elastic Collision:1 2m1v12+12m2v22 = 12m1v′ 2+12m2v′ 2 Potential energy: U = −GMm
Coefficient of restitution:
Gravitational acceleration: g =GM
e =−(v′ 1−v′2) = 1, completely elastic
v1 −v2 0, completely in-elastic Variation of g with depth: ginside ≈g 1 −hR
Variation of g with height: goutside ≈g 1 −2h
R
If v2 = 0 and m1 ≪m2 then v′ 1= −v1.
If v2 = 0 and m1 ≫m2 then v′ 2= 2v1. Effect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : v′ 1= v2and v′ 2= v1. gatpole > gatequator (∵Re −Rp ≈21 km)
Effect of earth rotation on apparent weight:
1.6: Rigid Body Dynamics
Angular velocity: ωav =∆θ ∆t, ω =dθ ⃗v = ⃗ω × ⃗r
dt,
Angular Accel.: αav =∆ω α =dω ⃗a = α
⃗ × ⃗r
dt,
v = u + at, s = ut +12at2, v2−u2= 2as
Relative Velocity: ⃗vA/B = ⃗vA −⃗vB
0.1: Physical Constants
Speed of light c 3 × 108m/s
u
Planck constant h 6.63 × 10−34J s y
hc 1242 eV-nm H
x
Gravitation constant G 6.67×10−11m3kg−1s−2 Projectile Motion: θ
Boltzmann constant k O
1.38 × 10−23J/K u cos θ
Molar gas constant R 8.314 J/(mol K) R
Avogadro’s number NA 6.023 × 1023mol−1
Charge of electron e 1.602 × 10−19C x = ut cos θ, y = ut sin θ −12gt2
Permeability of vac- µ0 4π × 10−7N/A2 y = x tan θ − 2u2cos2θx2
uum
Permitivity of vacuum ϵ0 8.85 × 10−12F/m T = 2u sin θ R =u2 sin 2θ H =u2 sin2 θ
Coulomb constant 1
9 × 109N m2/C2 , , 2g
4πϵ0 g g
Faraday constant F 96485 C/mol
Mass of electron me 9.1 × 10−31kg 1.3: Newton’s Laws and Friction
Mass of proton mp 1.6726 × 10−27kg
Mass of neutron mn 1.6749 × 10−27kg Linear momentum: ⃗p = m⃗v
Atomic mass unit u 1.66 × 10−27kg
Atomic mass unit u 931.49 MeV/c2 Newton’s first law: inertial frame.
Stefan-Boltzmann σ 5.67×10−8W/(m2K4) Newton’s second law:⃗F =d⃗p ⃗F = m⃗a
constant dt,
Rydberg constant R∞ 1.097 × 107m−1 Newton’s third law:⃗FAB = −⃗FBA
Bohr magneton µB 9.27 × 10−24J/T
Bohr radius a0 0.529 × 10−10m Frictional force: fstatic,max = µsN, fkinetic = µkN
Standard atmosphere atm 1.01325 × 105Pa
2.9 × 10−3m K µ+tan θ
Wien displacement b Banking angle: rg= tan θ, v2 rg= 1−µ tan θ
constant
Centripetal force: Fc =mv2 ac =v2
r, r
Pseudo force:⃗Fpseudo = −m⃗a0, Fcentrifugal = −mv2
1 MECHANICS
Minimum speed to complete vertical circle:
1.1: Vectors
vmin,bottom = 5gl, vmin,top = gl
Notation: ⃗a = ax ˆı + ay ˆ+ azˆk
Magnitude: a = |⃗a| = a2x + a2y + a2z l
θ
l cos θ
Conical pendulum: T = 2π θ T
Dot product: ⃗a ·⃗b = axbx + ayby + azbz = ab cos θ g
ˆı mg
⃗a ×⃗b ⃗b
Cross product:
θ ˆk ˆ
⃗a
1.4: Work, Power and Energy
⃗a×⃗b = (aybz −azby)ˆı+(azbx −axbz)ˆ+(axby −aybx)ˆk Work: W =⃗F ·⃗S = FS cos θ, W = ⃗F · d⃗S
|⃗a ×⃗b| = ab sin θ Kinetic energy: K =1 2mv2 = p2
Potential energy: F = −∂U/∂x for conservative forces.
1.2: Kinematics
Ugravitational = mgh, Uspring =1 2kx2
Average and Instantaneous Vel. and Accel.:
⃗vav = ∆⃗r/∆t, ⃗
⃗vinst = dr/dt Work done by conservative forces is path indepen-
dent and depends only on initial and final points:
⃗aav = ∆⃗v/∆t ⃗
⃗ainst = dv/dt ⃗Fconservative · d⃗r = 0.
Work-energy theorem: W = ∆K
, Mechanical energy: E = U +K. Conserved if forces are Rotation about an axis with constant α:
conservative in nature.
ω = ω0 + αt, θ = ωt +12αt2, ω2−ω0 2 = 2αθ
Power Pav =∆W∆t, Pinst =⃗F ·⃗v
Moment of Inertia: I = imiri2, I= r2dm
1.5: Centre of Mass and Collision
ximi xdm 2mr2
Centre of mass: xcm = mi , xcm = dm
mr2 2mr2 3mr2 5mr2 12ml2
mr2 m(a2+b2)
12
b
a
CM of few useful configurations: ring disk shell sphere rod hollow solid rectangle
m1 r m2
1. m1, m2 separated by r: C
m2 r m 1r I∥ Ic
m1+m2 m 1+m2
Theorem of Parallel Axes: I∥= Icm + md2 d
cm
2. Triangle (CM ≡Centroid) yc =h h
C
h
3 z y
Theorem of Perp. Axes: Iz = Ix + Iy
x
3. Semicircular ring: yc =2r C
2r
r π
Radius of Gyration: k = I/m
4. Semicircular disc: yc =4r C
r
4r
3π Angular Momentum:⃗L = ⃗r × ⃗p, ⃗
⃗L = Iω
5. Hemispherical shell: yc =r C r y
P θ ⃗F
r 2
Torque: ⃗τ = ⃗r ×⃗F, ⃗τ =d⃗L τ = Iα
dt, ⃗r
O x
6. Solid Hemisphere: yc =3r C 3r
r 8
Conservation of⃗L: ⃗τext = 0 =⇒⃗L = const.
7. Cone: the height of CM from the base is h/4 for Equilibrium condition:⃗F =⃗0, ⃗τ = 0
⃗
the solid cone and h/3 for the hollow cone.
Kinetic Energy: Krot =1 2Iω2
Motion of the CM: M =mi Dynamics:
⃗Fext ⃗τcm = Icm⃗α, ⃗Fext = m⃗acm, ⃗pcm = m⃗vcm
miv⃗ i
⃗vcm = ⃗ cm, ⃗acm =
⃗pcm = Mv
M , M K =1 2mvcm 2 + 12Icmω2, ⃗L = Icm⃗ω + ⃗rcm × m⃗vcm
Impulse:⃗J = ⃗F dt = ∆⃗p
1.7: Gravitation
Before collision After collision
Collision: m1 m2 m1 m2 m1 F F m2
Gravitational force: F = Gm1m2r2
v1 v2 v1′ v2′
r
Momentum conservation: m1v1+m2v2 = m1v′ 1+m2v′
Elastic Collision:1 2m1v12+12m2v22 = 12m1v′ 2+12m2v′ 2 Potential energy: U = −GMm
Coefficient of restitution:
Gravitational acceleration: g =GM
e =−(v′ 1−v′2) = 1, completely elastic
v1 −v2 0, completely in-elastic Variation of g with depth: ginside ≈g 1 −hR
Variation of g with height: goutside ≈g 1 −2h
R
If v2 = 0 and m1 ≪m2 then v′ 1= −v1.
If v2 = 0 and m1 ≫m2 then v′ 2= 2v1. Effect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : v′ 1= v2and v′ 2= v1. gatpole > gatequator (∵Re −Rp ≈21 km)
Effect of earth rotation on apparent weight:
1.6: Rigid Body Dynamics
Angular velocity: ωav =∆θ ∆t, ω =dθ ⃗v = ⃗ω × ⃗r
dt,
Angular Accel.: αav =∆ω α =dω ⃗a = α
⃗ × ⃗r
dt,
