, Seep Pahuja
Motion in a straight line with constant a:
v = u + at, s = ut + 12 at2 , v2 u2 = 2as
Relative Velocity: ~vA/B = ~vA ~vB
0.1: Physical Constants
Speed of light c 3 ⇥ 108 m/s
u
Planck constant h 6.63 ⇥ 10 34 J s y
u sin ✓
hc 1242 eV-nm H
x
Projectile Motion:
Gravitation constant G 6.67⇥10 11 m3 kg 1 s 2
✓
Boltzmann constant k 1.38 ⇥ 10 23 J/K O u cos ✓
Molar gas constant R 8.314 J/(mol K) R
Avogadro’s number NA 6.023 ⇥ 1023 mol 1
Charge of electron e 1.602 ⇥ 10 19 C x = ut cos ✓, y = ut sin ✓ 12 gt2
Permeability of vac- µ0 4⇡ ⇥ 10 7 N/A2 g
y = x tan ✓ x2
uum 2u cos2 ✓
2
Permitivity of vacuum ✏0 8.85 ⇥ 10 12 F/m 2u sin ✓ u2 sin 2✓ u2 sin2 ✓
T = , R= , H=
Coulomb constant 1
4⇡✏0 9 ⇥ 109 N m2 /C2 g g 2g
Faraday constant F 96485 C/mol
Mass of electron me 9.1 ⇥ 10 31 kg 1.3: Newton’s Laws and Friction
Mass of proton mp 1.6726 ⇥ 10 27 kg
Mass of neutron mn 1.6749 ⇥ 10 27 kg Linear momentum: p~ = m~v
Atomic mass unit u 1.66 ⇥ 10 27 kg
Atomic mass unit u 931.49 MeV/c2 Newton’s first law: inertial frame.
Stefan-Boltzmann 5.67⇥10 8 W/(m2 K4 )
Newton’s second law: F~ = d~
p
dt , F~ = m~a
constant
Rydberg constant R1 1.097 ⇥ 107 m 1 Newton’s third law: F~AB = F~BA
Bohr magneton µB 9.27 ⇥ 10 24 J/T
Bohr radius a0 0.529 ⇥ 10 10 m Frictional force: fstatic, max = µs N, fkinetic = µk N
Standard atmosphere atm 1.01325 ⇥ 105 Pa
v2 v2 µ+tan ✓
Wien displacement b 2.9 ⇥ 10 3 m K Banking angle: rg = tan ✓, rg = 1 µ tan ✓
constant
mv 2 v2
Centripetal force: Fc = r , ac = r
mv 2
1 MECHANICS Pseudo force: F~pseudo = m~a0 , Fcentrifugal = r
Minimum speed to complete vertical circle:
1.1: Vectors
p p
vmin, bottom = 5gl, vmin, top = gl
Notation: ~a = ax ı̂ + ay |ˆ + az k̂
q
Magnitude: a = |~a| = a2x + a2y + a2z ✓
l
q
l cos ✓
Conical pendulum: T = 2⇡ ✓ T
Dot product: ~a · ~b = ax bx + ay by + az bz = ab cos ✓ g
ı̂ mg
a ⇥ ~b
~ ~b
Cross product:
✓ k̂ |ˆ
~
a
1.4: Work, Power and Energy
~a ⇥~b = (ay bz az by )ı̂ + (az bx ax bz )ˆ
| + (ax by ay bx )k̂ R
Work: W = F~ · S
~ = F S cos ✓, W = F~ · dS
~
|~a ⇥ ~b| = ab sin ✓ Kinetic energy: K = 12 mv 2 = p2
2m
Potential energy: F = @U/@x for conservative forces.
1.2: Kinematics
Ugravitational = mgh, Uspring = 12 kx2
Average and Instantaneous Vel. and Accel.:
~vav = ~r/ t, ~vinst = d~r/dt Work done by conservative forces is path indepen-
dent
H and depends only on initial and final points:
~aav = ~v / t ~ainst = d~v /dt F~conservative · d~r = 0.
Work-energy theorem: W = K
, Seep Pahuja
Mechanical energy: E = U + K. Conserved if forces are Rotation about an axis with constant ↵:
conservative in nature.
! = !0 + ↵t, ✓ = !t + 12 ↵t2 , !2 !0 2 = 2↵✓
Power Pav = W
t , Pinst = F~ · ~v
P R
Moment of Inertia: I = i mi ri 2 , I= r2 dm
1.5: Centre of Mass and Collision
P R
Pxi mi , R xdm
1 2
2 mr m(a +b )
2 2
Centre of mass: xcm = xcm = mr 2 1 2 2 2 2 2 1 2 mr 2
mi dm 2 mr 3 mr 5 mr 12 ml 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 m1 r Ik Ic
m1 +m2 m1 +m2 2
Theorem of Parallel Axes: Ik = Icm + md d
cm
h
2. Triangle (CM ⌘ Centroid) yc = 3 h
C
h
3
z y
Theorem of Perp. Axes: Iz = Ix + Iy
2r
x
3. Semicircular ring: yc = ⇡
C
2r
r ⇡
p
Radius of Gyration: k = I/m
4r
4. Semicircular disc: yc = 3⇡ C 4r
r ~ = ~r ⇥ p~, ~ = I~
3⇡ Angular Momentum: L L !
r y
5. Hemispherical shell: yc = 2 C r ~ P ✓ ~
r 2
Torque: ~⌧ = ~r ⇥ F~ , ~⌧ = dL
dt , ⌧ = I↵ F
~
r x
O
3r
6. Solid Hemisphere: yc = 8 C 3r
r 8 ~ ~⌧ext = 0 =) L
Conservation of L: ~ = const.
7. Cone: the height of CM from the base is h/4 for P~ P
Equilibrium condition: F = ~0, ~⌧ = ~0
the solid cone and h/3 for the hollow cone.
Kinetic Energy: Krot = 12 I! 2
P Dynamics:
Motion of the CM: M = mi
P ~⌧cm = Icm ↵
~, F~ext = m~acm , p~cm = m~vcm
mi~vi F~ext
~vcm = , p~cm = M~vcm , ~acm = 1 2 1 2 ~
M M ~ + ~rcm ⇥ m~vcm
K = 2 mvcm + 2 Icm ! , L = Icm !
R
Impulse: J~ = F~ dt = p~
1.7: Gravitation
Before collision After collision
Collision: m1 m2 m1 m2 m1 F F m2
Gravitational force: F = G mr1 m
2
2
v1 v2 v10 v20 r
Momentum conservation: m1 v1 +m2 v2 = m1 v10 +m2 v20
2 2 GM m
Elastic Collision: 12 m1 v1 2+ 12 m2 v2 2 = 12 m1 v10 + 12 m2 v20 Potential energy: U = r
Coefficient of restitution: GM
⇢ Gravitational acceleration: g = R2
(v10 v20 ) 1, completely elastic
e= = h
v1 v2 0, completely in-elastic Variation of g with depth: ginside ⇡ g 1 R
2h
Variation of g with height: goutside ⇡ g 1
If v2 = 0 and m1 ⌧ m2 then v10 = v1 . R
If v2 = 0 and m1 m2 then v20 = 2v1 . E↵ect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : v10 = v2 and v20 = v1 . gat pole > gat equator (* Re Rp ⇡ 21 km)
E↵ect of earth rotation on apparent weight:
1.6: Rigid Body Dynamics
✓ d✓
Angular velocity: !av = t, != dt , ~ ⇥ ~r
~v = !
! d!
Angular Accel.: ↵av = t, ↵= dt , ~ ⇥ ~r
~a = ↵
, Seep Pahuja
!
~ ~
A
~2
A
2
Superposition of two SHM’s: ✏
mg m! R cos ✓
mg✓0 = mg m! 2 R cos2 ✓ ~1
A
✓
R
x1 = A1 sin !t, x2 = A2 sin(!t + )
x = x1 + x2 = A sin(!t + ✏)
q q
GM
Orbital velocity of satellite: vo = R A = A1 2 + A2 2 + 2A1 A2 cos
q A2 sin
Escape velocity: ve = 2GM tan ✏ =
R A1 + A2 cos
vo
Kepler’s laws: 1.9: Properties of Matter
a
F/A P F
First: Elliptical orbit with sun at one of the focus. Modulus of rigidity: Y = l/l , B= V V , ⌘= A✓
~
Second: Areal velocity is constant. (* dL/dt = 0). 1 1 dV
2 3 2 4⇡ 2 3 Compressibility: K = =
Third: T / a . In circular orbit T = GM a . B V dP
lateral strain D/D
Poisson’s ratio: = longitudinal strain = l/l
1.8: Simple Harmonic Motion 1
Elastic energy: U = 2 stress ⇥ strain ⇥ volume
Hooke’s law: F = kx (for small elongation x.)
d2 x k
Acceleration: a = dt2 = mx = !2 x Surface tension: S = F/l
2⇡
pm
Time period: T = ! = 2⇡ k
Surface energy: U = SA
Displacement: x = A sin(!t + ) Excess pressure in bubble:
p
Velocity: v = A! cos(!t + ) = ±! A2 x2 pair = 2S/R, psoap = 4S/R
2S cos ✓
Capillary rise: h = r⇢g
Potential energy: U = 12 kx2 U
x
A 0 A
Hydrostatic pressure: p = ⇢gh
Kinetic energy K = 12 mv 2 K
x Buoyant force: FB = ⇢V g = Weight of displaced liquid
A 0 A
Equation of continuity: A1 v1 = A2 v2 v2
v1
Total energy: E = U + K = 12 m! 2 A2
Bernoulli’s equation: p + 12 ⇢v 2 + ⇢gh = constant
p
q Torricelli’s theorem: ve✏ux = 2gh
l
Simple pendulum: T = 2⇡ g l
dv
Viscous force: F = ⌘A dx
q F
I
Physical Pendulum: T = 2⇡ mgl Stoke’s law: F = 6⇡⌘rv
v
q Volume flow ⇡pr 4 r
I Poiseuilli’s equation: =
Torsional Pendulum T = 2⇡ k
time 8⌘l
l
2r 2 (⇢ )g
Terminal velocity: vt = 9⌘
1 1 1
Springs in series: keq = k1 + k2
k1 k2
Springs in parallel: keq = k1 + k2 k2
k1
Motion in a straight line with constant a:
v = u + at, s = ut + 12 at2 , v2 u2 = 2as
Relative Velocity: ~vA/B = ~vA ~vB
0.1: Physical Constants
Speed of light c 3 ⇥ 108 m/s
u
Planck constant h 6.63 ⇥ 10 34 J s y
u sin ✓
hc 1242 eV-nm H
x
Projectile Motion:
Gravitation constant G 6.67⇥10 11 m3 kg 1 s 2
✓
Boltzmann constant k 1.38 ⇥ 10 23 J/K O u cos ✓
Molar gas constant R 8.314 J/(mol K) R
Avogadro’s number NA 6.023 ⇥ 1023 mol 1
Charge of electron e 1.602 ⇥ 10 19 C x = ut cos ✓, y = ut sin ✓ 12 gt2
Permeability of vac- µ0 4⇡ ⇥ 10 7 N/A2 g
y = x tan ✓ x2
uum 2u cos2 ✓
2
Permitivity of vacuum ✏0 8.85 ⇥ 10 12 F/m 2u sin ✓ u2 sin 2✓ u2 sin2 ✓
T = , R= , H=
Coulomb constant 1
4⇡✏0 9 ⇥ 109 N m2 /C2 g g 2g
Faraday constant F 96485 C/mol
Mass of electron me 9.1 ⇥ 10 31 kg 1.3: Newton’s Laws and Friction
Mass of proton mp 1.6726 ⇥ 10 27 kg
Mass of neutron mn 1.6749 ⇥ 10 27 kg Linear momentum: p~ = m~v
Atomic mass unit u 1.66 ⇥ 10 27 kg
Atomic mass unit u 931.49 MeV/c2 Newton’s first law: inertial frame.
Stefan-Boltzmann 5.67⇥10 8 W/(m2 K4 )
Newton’s second law: F~ = d~
p
dt , F~ = m~a
constant
Rydberg constant R1 1.097 ⇥ 107 m 1 Newton’s third law: F~AB = F~BA
Bohr magneton µB 9.27 ⇥ 10 24 J/T
Bohr radius a0 0.529 ⇥ 10 10 m Frictional force: fstatic, max = µs N, fkinetic = µk N
Standard atmosphere atm 1.01325 ⇥ 105 Pa
v2 v2 µ+tan ✓
Wien displacement b 2.9 ⇥ 10 3 m K Banking angle: rg = tan ✓, rg = 1 µ tan ✓
constant
mv 2 v2
Centripetal force: Fc = r , ac = r
mv 2
1 MECHANICS Pseudo force: F~pseudo = m~a0 , Fcentrifugal = r
Minimum speed to complete vertical circle:
1.1: Vectors
p p
vmin, bottom = 5gl, vmin, top = gl
Notation: ~a = ax ı̂ + ay |ˆ + az k̂
q
Magnitude: a = |~a| = a2x + a2y + a2z ✓
l
q
l cos ✓
Conical pendulum: T = 2⇡ ✓ T
Dot product: ~a · ~b = ax bx + ay by + az bz = ab cos ✓ g
ı̂ mg
a ⇥ ~b
~ ~b
Cross product:
✓ k̂ |ˆ
~
a
1.4: Work, Power and Energy
~a ⇥~b = (ay bz az by )ı̂ + (az bx ax bz )ˆ
| + (ax by ay bx )k̂ R
Work: W = F~ · S
~ = F S cos ✓, W = F~ · dS
~
|~a ⇥ ~b| = ab sin ✓ Kinetic energy: K = 12 mv 2 = p2
2m
Potential energy: F = @U/@x for conservative forces.
1.2: Kinematics
Ugravitational = mgh, Uspring = 12 kx2
Average and Instantaneous Vel. and Accel.:
~vav = ~r/ t, ~vinst = d~r/dt Work done by conservative forces is path indepen-
dent
H and depends only on initial and final points:
~aav = ~v / t ~ainst = d~v /dt F~conservative · d~r = 0.
Work-energy theorem: W = K
, Seep Pahuja
Mechanical energy: E = U + K. Conserved if forces are Rotation about an axis with constant ↵:
conservative in nature.
! = !0 + ↵t, ✓ = !t + 12 ↵t2 , !2 !0 2 = 2↵✓
Power Pav = W
t , Pinst = F~ · ~v
P R
Moment of Inertia: I = i mi ri 2 , I= r2 dm
1.5: Centre of Mass and Collision
P R
Pxi mi , R xdm
1 2
2 mr m(a +b )
2 2
Centre of mass: xcm = xcm = mr 2 1 2 2 2 2 2 1 2 mr 2
mi dm 2 mr 3 mr 5 mr 12 ml 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 m1 r Ik Ic
m1 +m2 m1 +m2 2
Theorem of Parallel Axes: Ik = Icm + md d
cm
h
2. Triangle (CM ⌘ Centroid) yc = 3 h
C
h
3
z y
Theorem of Perp. Axes: Iz = Ix + Iy
2r
x
3. Semicircular ring: yc = ⇡
C
2r
r ⇡
p
Radius of Gyration: k = I/m
4r
4. Semicircular disc: yc = 3⇡ C 4r
r ~ = ~r ⇥ p~, ~ = I~
3⇡ Angular Momentum: L L !
r y
5. Hemispherical shell: yc = 2 C r ~ P ✓ ~
r 2
Torque: ~⌧ = ~r ⇥ F~ , ~⌧ = dL
dt , ⌧ = I↵ F
~
r x
O
3r
6. Solid Hemisphere: yc = 8 C 3r
r 8 ~ ~⌧ext = 0 =) L
Conservation of L: ~ = const.
7. Cone: the height of CM from the base is h/4 for P~ P
Equilibrium condition: F = ~0, ~⌧ = ~0
the solid cone and h/3 for the hollow cone.
Kinetic Energy: Krot = 12 I! 2
P Dynamics:
Motion of the CM: M = mi
P ~⌧cm = Icm ↵
~, F~ext = m~acm , p~cm = m~vcm
mi~vi F~ext
~vcm = , p~cm = M~vcm , ~acm = 1 2 1 2 ~
M M ~ + ~rcm ⇥ m~vcm
K = 2 mvcm + 2 Icm ! , L = Icm !
R
Impulse: J~ = F~ dt = p~
1.7: Gravitation
Before collision After collision
Collision: m1 m2 m1 m2 m1 F F m2
Gravitational force: F = G mr1 m
2
2
v1 v2 v10 v20 r
Momentum conservation: m1 v1 +m2 v2 = m1 v10 +m2 v20
2 2 GM m
Elastic Collision: 12 m1 v1 2+ 12 m2 v2 2 = 12 m1 v10 + 12 m2 v20 Potential energy: U = r
Coefficient of restitution: GM
⇢ Gravitational acceleration: g = R2
(v10 v20 ) 1, completely elastic
e= = h
v1 v2 0, completely in-elastic Variation of g with depth: ginside ⇡ g 1 R
2h
Variation of g with height: goutside ⇡ g 1
If v2 = 0 and m1 ⌧ m2 then v10 = v1 . R
If v2 = 0 and m1 m2 then v20 = 2v1 . E↵ect of non-spherical earth shape on g:
Elastic collision with m1 = m2 : v10 = v2 and v20 = v1 . gat pole > gat equator (* Re Rp ⇡ 21 km)
E↵ect of earth rotation on apparent weight:
1.6: Rigid Body Dynamics
✓ d✓
Angular velocity: !av = t, != dt , ~ ⇥ ~r
~v = !
! d!
Angular Accel.: ↵av = t, ↵= dt , ~ ⇥ ~r
~a = ↵
, Seep Pahuja
!
~ ~
A
~2
A
2
Superposition of two SHM’s: ✏
mg m! R cos ✓
mg✓0 = mg m! 2 R cos2 ✓ ~1
A
✓
R
x1 = A1 sin !t, x2 = A2 sin(!t + )
x = x1 + x2 = A sin(!t + ✏)
q q
GM
Orbital velocity of satellite: vo = R A = A1 2 + A2 2 + 2A1 A2 cos
q A2 sin
Escape velocity: ve = 2GM tan ✏ =
R A1 + A2 cos
vo
Kepler’s laws: 1.9: Properties of Matter
a
F/A P F
First: Elliptical orbit with sun at one of the focus. Modulus of rigidity: Y = l/l , B= V V , ⌘= A✓
~
Second: Areal velocity is constant. (* dL/dt = 0). 1 1 dV
2 3 2 4⇡ 2 3 Compressibility: K = =
Third: T / a . In circular orbit T = GM a . B V dP
lateral strain D/D
Poisson’s ratio: = longitudinal strain = l/l
1.8: Simple Harmonic Motion 1
Elastic energy: U = 2 stress ⇥ strain ⇥ volume
Hooke’s law: F = kx (for small elongation x.)
d2 x k
Acceleration: a = dt2 = mx = !2 x Surface tension: S = F/l
2⇡
pm
Time period: T = ! = 2⇡ k
Surface energy: U = SA
Displacement: x = A sin(!t + ) Excess pressure in bubble:
p
Velocity: v = A! cos(!t + ) = ±! A2 x2 pair = 2S/R, psoap = 4S/R
2S cos ✓
Capillary rise: h = r⇢g
Potential energy: U = 12 kx2 U
x
A 0 A
Hydrostatic pressure: p = ⇢gh
Kinetic energy K = 12 mv 2 K
x Buoyant force: FB = ⇢V g = Weight of displaced liquid
A 0 A
Equation of continuity: A1 v1 = A2 v2 v2
v1
Total energy: E = U + K = 12 m! 2 A2
Bernoulli’s equation: p + 12 ⇢v 2 + ⇢gh = constant
p
q Torricelli’s theorem: ve✏ux = 2gh
l
Simple pendulum: T = 2⇡ g l
dv
Viscous force: F = ⌘A dx
q F
I
Physical Pendulum: T = 2⇡ mgl Stoke’s law: F = 6⇡⌘rv
v
q Volume flow ⇡pr 4 r
I Poiseuilli’s equation: =
Torsional Pendulum T = 2⇡ k
time 8⌘l
l
2r 2 (⇢ )g
Terminal velocity: vt = 9⌘
1 1 1
Springs in series: keq = k1 + k2
k1 k2
Springs in parallel: keq = k1 + k2 k2
k1
