Equation Sheet – PHYS–2350
1
Kinematics – One & two dimensions
Average speed
Average acceleration 1-D Kinematics
Relative velocity:
#
βπ₯
βπ£!
π₯ = π₯" + π£"! π‘ + $π! π‘ $
$
'
⇒ π£!$ = π£"!
+ 2π! βπ₯
π―-β%# = π―-β%$ + π―-β$#
π£! =
π! =
βπ‘
βπ‘
π£! = π£"! + π! π‘
Quadratic Formula
Projectile with βπ¦ = 0
2π£" sin π
π£"$ sin 2π
π£"$ sin$ π
−π ± √π$ − 4ππ
π‘)*+,-. =
π
= π₯&'( =
π»=
ππ₯ $ + ππ₯ + π = 0 π₯ =
π
π
2π
2π
Dynamics
Net Force
Weight
Acc. gravity on earth, π
Kinetic friction
Static friction
$
β
πΉ
=
ππ
π
=
9.8
m/s
πΉ
=
π
πΉ
πΉ3 ≤ π3 πΉ2
∑π
/ = ππ-β
0
1
1 2
Uniform Circular motion & Gravitation
Frequency
Tangent Velocity & acceleration Centripetal acceleration
Centripetal force
$
1
2ππ
π£.'4,54.
π=
π = 2ππ
π£.'4,54. =
π.'4,54. = 0
π6'7+'* = −
π£6'7+'* = 0 πΉ6'7+'* = ππ6'7+'*
π
π
π
Gravitational Force
Universal G constant
Orbits
Kepler’s 3rd law
π# ⋅ π$
π:$ π:%
π
9##
$
$
⁄
πΉ8 = πΊ
πΊ
=
6.67
×
10
N
⋅
m
kg
=
π£
=
πΊ
π$
π;$ π;%
π
Work and Energy
Work
Power – Intensity
βπΈ
βπΈ
= πΉπ£ cos π
πΌ=
π< = π
β45. ⋅ πβ = πΉ45. βπ₯ cos π π =
βπ‘
βπ‘ βπ
Mechanical energy – Conservation of energy Conservative work
πΈ = πΎπΈ + π
(πΎπΈ + π)+4+.+'* + π ∗ = (πΎπΈ + π))+4'*
π>A4 = −Δπ>A4
Escape speed
2πΊπ
$
π£5=>
=
π
Work – Energy Theo.; Kinetic Energy
#
πΎπΈ? = $ππ£ $
π45. = ΔπΎπΈ
πΊππ G≈I!
πΊππ
π,6'C+.D = −
lβ―β―n πππ¦ + const.
π
π$ βΉ
#
π=F6+4, = $ππ₯ $
= −ππ₯
(,6'C+.D)
=
→ iπΉ
πΉ(=F6+4,)
Linear Momentum
Momentum
Impulse
Conservation of momentum (π
β45. = 0) Center of mass = (π₯>& , π¦>& , π¦>& )
βπ©
-β
π©
-β = ππ―-β
βπ‘
Coefficient of restitution
π£$ − π£#
π=
π’# − π’$
π’ → initial ; π£ → final
πβ = βπ©
-β = π
ββπ‘
π©
-β+4+.+'* = π©
-β)+4'*
Total elastic collision
Inelastic collision
πΎπΈ+4+.+'* = πΎπΈ)+4'*
π=1
1-D/2-body
πΎπΈ# ≠ πΎπΈ$
π=0
π£$ = π£#
π
β =
∑ π/ π₯/
∑ π/
∑ π/ π¦/
=
∑ π/
∑ π/ π§/
=
∑ π/
π₯>& =
π¦>&
π§>&
Rotational Motion
Angular velocity
π₯π
π
z=
= 2ππ
π₯π‘
Angular acceleration
π₯π
πΌ} =
π₯π‘
βπ = π" + π" π‘ +
'
π = π" + πΌπ‘
Moment of Inertia
Parallel Axes
Angle–position
π −velocity
πΌ–acc.
Radial acc
βπ = πβπ
π£.'4,54. = ππ
π.'4,54. = ππΌ
π6'7+'* = −ππ$
πΌ45. = πΌ# + πΌ$ + β―
πΌ|| = πΌ>& + ππ
$
Kinetic Eqs.
Torque
#
πΌπ‘ $
$
π = ±ππΉ sinπ
π = ππΉJ or π = πJ πΉ
⇒ π$ = π"$ + 2πΌβπ
Torque & L
Rigid rotator about a static point O or the c.m.
ΔπΏ-β
† πΏ+4+.+'* = πΏ)+4'* ‡if ˆ π = 0‰
Δπ‘
πΏ = ππ sin π = ππJ = πJ π
π (I) = πβπ cos π
-πβ =
(?)
πA4 M
π=
β = πΌπ
π
---β
+
(I)
π'NAO. M
(?)
Momenta of Inertia examples
π >& = π
π₯π'NAO. >& π£>& = π
π'NAO. >&
π>& = π
πΌ'NAO. >&
πΌ=F-565 = Pππ
$
$
(I)
πΎπΈ = πΎπΈA) M + πΎπΈ'NAO. M (πΎπΈ(?) = "#ππ£ $ )
-β = πΌM π
π
-β
Rolling without slipping
π = ππ cos π
#
πΎπΈ (I) = $πΌπ$
#
πΌ>54.56 6A7 = #$πβ$
πΌ>D*+4756 = ππ
$
#
πΌ>D*+4756 = $ππ
$
Equation Sheet – PHYS–2350
Fluids
Density & Specific gravity Pressure
π
π)*O+7
πΉJ
π=
π 8 =
π
=
π
πQ#M
π΄
Archimedes
Density of Water
Absolute pressure
Atmospheric P
Vibrations
Max. velocity
Continuity
Acceleration
Vol. flow rate
βπ
π=
= π΄π£
βπ‘
Bernoulli’s equation
#
#
π# + $ππ£#$ + πππ¦# = π$ + $ππ£$$ + πππ¦$
Simple pendulum
$
π£&'( = ππ΄
Pressure in fluids
π('N=A*O.5) = π'.& + βπ 1.013 × 10P Pa Δπ)*O+7 = π)*O+7 πβ
πQ#M = 10% kg/m% π# π΄# π£# = π$ π΄$ π£$
π΅ = π)*O+7 π π7+=F*'>57
2
Physical pendulum
π=π π₯
π&'( = π$ π΄
π=–
π
→ π = 2π—β/π
β
π
π = —π/π → π = 2π–
π
π₯ = π΄ sin(ππ‘ + πΏ)
π£ = π΄π cos(ππ‘ + πΏ)
π = —πππ/πΌM → π = 2π—πΌM /πππ
Mass-spring
#
#
#
πΈ = $ππ£ $ + $ππ₯ $ = $ππ΄$
π£ = ±π£&'( —1 − (π₯/π΄)$
Waves
Wave velocity
π
π£ = ππ =
π
π = 2π/π
String
π£ = —π/π (π = π/πΏ)
π£
πR = π ‡ ‰ , π = 1, 2, 3, … standing waves
2πΏ
Sound
Decibel Intensity
Velocity of sound
S
π½ = 10log ‡S ‰ πΌ" = 109#$ W/m$
π£ = 331¢
$
Beats
πN5'. = |π# − π$ |
Z
$
$
T
π
[m/s] πR = π ‡$U‰ , π = 1, 2, 3, … open
273 K
9$%
Ideal Gas
Universal gas constant
ππ = ππ
π
ππ = πππ
π
π
ππ = =
π β³
Thermodynamics
1st Law
π
= 8.314 J/mol ⋅ K
Avogadro’s Number
π: = 6.022 × 10$%
π = 2π $ (ππ΄)$ ππ£
πΌ = 2π $ (ππ΄)$ ππ£
T
πR = π VU , π = 1, 3, 5, …closed
Expansion
Number of moles
π (g)
π
π£6&= = —}}}
π£ $ = —3ππ/β³ ΔπΏ = πΌπΏ" Δπ linear
π=
=
(g/mol)
Δπ = π½π" Δπ volumetric
β³
π:
J/K β³ = ππ:
Heat flow
βπ
βπ
= βπ΄(π\ − π=O6) )
πΜ>A47 = π
π΄
πΜ6'7
Work (area under ππ curve)
Δπ = π − π
%
π = $ππ
π
πβ_`" = ππ₯π
π β?`" = ππ
π ln(π)+4 /π+4+ )
Ideal gas
Processes (apply to 1st law)
π₯π = ππΆb βπ
#
πΆb = $π75, A) )6557A& π
Isothermal:
Isobaric:
Isochoric:
Adiabatic:
Cyclic:
e&
Power & Intensity
X
π£=AO47 ± π£AN=56C56
πW = π ¨
© obs ln source
π£=AO47
rms speed
π[ = πY + 273.15 π; = π
/π: = 1.38 × 10
πΆc = πΆb + π
ππ d = constant
e
πΎ = % adiabats
Boundary conditions
π# π$
=
π£# π£$
π# = π$
Pipe standing waves
Doppler Effect
X
π£=AO47
πW = π ¨
© source ln obs
π£=AO47 ± π£=AO6>5
Temperature & Kinetic Theory
Scales
Average kinetic energy
P
# }}}
$ = %π π
πY = (π< − 32) }}}}
πΎπΈ = ππ£
;
π = −π΄π$ sin(ππ‘ + πΏ)
ππ₯"
tan(πΏ) =
π£"
+75'* ,'=
By system π > 0
On system π < 0
Carnot engine
V
V
πΜ6'7 = πππ΄³π=O66
− πNA7D
´
9]
$ V
π = 5.67 × 10 W/m K
Heat
Specific Heat
πβa`" = ππΆa βπ
πβ_`" = ππΆ_ βπ
π = ππβπ
π = ππΏ Latent heat
∑R πR = 0 isolated system
Heat engine
Entropy (2nd Law)
βπ = 0 → βπ = 0
π = πf − πU
πΏπ π
π
π
βπ = 0 → π = πβπ − U = U
=
π
πU ΔπgO'=+ =.'.+> = ˆ
π
πf πf
π
π
=
=
1
−
βπ = 0 → π = 0
πf
πf Δπ
πU
≥
0
+=A*'.57 =D=.5&
πY'64A. = 1 −
π
π = 0 → βπ =– π
πf π
= − U
Δπ = πU − πf engine
βπ = 0 → π = π>D>*5
π
0
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