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Equation Sheet-2350

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Equation Sheet – PHYS–2350
1
Kinematics – One & two dimensions
Average speed
Average acceleration 1-D Kinematics
Relative velocity:
#
βˆ†π‘₯
βˆ†π‘£!
π‘₯ = π‘₯" + 𝑣"! 𝑑 + $π‘Ž! 𝑑 $
$
'
⇒ 𝑣!$ = 𝑣"!
+ 2π‘Ž! βˆ†π‘₯
𝐯-βƒ—%# = 𝐯-βƒ—%$ + 𝐯-βƒ—$#
𝑣! =
π‘Ž! =
βˆ†π‘‘
βˆ†π‘‘
𝑣! = 𝑣"! + π‘Ž! 𝑑
Quadratic Formula
Projectile with βˆ†π‘¦ = 0
2𝑣" sin πœƒ
𝑣"$ sin 2πœƒ
𝑣"$ sin$ πœƒ
−𝑏 ± √𝑏$ − 4π‘Žπ‘
𝑑)*+,-. =
𝑅 = π‘₯&'( =
𝐻=
π‘Žπ‘₯ $ + 𝑏π‘₯ + 𝑐 = 0 π‘₯ =
𝑔
𝑔
2𝑔
2π‘Ž
Dynamics
Net Force
Weight
Acc. gravity on earth, 𝑔
Kinetic friction
Static friction
$
βƒ—
𝐹
=
π‘šπ‘”
𝑔
=
9.8
m/s
𝐹
=
πœ‡
𝐹
𝐹3 ≤ πœ‡3 𝐹2
∑𝐅/ = π‘šπš-βƒ—
0
1
1 2
Uniform Circular motion & Gravitation
Frequency
Tangent Velocity & acceleration Centripetal acceleration
Centripetal force
$
1
2πœ‹π‘Ÿ
𝑣.'4,54.
𝑓=
πœ” = 2πœ‹π‘“
𝑣.'4,54. =
π‘Ž.'4,54. = 0
π‘Ž6'7+'* = −
𝑣6'7+'* = 0 𝐹6'7+'* = π‘šπ‘Ž6'7+'*
𝑇
𝑇
π‘Ÿ
Gravitational Force
Universal G constant
Orbits
Kepler’s 3rd law
π‘š# ⋅ π‘š$
𝑇:$ 𝑆:%
𝑀
9##
$
$
⁄
𝐹8 = 𝐺
𝐺
=
6.67
×
10
N
⋅
m
kg
=
𝑣
=
𝐺
π‘Ÿ$
𝑇;$ 𝑆;%
π‘Ÿ
Work and Energy
Work
Power – Intensity
βˆ†πΈ
βˆ†πΈ
= 𝐹𝑣 cos πœ™
𝐼=
π‘Š< = 𝐅⃗45. ⋅ 𝐝⃗ = 𝐹45. βˆ†π‘₯ cos πœ™ 𝑃 =
βˆ†π‘‘
βˆ†π‘‘ βˆ†π‘Ž
Mechanical energy – Conservation of energy Conservative work
𝐸 = 𝐾𝐸 + π‘ˆ
(𝐾𝐸 + π‘ˆ)+4+.+'* + π‘Š ∗ = (𝐾𝐸 + π‘ˆ))+4'*
π‘Š>A4 = −Δπ‘ˆ>A4
Escape speed
2𝐺𝑀
$
𝑣5=>
=
𝑅
Work – Energy Theo.; Kinetic Energy
#
𝐾𝐸? = $π‘šπ‘£ $
π‘Š45. = Δ𝐾𝐸
πΊπ‘€π‘š G≈I!
πΊπ‘€π‘š
π‘ˆ,6'C+.D = −
l⎯⎯n π‘šπ‘”π‘¦ + const.
π‘Ÿ
π‘Ÿ$ ⟹
#
π‘ˆ=F6+4, = $π‘˜π‘₯ $
= −π‘˜π‘₯
(,6'C+.D)
=
→ i𝐹
𝐹(=F6+4,)
Linear Momentum
Momentum
Impulse
Conservation of momentum (𝐅⃗45. = 0) Center of mass = (π‘₯>& , 𝑦>& , 𝑦>& )
βˆ†π©
-βƒ—
𝐩
-βƒ— = π‘šπ―-βƒ—
βˆ†π‘‘
Coefficient of restitution
𝑣$ − 𝑣#
πœ–=
𝑒# − 𝑒$
𝑒 → initial ; 𝑣 → final
𝐉⃗ = βˆ†π©
-βƒ— = π…βƒ—βˆ†π‘‘
𝐩
-βƒ—+4+.+'* = 𝐩
-βƒ—)+4'*
Total elastic collision
Inelastic collision
𝐾𝐸+4+.+'* = 𝐾𝐸)+4'*
πœ–=1
1-D/2-body
𝐾𝐸# ≠ 𝐾𝐸$
πœ–=0
𝑣$ = 𝑣#
𝐅⃗ =
∑ π‘š/ π‘₯/
∑ π‘š/
∑ π‘š/ 𝑦/
=
∑ π‘š/
∑ π‘š/ 𝑧/
=
∑ π‘š/
π‘₯>& =
𝑦>&
𝑧>&
Rotational Motion
Angular velocity
π›₯πœƒ
πœ”
z=
= 2πœ‹π‘“
π›₯𝑑
Angular acceleration
π›₯πœ”
𝛼} =
π›₯𝑑
βˆ†πœƒ = πœƒ" + πœ”" 𝑑 +
'
πœ” = πœ”" + 𝛼𝑑
Moment of Inertia
Parallel Axes
Angle–position
πœ” −velocity
𝛼–acc.
Radial acc
βˆ†π‘  = π‘Ÿβˆ†πœƒ
𝑣.'4,54. = π‘Ÿπœ”
π‘Ž.'4,54. = π‘Ÿπ›Ό
π‘Ž6'7+'* = −π‘Ÿπœ”$
𝐼45. = 𝐼# + 𝐼$ + β‹―
𝐼|| = 𝐼>& + 𝑀𝑑
$
Kinetic Eqs.
Torque
#
𝛼𝑑 $
$
𝜏 = ±π‘ŸπΉ sinπœ™
𝜏 = π‘ŸπΉJ or 𝜏 = π‘ŸJ 𝐹
⇒ πœ”$ = πœ”"$ + 2π›Όβˆ†πœƒ
Torque & L
Rigid rotator about a static point O or the c.m.
Δ𝐿-βƒ—
† 𝐿+4+.+'* = 𝐿)+4'* ‡if ˆ 𝜏 = 0‰
Δ𝑑
𝐿 = π‘Ÿπ‘ sin πœ™ = π‘Ÿπ‘J = π‘ŸJ 𝑝
π‘Š (I) = πœβˆ†πœƒ cos πœ™
-𝝉⃗ =
(?)
π‘ŠA4 M
π‘Š=
βƒ— = πΌπ›š
𝐋
---βƒ—
+
(I)
π‘Š'NAO. M
(?)
Momenta of Inertia examples
𝑠>& = 𝑅π›₯πœƒ'NAO. >& 𝑣>& = π‘…πœ”'NAO. >&
π‘Ž>& = 𝑅𝛼'NAO. >&
𝐼=F-565 = Pπ‘šπ‘…$
$
(I)
𝐾𝐸 = 𝐾𝐸A) M + 𝐾𝐸'NAO. M (𝐾𝐸(?) = "#𝑀𝑣 $ )
-βƒ— = 𝐼M 𝛂
𝝉
-βƒ—
Rolling without slipping
𝑃 = πœπœ” cos πœ™
#
𝐾𝐸 (I) = $πΌπœ”$
#
𝐼>54.56 6A7 = #$π‘šβ„“$
𝐼>D*+4756 = π‘šπ‘…$
#
𝐼>D*+4756 = $π‘šπ‘…$
Equation Sheet – PHYS–2350
Fluids
Density & Specific gravity Pressure
𝑀
𝜌)*O+7
𝐹J
𝜌=
𝑠8 =
𝑃
=
𝑉
𝜌Q#M
𝐴
Archimedes
Density of Water
Absolute pressure
Atmospheric P
Vibrations
Max. velocity
Continuity
Acceleration
Vol. flow rate
βˆ†π‘‰
𝑄=
= 𝐴𝑣
βˆ†π‘‘
Bernoulli’s equation
#
#
𝑃# + $πœŒπ‘£#$ + πœŒπ‘”π‘¦# = 𝑃$ + $πœŒπ‘£$$ + πœŒπ‘”π‘¦$
Simple pendulum
$
𝑣&'( = πœ”π΄
Pressure in fluids
𝑃('N=A*O.5) = 𝑃'.& + βˆ†π‘ƒ 1.013 × 10P Pa Δ𝑃)*O+7 = 𝜌)*O+7 π‘”β„Ž
𝜌Q#M = 10% kg/m% 𝜌# 𝐴# 𝑣# = 𝜌$ 𝐴$ 𝑣$
𝐡 = 𝜌)*O+7 𝑔 𝑉7+=F*'>57
2
Physical pendulum
π‘Ž=πœ” π‘₯
π‘Ž&'( = πœ”$ 𝐴
πœ”=–
𝑔
→ 𝑇 = 2πœ‹—β„“/𝑔
β„“
π‘š
πœ” = —π‘˜/π‘š → 𝑇 = 2πœ‹–
π‘˜
π‘₯ = 𝐴 sin(πœ”π‘‘ + 𝛿)
𝑣 = π΄πœ” cos(πœ”π‘‘ + 𝛿)
πœ” = —π‘šπ‘”π‘‘/𝐼M → 𝑇 = 2πœ‹—𝐼M /π‘šπ‘”π‘‘
Mass-spring
#
#
#
𝐸 = $π‘šπ‘£ $ + $π‘˜π‘₯ $ = $π‘˜π΄$
𝑣 = ±π‘£&'( —1 − (π‘₯/𝐴)$
Waves
Wave velocity
πœ”
𝑣 = πœ†π‘“ =
π‘˜
π‘˜ = 2πœ‹/πœ†
String
𝑣 = —𝑇/πœ‡ (πœ‡ = π‘š/𝐿)
𝑣
𝑓R = 𝑛 ‡ ‰ , 𝑛 = 1, 2, 3, … standing waves
2𝐿
Sound
Decibel Intensity
Velocity of sound
S
𝛽 = 10log ‡S ‰ 𝐼" = 109#$ W/m$
𝑣 = 331¢
$
Beats
𝑓N5'. = |𝑓# − 𝑓$ |
Z
$
$
T
𝑇
[m/s] 𝑓R = 𝑛 ‡$U‰ , 𝑛 = 1, 2, 3, … open
273 K
9$%
Ideal Gas
Universal gas constant
𝑃𝑉 = 𝑛𝑅𝑇
𝑃𝑉 = π‘π‘˜π‘‡
𝑛
𝜌
𝑃𝑉 = =
𝑉 β„³
Thermodynamics
1st Law
𝑅 = 8.314 J/mol ⋅ K
Avogadro’s Number
𝑁: = 6.022 × 10$%
𝑃 = 2πœ‹ $ (𝑓𝐴)$ πœ‡π‘£
𝐼 = 2πœ‹ $ (𝑓𝐴)$ πœŒπ‘£
T
𝑓R = π‘š VU , π‘š = 1, 3, 5, …closed
Expansion
Number of moles
π‘š (g)
𝑁
𝑣6&= = —}}}
𝑣 $ = —3π‘˜π‘‡/β„³ Δ𝐿 = 𝛼𝐿" Δ𝑇 linear
𝑛=
=
(g/mol)
Δ𝑉 = 𝛽𝑉" Δ𝑇 volumetric
β„³
𝑁:
J/K β„³ = π‘šπ‘:
Heat flow
βˆ†π‘‡
βˆ†π‘ 
= β„Žπ΄(𝑇\ − 𝑇=O6) )
𝑄̇>A47 = πœ…π΄
𝑄̇6'7
Work (area under 𝑃𝑉 curve)
Δπ‘ˆ = 𝑄 − π‘Š
%
π‘ˆ = $𝑛𝑅𝑇
π‘Šβˆ†_`" = 𝑃π›₯𝑉
π‘Š βˆ†?`" = 𝑛𝑅𝑇 ln(𝑉)+4 /𝑉+4+ )
Ideal gas
Processes (apply to 1st law)
π›₯π‘ˆ = 𝑛𝐢b βˆ†π‘‡
#
𝐢b = $𝑁75, A) )6557A& 𝑅
Isothermal:
Isobaric:
Isochoric:
Adiabatic:
Cyclic:
e&
Power & Intensity
X
𝑣=AO47 ± 𝑣AN=56C56
𝑓W = 𝑓 ¨
© obs ln source
𝑣=AO47
rms speed
𝑇[ = 𝑇Y + 273.15 π‘˜; = 𝑅/𝑁: = 1.38 × 10
𝐢c = 𝐢b + 𝑅
𝑃𝑉 d = constant
e
𝛾 = % adiabats
Boundary conditions
πœ†# πœ†$
=
𝑣# 𝑣$
𝑓# = 𝑓$
Pipe standing waves
Doppler Effect
X
𝑣=AO47
𝑓W = 𝑓 ¨
© source ln obs
𝑣=AO47 ± 𝑣=AO6>5
Temperature & Kinetic Theory
Scales
Average kinetic energy
P
# }}}
$ = %π‘˜ 𝑇
𝑇Y = (𝑇< − 32) }}}}
𝐾𝐸 = π‘šπ‘£
;
π‘Ž = −π΄πœ”$ sin(πœ”π‘‘ + 𝛿)
πœ”π‘₯"
tan(𝛿) =
𝑣"
+75'* ,'=
By system π‘Š > 0
On system π‘Š < 0
Carnot engine
V
V
𝑄̇6'7 = π‘’πœŽπ΄³π‘‡=O66
− 𝑇NA7D
´
9]
$ V
𝜎 = 5.67 × 10 W/m K
Heat
Specific Heat
π‘„βˆ†a`" = 𝑛𝐢a βˆ†π‘‡
π‘„βˆ†_`" = 𝑛𝐢_ βˆ†π‘‡
𝑄 = π‘šπ‘βˆ†π‘‡
𝑄 = π‘šπΏ Latent heat
∑R 𝑄R = 0 isolated system
Heat engine
Entropy (2nd Law)
βˆ†π‘‡ = 0 → βˆ†π‘ˆ = 0
π‘Š = 𝑄f − 𝑄U
𝛿𝑄 𝑄
𝑄
𝑇
βˆ†π‘ƒ = 0 → π‘Š = π‘ƒβˆ†π‘‰ − U = U
=
π‘Š
𝑄U Δ𝑆gO'=+ =.'.+> = ˆ
𝑇
𝑄f 𝑇f
𝑇
𝑒
=
=
1
−
βˆ†π‘‰ = 0 → π‘Š = 0
𝑄f
𝑄f Δ𝑆
𝑇U
≥
0
+=A*'.57 =D=.5&
𝑒Y'64A. = 1 −
𝑄
𝑄 = 0 → βˆ†π‘ˆ =– π‘Š
𝑇f πœ… = − U
Δ𝑆 = 𝑆U − 𝑆f engine
βˆ†π‘ˆ = 0 → 𝑄 = π‘Š>D>*5
π‘Š
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