Lead Article: Tables of Physics Formulae
Mass and Inertia
edit
Mass can be considered to be inertial or gravitational.
Inertial mass is the mass associated with the inertia of a body. By Newton's 3rd Law of Motion, the acceleration of a body is proportional to the force applied to it. Force divided by acceleration is the inertial mass.
Gravitational mass is that mass associated with gravitational attraction. By Newton's Law of Gravity, the gravitational force exerted by or on a body is proportional to its gravitational mass.
By Einstein's Principle of Equivalence, inertial and gravitational mass are always equal.
M
i
n
e
r
t
i
a
l
a
=
M
g
r
a
v
i
t
y
g
{\displaystyle M_{\mathrm {inertial} }\mathbf {a} =M_{\mathrm {gravity} }\mathbf {g} \,\!}
not different concepts, but the physical situation may demand the calculation either as summation (discrete) or integration (continuous). Centre of mass is not to be confused with centre of gravity (see Gravitation section).
Note the convenient generalisation of mass density through an n-space, since mass density is simply the amount of mass per unit length, area or volume; there is only a change in dimension number between them.
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Mass density of dimension n
(
V
n
{\displaystyle V_{n}\,\!}
n = 1 for linear mass density,
n = 2 for surface mass density,
n = 3 for volume mass density,
etc
linear mass density
λ
{\displaystyle \lambda \,\!}
surface mass density
σ
{\displaystyle \sigma \,\!}
volume mass density
ρ
{\displaystyle \rho \,\!}
any dimension
n-space mass density:
ρ
n
=
∂
n
m
∂
x
n
⋯
∂
x
2
∂
x
1
=
∂
m
∂
V
n
{\displaystyle \rho _{n}={\frac {\partial ^{n}m}{\partial x_{n}\cdots \partial x_{2}\partial x_{1}}}={\frac {\partial m}{\partial V_{n}}}\,\!}
special cases are:
λ
=
∂
m
∂
x
{\displaystyle \lambda ={\frac {\partial m}{\partial x}}\,\!}
σ
=
∂
2
m
∂
x
2
∂
x
1
=
∂
2
m
∂
S
{\displaystyle \sigma ={\frac {\partial ^{2}m}{\partial x_{2}\partial x_{1}}}={\frac {\partial ^{2}m}{\partial S}}\,\!}
ρ
=
∂
3
m
∂
x
3
∂
x
2
∂
x
1
=
∂
m
∂
V
{\displaystyle \rho ={\frac {\partial ^{3}m}{\partial x_{3}\partial x_{2}\partial x_{1}}}={\frac {\partial m}{\partial V}}\,\!}
kg m-n
[M][L]-n
Total descrete mass
M
{\displaystyle M\,\!}
M
=
∑
i
m
i
{\displaystyle M=\sum _{i}m_{i}\,\!}
kg m
[M][L]
Total continuum mass
M
{\displaystyle M\,\!}
n-space mass density
M
=
∫
ρ
n
d
n
x
=
∫
⋯
∫
∫
ρ
n
d
x
1
d
x
2
⋯
d
x
n
{\displaystyle M=\int \rho _{n}\mathrm {d} ^{n}x=\int \cdots \int \int \rho _{n}\mathrm {d} x_{1}\mathrm {d} x_{2}\cdots \mathrm {d} x_{n}\,\!}
special cases are:
M
=
∫
λ
d
x
{\displaystyle M=\int \lambda \mathrm {d} x\,\!}
M
=
∫
σ
d
A
=
∬
σ
d
x
1
d
x
2
{\displaystyle M=\int \sigma \mathrm {d} A=\iint \sigma \mathrm {d} x_{1}\mathrm {d} x_{2}\,\!}
M
=
∫
ρ
d
V
=
∭
ρ
d
x
1
d
x
2
d
x
3
{\displaystyle M=\int \rho \mathrm {d} V=\iiint \rho \mathrm {d} x_{1}\mathrm {d} x_{2}\mathrm {d} x_{3}\,\!}
kg
[M]
Moment of Mass
(No common symbol)
m
=
r
m
{\displaystyle \mathbf {m} =\mathbf {r} m\,\!}
kg m
[M][L]
Centre of Mass
r
c
o
m
{\displaystyle \mathbf {r} _{\mathrm {com} }\,\!}
(Symbols can vary
enourmously)
i th moment of mass
m
i
=
r
i
m
i
{\displaystyle \mathbf {m} _{i}=\mathbf {r} _{i}m_{i}\,\!}
r
c
o
m
=
1
M
∑
i
r
i
m
i
=
1
M
∑
i
{\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}={\frac {1}{M}}\sum _{i}\,\!}
Centre of a mass for a continuum of mass
r
c
o
m
=
1
M
∫
r
d
m
=
1
M
∫
r
d
n
m
=
1
M
∫
r
ρ
n
d
n
x
{\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\int \mathbf {r} \mathrm {d} \mathbf {m} ={\frac {1}{M}}\int \mathbf {r} \mathrm {d} ^{n}m={\frac {1}{M}}\int \mathbf {r} \rho _{n}\mathrm {d} ^{n}x\,\!}
m
[L]
Moment of Inertia (M.O.I.)
I
{\displaystyle I\,\!}
M.O.I. for Descrete Masses
I
=
∑
i
|
r
|
2
m
{\displaystyle I=\sum _{i}\left|\mathbf {r} \right|^{2}m\,\!}
M.O.I. for a Continuum of Mass
I
=
∫
|
r
|
2
d
m
=
∫
|
r
|
2
ρ
n
d
n
x
{\displaystyle I=\int \left|\mathbf {r} \right|^{2}\mathrm {d} m=\int \left|\mathbf {r} \right|^{2}\rho _{n}\mathrm {d} ^{n}x\,\!}
kg m2 s-1
[M][L]2
Mass Tensor
Components
m
i
j
{\displaystyle m_{\mathrm {ij} }\,\!}
Contraction of the tensor with itself yeilds the more familiar scalar
kg
[M]
M.O.I. Tensor
I
{\displaystyle \mathbf {\mathrm {I} } \,\!}
Components
I
i
j
{\displaystyle I_{\mathrm {ij} }\,\!}
2nd-Order Tensor Matrix form
I
=
(
I
11
I
12
I
13
I
21
I
22
I
23
I
31
I
32
I
33
)
{\displaystyle \mathbf {\mathrm {I} } ={\begin{pmatrix}I_{11}&I_{12}&I_{13}\\I_{21}&I_{22}&I_{23}\\I_{31}&I_{32}&I_{33}\end{pmatrix}}\,\!}
Contraction of the tensor with itself yeilds the more familiar scalar
kg m2 s-1
[M][L]2
Moment of Inertia Theorems
edit
Often the calculations for the M.O.I. of a body are not easy; fortunatley there are theorems which can simplify the calculation.
Theorem
Nomenclature
Equation
Superposition Principle for
M.O.I. about any chosen Axis
I
n
e
t
{\displaystyle I_{\mathrm {net} }\,\!}
I
n
e
t
=
∑
j
I
j
{\displaystyle I_{\mathrm {net} }=\sum _{j}I_{j}\,\!}
Parallel Axis Theorem
M
{\displaystyle M\,\!}
d
{\displaystyle d\,\!}
through the C.O.M. to another parallel axis
I
c
o
m
{\displaystyle I_{\mathrm {com} }\,\!}
the C.O.M.
I
{\displaystyle I\,\!}
I
=
I
c
o
m
+
M
r
2
{\displaystyle I=I_{\mathrm {com} }+Mr^{2}\,\!}
Perpendicular Axis Theorem
i, j, k refer to M.O.I. about any three mutually
perpendicular axes:
the sum of M.O.I. about any two is the third.
I
k
=
I
i
+
I
j
{\displaystyle I_{\mathrm {k} }=I_{\mathrm {i} }+I_{\mathrm {j} }\,\!}
Galilean Transforms
edit
The transformation law from one inertial frame (reference frame travelling at constant velocity - including zero) to another is the Galilean transform. It is only true for classical (Galilei-Newtonian) mechanics.
Unprimed quantites refer to position, velocity and acceleration in one frame F ; primed quantites refer to position, velocity and acceleration in another frame F' moving at velocity V relative to F. Conversely F moves at velocity (—V) relative to F' .
Galilean Inertial Frames
V
{\displaystyle \mathbf {V} \,\!}
two frames F and F' .
r
,
v
,
a
{\displaystyle \mathbf {r} ,\mathbf {v} ,\mathbf {a} \,\!}
as measured in frame F .
r
′
,
v
′
,
a
′
{\displaystyle \mathbf {r} ',\mathbf {v} ',\mathbf {a} '\,\!}
as measured in frame F' .
Relative Position
r
′
=
r
+
V
t
{\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t\,\!}
v
′
=
v
+
V
{\displaystyle \mathbf {v} '=\mathbf {v} +\mathbf {V} \,\!}
Equivalent Accelerations
a
′
=
a
{\displaystyle \mathbf {a} '=\mathbf {a} }
Laws of Classical Mechanics
edit
The following general approaches to classical mechanics are summarized below in the order of establishment. They are equivalent formulations, Newton's is very commonly used due to simplicity, but Hamilton's and Lagrange's equations are more general, and their range can extend into other branches of physics with suitable modifications.
Newton's Formulation (1687)
edit
Force, acceleration, and the momentum rate of change are all equated neatly in Newton's Laws .
1st Law : A zero resultant force acting ON a body BY an external agent causes
zero change in momentum. The effect is a constant momentum vector and therefore
velocity (including zero).
2nd Law : A resultant force acting ON a body BY an external agent causes
change in momentum.
3rd Law : Two bodies i and j mutually exert forces ON each other BY each other,
when in contact.
The 1st law is a special case of the 2nd law. The laws summarized in two
equations (rather than three where one is a corollary). One is an ordinary
differential equation used to summarize the dynamics of the system, the other
is an equivalance between any two agents in the system. F ij =
force ON body i BY body j, F ij = force ON body j BY body i.
F
=
d
p
d
t
{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\!}
F
i
j
=
−
F
j
i
{\displaystyle \mathbf {F} _{\mathrm {ij} }=-\mathbf {F} _{\mathrm {ji} }\,\!}
In applications to a dynamical system of bodies the two equations (effectively)
combine into one. p i = momentum of body i, and F E
resultant external force (due to any agent not part of system). Body i does not
exert a force on itself.
d
p
i
d
t
=
F
E
+
∑
i
≠
j
F
i
j
{\displaystyle {\frac {\mathrm {d} \mathbf {p} _{\mathrm {i} }}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{\mathrm {i} \neq \mathrm {j} }\mathbf {F} _{\mathrm {ij} }\,\!}
Euler-Lagrange Formulation (1750s)
edit
The generalized coordinates and generalized momenta of any classical
dynamical system satisfy the Euler-Lagrange Equation
of (partial) differential equations describing the minimization of the system.
p
α
=
∂
L
∂
q
˙
α
{\displaystyle p_{\alpha }={\frac {\partial L}{\partial {\dot {q}}_{\alpha }}}}
p
˙
α
=
∂
L
∂
q
α
{\displaystyle {\dot {p}}_{\alpha }={\frac {\partial L}{\partial {q}_{\alpha }}}}
Written as a single equation:
d
d
t
(
∂
L
∂
q
˙
α
)
=
−
∂
L
∂
q
α
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{\alpha }}}\right)=-{\frac {\partial L}{\partial q_{\alpha }}}}
Hamilton's Formulation (1833)
edit
The generalized coordinates and generalized momenta of any classical dynamical
system also satisfy Hamilton's equations , which are a set of (partial) differential
equations describing the time development of the system.
∂
p
∂
t
=
−
∂
H
∂
q
{\displaystyle {\frac {\partial \mathbf {p} }{\partial t}}=-{\frac {\partial H}{\partial \mathbf {q} }}}
∂
q
∂
t
=
∂
H
∂
p
{\displaystyle {\frac {\partial \mathbf {q} }{\partial t}}={\frac {\partial H}{\partial \mathbf {p} }}}
The Hamiltonian as a function of generalized coordinates and momenta has the
general form:
H
=
[
q
(
t
)
,
p
(
t
)
,
t
]
{\displaystyle H=\left[\mathbf {q} (t),\mathbf {p} (t),t\right]}
— The value of the Hamiltonian H is the total energy of the dynamical system. For an isolated system, it generally equals the total kinetic T and potential energy V .
— Hamiltonians can be used to analyze energy changes of many classical systems; as diverse as the simplist one-body motion to complex many-body systems. They also apply in non-relativistic quantum mechanics; in the relativistic formulation the hamiltonian can be modified to be relativistic like many other quantities.
Derived Kinematic Quantities
edit
For rotation the vectors are axial vectors (also known as pseudovectors), the direction is perpendicular to the plane of the position vector and tangential direction of rotation, and the sense of rotation is determined by a right hand screw system.
For the inclusion of the scalar angle of rotational position
θ
{\displaystyle \theta \,\!}
n
^
{\displaystyle \mathbf {\hat {n}} \,\!}
Using the basis vectors for polar coordinates, which are
r
^
,
θ
^
,
ϕ
^
{\displaystyle {\boldsymbol {\hat {r}}},{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}\,\!}
n
^
=
θ
^
×
r
^
{\displaystyle \mathbf {\hat {n}} ={\boldsymbol {\hat {\theta }}}\times {\boldsymbol {\hat {r}}}\,\!}
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Velocity
v
{\displaystyle \mathbf {v} \,\!}
v
=
d
r
d
t
{\displaystyle \mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\!}
m s-1
[L][T]-1
Acceleration
a
{\displaystyle \mathbf {a} \,\!}
a
=
d
v
d
t
=
d
2
v
d
t
2
{\displaystyle \mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {v} }{\mathrm {d} t^{2}}}\,\!}
m s-2
[L][T]-2
Jerk
j
{\displaystyle \mathbf {j} \,\!}
j
=
d
a
d
t
=
d
3
v
d
t
3
{\displaystyle \mathbf {j} ={\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{3}\mathbf {v} }{\mathrm {d} t^{3}}}\,\!}
m s-3
[L][T]-3
Angular Velocity
ω
{\displaystyle {\boldsymbol {\omega }}\,\!}
ω
=
n
^
d
θ
d
t
{\displaystyle {\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\,\!}
rad s-1
[T]-1
Angular Acceleration
α
{\displaystyle {\boldsymbol {\alpha }}\,\!}
α
=
d
ω
d
t
=
n
^
d
2
θ
d
t
2
{\displaystyle {\boldsymbol {\alpha }}={\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\,\!}
rad s-2
[T]-2
v
=
ω
×
r
{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!}
a
=
α
×
r
+
ω
×
v
{\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} \,\!}
Derived Dynamic Quantities
edit
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Momentum
p
{\displaystyle \mathbf {p} \,\!}
p
=
m
v
{\displaystyle \mathbf {p} =m\mathbf {v} \,\!}
kg m s-1
[M][L][T]-1
Force
F
{\displaystyle \mathbf {F} \,\!}
F
=
d
p
d
t
{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\!}
N = kg m s-2
[M][L][T]-2
Impulse
Δ
p
,
I
{\displaystyle \Delta \mathbf {p} ,\mathbf {I} \,\!}
I
=
Δ
p
=
∫
t
1
t
2
F
d
t
{\displaystyle \mathbf {I} =\Delta \mathbf {p} =\int _{t_{1}}^{t_{2}}\mathbf {F} \mathrm {d} t\,\!}
kg m s-1
[M][L][T]-1
Angular Momentum
about a position point
R
{\displaystyle \mathbf {R} \,\!}
L
,
J
,
S
{\displaystyle \mathbf {L} ,\mathbf {J} ,\mathbf {S} \,\!}
L
=
(
r
−
R
)
×
p
{\displaystyle \mathbf {L} =\left(\mathbf {r} -\mathbf {R} \right)\times \mathbf {p} \,\!}
kg m2 s-1
[M][L]2 [T]-1
Total, Spin and Orbital
Angular Momentum
L
,
J
,
S
{\displaystyle \mathbf {L} ,\mathbf {J} ,\mathbf {S} \,\!}
L
t
o
t
a
l
=
L
s
p
i
n
+
L
o
r
b
i
t
a
l
{\displaystyle \mathbf {L} _{\mathrm {total} }=\mathbf {L} _{\mathrm {spin} }+\mathbf {L} _{\mathrm {orbital} }\,\!}
kg m2 s-1
[M][L]2 [T]-1
Moment of a Force
about a position point
R
{\displaystyle \mathbf {R} \,\!}
Torque
τ
,
M
{\displaystyle {\boldsymbol {\tau }},\mathbf {M} \,\!}
τ
=
(
r
−
R
)
×
F
=
d
L
d
t
{\displaystyle {\boldsymbol {\tau }}=\left(\mathbf {r} -\mathbf {R} \right)\times \mathbf {F} ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}\,\!}
N m = kg m2 s-2
[M][L]2 [T]-2
Angular Impulse
Δ
L
{\displaystyle \Delta \mathbf {L} \,\!}
No common symbol
Δ
L
=
∫
t
1
t
2
L
d
t
{\displaystyle \Delta \mathbf {L} =\int _{t_{1}}^{t_{2}}\mathbf {L} \mathrm {d} t\,\!}
kg m2 s-1
[M][L]2 [T]-1
Coefficeint of Restitution
e
,
ϵ
{\displaystyle e,\epsilon \,\!}
e
=
|
v
|
s
e
p
a
r
a
t
i
o
n
|
v
|
a
p
p
r
o
a
c
h
{\displaystyle e={\frac {\left|\mathbf {v} \right|_{\mathrm {separation} }}{\left|\mathbf {v} \right|_{\mathrm {approach} }}}\,\!}
usually
0
⩽
e
⩽
1
{\displaystyle 0\leqslant e\leqslant 1\,\!}
but it is possible that
e
⩾
1
{\displaystyle e\geqslant 1\,\!}
Dimensionless
Dimensionless
Translational Collisions
edit
For conservation of mass and momentum see Conservation and Continuity Equations .
General Planar Motion
edit
The plane of motion is considered in a the cartesian x -y plane using basis vectors (i , j ), or alternativley the polar plane containing the (r, θ) coordinates using the basis vectors
(
r
^
,
θ
^
)
{\displaystyle \left(\mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}}\right)\,\!}
For any object moving in any path
r
=
r
(
t
)
{\displaystyle \mathbf {r} =\mathbf {r} \left(t\right)\,\!}
[1]
Quantity
Nomenclature
Equation
Position
r
=
r
(
t
)
{\displaystyle r=r\left(t\right)\,\!}
θ
=
θ
(
t
)
{\displaystyle \theta =\theta \left(t\right)\,\!}
R
=
R
(
t
)
{\displaystyle R=R\left(t\right)\,\!}
curvature at
r
{\displaystyle \mathbf {r} \,\!}
R
^
{\displaystyle \mathbf {\hat {R}} \,\!}
circle of curvature
r
=
r
(
t
)
=
r
r
^
+
θ
θ
^
{\displaystyle \mathbf {r} =\mathbf {r} \left(t\right)=r\mathbf {\hat {r}} +\theta {\hat {\boldsymbol {\theta }}}\,\!}
Velocity
ω
{\displaystyle \omega \,\!}
v
=
r
^
d
2
r
d
t
2
+
r
ω
θ
^
{\displaystyle \mathbf {v} =\mathbf {\hat {r}} {\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}+r\omega {\hat {\boldsymbol {\theta }}}\,\!}
Acceleration
α
{\displaystyle \alpha \,\!}
a
=
(
d
2
r
d
t
2
−
r
ω
2
)
r
^
+
(
r
α
+
2
ω
d
r
d
t
)
θ
^
{\displaystyle \mathbf {a} =\left({\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\omega ^{2}\right)\mathbf {\hat {r}} +\left(r\alpha +2\omega {\frac {\mathrm {d} r}{\mathrm {d} t}}\right){\hat {\boldsymbol {\theta }}}\,\!}
Centripetal Force
m
=
m
R
R
^
{\displaystyle \mathbf {m} =mR\mathbf {\hat {R}} \,\!}
F
⊥
=
−
m
ω
2
R
R
^
{\displaystyle \mathbf {F} _{\bot }=-m\omega ^{2}R\mathbf {\hat {R}} \,\!}
F
⊥
=
−
ω
2
m
{\displaystyle \mathbf {F} _{\bot }=-\omega ^{2}\mathbf {m} \,\!}
Common special cases are:
— the angular components are constant, so these represent equations of motion in a streight line
— the radial components i.e.
|
r
|
{\displaystyle \left|\mathbf {r} \right|\,\!}
not neccersarily a circle, osscilations on an arc of a circle are possible)
—
ω
{\displaystyle \omega \,\!}
|
r
|
{\displaystyle \left|\mathbf {r} \right|\,\!}
α
=
0
{\displaystyle \alpha =0\,\!}
ω
=
0
{\displaystyle \omega =0\,\!}
d
2
r
d
t
2
{\displaystyle {\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\,\!}
Mechanical Energy
edit
General Definitions
edit
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Mechanical Work due
to a Resultant Force
W
{\displaystyle W\,\!}
W
=
∫
C
F
⋅
d
r
{\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r} \,\!}
J = N m = kg m2 s-2
[M][L]2 [T]-2
Work done ON mechanical
system, Work done BY
W
O
N
,
W
B
Y
{\displaystyle W_{\mathrm {ON} },W_{\mathrm {BY} }\,\!}
Δ
W
O
N
=
−
Δ
W
B
Y
{\displaystyle \Delta W_{\mathrm {ON} }=-\Delta W_{\mathrm {BY} }\,\!}
J = N m = kg m2 s-2
[M][L]2 [T]-2
Potential Energy
ϕ
,
Φ
,
U
,
V
,
E
p
{\displaystyle \phi ,\Phi ,U,V,E_{\mathrm {p} }\,\!}
Δ
W
=
−
Δ
V
{\displaystyle \Delta W=-\Delta V\,\!}
J = N m = kg m2 s-2
[M][L]2 [T]-2
Mechanical Power
P
{\displaystyle P\,\!}
P
=
d
E
d
t
{\displaystyle P={\frac {\mathrm {d} E}{\mathrm {d} t}}\,\!}
W = J s-1
[M][L]2 [T]-3
Lagrangian
L
{\displaystyle L\,\!}
L
=
T
−
V
{\displaystyle L=T-V\,\!}
J
[M][L]2 [T]-2
Action
S
{\displaystyle S\,\!}
S
=
∫
L
d
t
{\displaystyle S=\int L{\mathrm {d} t}\,\!}
J s
[M][L]2 [T]-1
Energy Theorems and Principles
edit
Work-Energy Equations
C , this type of integration is a typical example of a line integral.
For formulae on energy conservation see Conservation and Continuity Equations .
Theorem/Principle (Common) Equation
Work-Energy Theorem for Translation
W
=
Δ
T
=
∫
C
F
⋅
d
r
{\displaystyle W=\Delta T=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r} \,\!}
Work-Energy Theorem for Rotation
W
=
Δ
T
=
∫
C
|
τ
|
d
θ
{\displaystyle W=\Delta T=\int _{C}{\left|{\boldsymbol {\tau }}\right|}{\mathrm {d} \theta }\,\!}
General Work-Energy Theorem
W
=
Δ
T
=
∫
C
(
F
⋅
d
r
+
|
τ
|
d
θ
)
{\displaystyle W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\left|{\boldsymbol {\tau }}\right|}{\mathrm {d} \theta }\right)\,\!}
Principle of Least Action
Various minimized quantity formulations are:
Maupertuis' Formulation
S
=
2
∫
T
d
t
{\displaystyle S=2\int T\mathrm {d} t\,\!}
Euler's Formulation
S
=
∫
|
p
|
d
r
{\displaystyle S=\int \left|\mathbf {p} \right|\mathrm {d} r\,\!}
Lagrangian Formulation
S
=
∫
L
d
t
{\displaystyle S=\int L\mathrm {d} t\,\!}
Potential Energy and Work
edit
Every conservative force potential energy (often incorrectly termed as "potential ", which is related to energy but not exactly the same quantity):
F
=
−
∇
W
=
∇
E
p
{\displaystyle \mathbf {F} =-\nabla W=\nabla E_{\mathrm {p} }\,\!}
∫
r
1
r
2
F
d
⋅
r
=
−
Δ
W
=
Δ
E
p
{\displaystyle \int _{\mathbf {r} _{1}}^{\mathbf {r} _{2}}\mathbf {F} \mathrm {d} \cdot \mathbf {r} =-\Delta W=\Delta E_{\mathrm {p} }\,\!}
U can be consistently assigned:
— Wherever the force is zero , its potential energy is defined to be zero as well.
— Whenever the force does positive work , potential energy decreases (becomes more negative ), and vice versa.
Useful Derived Equations
edit
Description (Common) Symbols General Vector/Scalar Equation
Kinetic Energy
T
,
E
k
{\displaystyle T,E_{\mathrm {k} }\,\!}
T
=
1
2
m
|
v
|
2
=
|
p
|
2
2
m
{\displaystyle T={\frac {1}{2}}m{\left|\mathbf {v} \right|^{2}}={\frac {\left|\mathbf {p} \right|^{2}}{2m}}\,\!}
Angular Kinetic Energy
T
,
E
k
{\displaystyle T,E_{\mathrm {k} }\,\!}
T
=
1
2
I
|
ω
|
2
=
|
L
|
2
2
I
{\displaystyle T={\frac {1}{2}}I{\left|{\boldsymbol {\omega }}\right|^{2}}={\frac {\left|\mathbf {L} \right|^{2}}{2I}}\,\!}
Total Kinetic Energy
Sum of translational and rotational kinetic energy
T
,
E
k
{\displaystyle T,E_{\mathrm {k} }\,\!}
T
=
1
2
m
|
v
|
2
+
1
2
I
|
ω
|
2
=
|
p
|
2
2
m
+
|
L
|
2
2
I
{\displaystyle T={\frac {1}{2}}m{\left|\mathbf {v} \right|^{2}}+{\frac {1}{2}}I{\left|{\boldsymbol {\omega }}\right|^{2}}={\frac {\left|\mathbf {p} \right|^{2}}{2m}}+{\frac {\left|\mathbf {L} \right|^{2}}{2I}}\,\!}
Mechanical Work due
to a Resultant Torque
W
{\displaystyle W\,\!}
W
=
∫
C
|
τ
|
d
θ
{\displaystyle W=\int _{C}{\left|{\boldsymbol {\tau }}\right|}{\mathrm {d} \theta }\,\!}
Total work done due to resultant forces and torques
Sum of work due to translational and rotational motion
W
{\displaystyle W\,\!}
W
=
∫
C
(
F
⋅
d
r
+
|
τ
|
d
θ
)
{\displaystyle W=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +\left|{\boldsymbol {\tau }}\right|\mathrm {d} \theta \right)\,\!}
Elastic Potential Energy
E
p
{\displaystyle E_{\mathrm {p} }\,\!}
E
p
=
1
2
k
x
2
=
1
2
F
Δ
(
x
2
)
{\displaystyle E_{\mathrm {p} }={\frac {1}{2}}kx^{2}={\frac {1}{2}}F\Delta \left(x^{2}\right)\,\!}
Power transfer by a resultant force
P
{\displaystyle P\,\!}
P
=
∫
C
F
⋅
d
v
{\displaystyle P=\int _{C}\mathbf {F} \cdot {\mathrm {d} \mathbf {v} }\,\!}
Power transfer by a resultant torque
P
{\displaystyle P\,\!}
P
=
∫
C
τ
⋅
d
ω
{\displaystyle P=\int _{C}{\boldsymbol {\tau }}\cdot {\mathrm {d} {\boldsymbol {\omega }}}\,\!}
Total power transfer due to resultant forces and torques
Sum of power transfer due to translational and rotational motion
P
{\displaystyle P\,\!}
P
=
∫
C
(
F
⋅
d
v
+
τ
⋅
d
ω
)
{\displaystyle P=\int _{C}\left(\mathbf {F} \cdot {\mathrm {d} \mathbf {v} }+{\boldsymbol {\tau }}\cdot {\mathrm {d} {\boldsymbol {\omega }}}\right)\,\!}
Transport Mechanics
edit
Here
A
{\displaystyle \mathbf {A} \,\!}
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Flow Velocity Vector Field
u
{\displaystyle \mathbf {u} \,\!}
u
=
u
(
x
,
t
)
{\displaystyle \mathbf {u} =\mathbf {u} \left(x,t\right)\,\!}
m s-1
[L][T]-1
Mass Current
I
m
{\displaystyle I_{\mathrm {m} }\,\!}
I
m
=
∂
m
∂
t
{\displaystyle I_{\mathrm {m} }={\frac {\partial m}{\partial t}}\,\!}
kg s-1
[M][T]-1
Mass Current Density
j
m
{\displaystyle \mathbf {j} _{\mathrm {m} }\,\!}
j
m
=
A
^
∂
I
m
∂
A
=
A
^
∂
2
m
∂
A
∂
t
{\displaystyle \mathbf {j} _{\mathrm {m} }=\mathbf {\hat {A}} {\frac {\partial I_{\mathrm {m} }}{\partial A}}=\mathbf {\hat {A}} {\frac {\partial ^{2}m}{\partial A\partial t}}\,\!}
kg m-2 s-1
[M][L]-2 [T]-1
Momentum Current
I
p
{\displaystyle I_{\mathrm {p} }\,\!}
I
p
=
∂
|
p
|
∂
t
{\displaystyle I_{\mathrm {p} }={\frac {\partial \left|\mathbf {p} \right|}{\partial t}}\,\!}
kg m s-2
[M][L][T]-2
Momentum Current Density
j
p
{\displaystyle \mathbf {j} _{\mathrm {p} }\,\!}
j
p
=
∂
I
p
∂
A
=
∂
2
p
∂
A
∂
t
{\displaystyle \mathbf {j} _{\mathrm {p} }={\frac {\partial I_{\mathrm {p} }}{\partial A}}={\frac {\partial ^{2}\mathbf {p} }{\partial A\partial t}}\,\!}
kg m s-2
[M][L][T]-2
Damping Parameters, Forces and Torques
edit
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Spring Constant
(Hooke's Law)
k
,
K
,
κ
,
k
H
{\displaystyle k,K,\kappa ,k_{H}\,\!}
k
H
=
Δ
F
⊥
Δ
x
⊥
{\displaystyle k_{H}={\frac {\Delta F_{\bot }}{\Delta x_{\bot }}}\,\!}
N m-1
[M][T]-2
Damping Coefficient
b
{\displaystyle b\,\!}
b
=
−
Δ
F
d
Δ
v
⊥
{\displaystyle b=-{\frac {\Delta F_{\mathrm {d} }}{\Delta v_{\bot }}}\,\!}
N s m-1
[L][T]-1
Damping Force
F
d
{\displaystyle F_{\mathrm {d} }\,\!}
N
[M][L][T]-2
Damping Ratio
ζ
{\displaystyle \zeta \,\!}
ζ
=
b
b
c
{\displaystyle \zeta ={\frac {b}{b_{c}}}\,\!}
dimensionless
dimensionless
Logarithmic decrement
δ
{\displaystyle \delta \,\!}
δ
=
1
n
ln
A
0
A
n
{\displaystyle \delta ={\frac {1}{n}}\ln {\frac {A_{0}}{A_{n}}}\,\!}
A
0
{\displaystyle A_{0}\,\!}
A
n
{\displaystyle A_{n}\,\!}
amplitude n successive peaks
later from
A
0
{\displaystyle A_{0}\,\!}
A
0
>
A
n
{\displaystyle A_{0}>A_{n}\,\!}
dimensionless
dimensionless
Torsion Constant
κ
{\displaystyle \kappa \,\!}
κ
=
−
Δ
τ
Δ
θ
{\displaystyle \kappa =-{\frac {\Delta \tau }{\Delta \theta }}\,\!}
N m rad-1
[M][L]2 [T]-2
Damping Torque
τ
d
{\displaystyle \tau _{\mathrm {d} }\,\!}
N m
[M][L]2 [T]-2
Rotational Damping Coefficient
β
{\displaystyle \beta \,\!}
β
=
−
τ
d
d
θ
/
d
t
{\displaystyle \beta =-{\frac {\tau _{\mathrm {d} }}{\mathrm {d} \theta /\mathrm {d} t}}\,\!}
N m s rad-1
[M][L]2 [T]-1
References
edit
↑ 3000 Solved Problems in Physics, Schaum Series, A. Halpern, Mc Graw Hill, 1988, ISBN 9-780070-257344
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