Classical mechanics is the branch of physics used to describe the motion of macroscopic objects.[1] It is the most familiar of the theories of physics. The concepts it covers, such as mass , acceleration , and force , are commonly used and known.[2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space.[3]
Classical mechanics utilises many equations —as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations , manifolds , Lie groups , and ergodic theory .[4] This article gives a summary of the most important of these.
This article lists equations from Newtonian mechanics , see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics ).
Classical mechanics [ edit ]
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Linear, surface, volumetric mass density
λ らむだ or μ みゅー (especially in acoustics , see below) for Linear, σ しぐま for surface, ρ ろー for volume.
m
=
∫
λ らむだ
d
ℓ
{\displaystyle m=\int \lambda \,\mathrm {d} \ell }
m
=
∬
σ しぐま
d
S
{\displaystyle m=\iint \sigma \,\mathrm {d} S}
m
=
∭
ρ ろー
d
V
{\displaystyle m=\iiint \rho \,\mathrm {d} V}
kg m−n , n = 1, 2, 3
M L−n
Moment of mass[5]
m (No common symbol)
Point mass:
m
=
r
m
{\displaystyle \mathbf {m} =\mathbf {r} m}
Discrete masses about an axis
x
i
{\displaystyle x_{i}}
:
m
=
∑
i
=
1
N
r
i
m
i
{\displaystyle \mathbf {m} =\sum _{i=1}^{N}\mathbf {r} _{i}m_{i}}
Continuum of mass about an axis
x
i
{\displaystyle x_{i}}
:
m
=
∫
ρ ろー
(
r
)
x
i
d
r
{\displaystyle \mathbf {m} =\int \rho \left(\mathbf {r} \right)x_{i}\mathrm {d} \mathbf {r} }
kg m
M L
Center of mass
r com
(Symbols vary)
i -th moment of mass
m
i
=
r
i
m
i
{\displaystyle \mathbf {m} _{i}=\mathbf {r} _{i}m_{i}}
Discrete masses:
r
c
o
m
=
1
M
∑
i
r
i
m
i
=
1
M
∑
i
m
i
{\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}={\frac {1}{M}}\sum _{i}\mathbf {m} _{i}}
Mass continuum:
r
c
o
m
=
1
M
∫
d
m
=
1
M
∫
r
d
m
=
1
M
∫
r
ρ ろー
d
V
{\displaystyle \mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\int \mathrm {d} \mathbf {m} ={\frac {1}{M}}\int \mathbf {r} \,\mathrm {d} m={\frac {1}{M}}\int \mathbf {r} \rho \,\mathrm {d} V}
m
L
2-Body reduced mass
m 12 , μ みゅー Pair of masses = m 1 and m 2
μ みゅー
=
m
1
m
2
m
1
+
m
2
{\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}
kg
M
Moment of inertia (MOI)
I
Discrete Masses:
I
=
∑
i
m
i
⋅
r
i
=
∑
i
|
r
i
|
2
m
{\displaystyle I=\sum _{i}\mathbf {m} _{i}\cdot \mathbf {r} _{i}=\sum _{i}\left|\mathbf {r} _{i}\right|^{2}m}
Mass continuum:
I
=
∫
|
r
|
2
d
m
=
∫
r
⋅
d
m
=
∫
|
r
|
2
ρ ろー
d
V
{\displaystyle I=\int \left|\mathbf {r} \right|^{2}\mathrm {d} m=\int \mathbf {r} \cdot \mathrm {d} \mathbf {m} =\int \left|\mathbf {r} \right|^{2}\rho \,\mathrm {d} V}
kg m2
M L2
Derived kinematic quantities [ edit ]
Kinematic quantities of a classical particle: mass m , position r , velocity v , acceleration a .
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Velocity
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
a
=
d
v
d
t
=
d
2
r
d
t
2
{\displaystyle \mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}}
m s−2
L T−2
Jerk
j
j
=
d
a
d
t
=
d
3
r
d
t
3
{\displaystyle \mathbf {j} ={\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{3}\mathbf {r} }{\mathrm {d} t^{3}}}}
m s−3
L T−3
Jounce
s
s
=
d
j
d
t
=
d
4
r
d
t
4
{\displaystyle \mathbf {s} ={\frac {\mathrm {d} \mathbf {j} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{4}\mathbf {r} }{\mathrm {d} t^{4}}}}
m s−4
L T−4
Angular velocity
ω おめが
ω おめが
=
n
^
d
θ しーた
d
t
{\displaystyle {\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}
rad s−1
T−1
Angular Acceleration
α あるふぁ
α あるふぁ
=
d
ω おめが
d
t
=
n
^
d
2
θ しーた
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
Angular jerk
ζ ぜーた
ζ ぜーた
=
d
α あるふぁ
d
t
=
n
^
d
3
θ しーた
d
t
3
{\displaystyle {\boldsymbol {\zeta }}={\frac {\mathrm {d} {\boldsymbol {\alpha }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{3}\theta }{\mathrm {d} t^{3}}}}
rad s−3
T−3
Derived dynamic quantities [ edit ]
Angular momenta of a classical object.Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point,right: extrinsic orbital angular momentum L about an axis,top: the moment of inertia tensor I and angular velocity ω おめが (L is not always parallel to ω おめが )[6] bottom: momentum p and its radial position r from the axis. The total angular momentum (spin + orbital) is J .
General energy definitions [ edit ]
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Mechanical work due to a Resultant Force
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 L2 T−2
Work done ON mechanical system, Work done BY
W ON , W BY
Δ でるた
W
O
N
=
−
Δ でるた
W
B
Y
{\displaystyle \Delta W_{\mathrm {ON} }=-\Delta W_{\mathrm {BY} }}
J = N m = kg m2 s−2
M L2 T−2
Potential energy
φ ふぁい , Φ ふぁい , U , V , Ep
Δ でるた
W
=
−
Δ でるた
V
{\displaystyle \Delta W=-\Delta V}
J = N m = kg m2 s−2
M L2 T−2
Mechanical power
P
P
=
d
E
d
t
{\displaystyle P={\frac {\mathrm {d} E}{\mathrm {d} t}}}
W = J s−1
M L2 T−3
Every conservative force has a potential energy . By following two principles one can consistently assign a non-relative value to U :
Wherever the force is zero, its potential energy is defined to be zero as well.
Whenever the force does work, potential energy is lost.
Generalized mechanics [ edit ]
Quantity (common name/s)
(Common) symbol/s
Defining equation
SI units
Dimension
Generalized coordinates
q, Q
varies with choice
varies with choice
Generalized velocities
q
˙
,
Q
˙
{\displaystyle {\dot {q}},{\dot {Q}}}
q
˙
≡
d
q
/
d
t
{\displaystyle {\dot {q}}\equiv \mathrm {d} q/\mathrm {d} t}
varies with choice
varies with choice
Generalized momenta
p, P
p
=
∂
L
/
∂
q
˙
{\displaystyle p=\partial L/\partial {\dot {q}}}
varies with choice
varies with choice
Lagrangian
L
L
(
q
,
q
˙
,
t
)
=
T
(
q
˙
)
−
V
(
q
,
q
˙
,
t
)
{\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} )-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}
where
q
=
q
(
t
)
{\displaystyle \mathbf {q} =\mathbf {q} (t)}
and p = p (t ) are vectors of the generalized coords and momenta, as functions of time
J
M L2 T−2
Hamiltonian
H
H
(
p
,
q
,
t
)
=
p
⋅
q
˙
−
L
(
q
,
q
˙
,
t
)
{\displaystyle H(\mathbf {p} ,\mathbf {q} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)}
J
M L2 T−2
Action , Hamilton's principal function
S ,
S
{\displaystyle \scriptstyle {\mathcal {S}}}
S
=
∫
t
1
t
2
L
(
q
,
q
˙
,
t
)
d
t
{\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\mathrm {d} t}
J s
M L2 T−1
In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ しーた , but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector
n
^
=
e
^
r
×
e
^
θ しーた
{\displaystyle \mathbf {\hat {n}} =\mathbf {\hat {e}} _{r}\times \mathbf {\hat {e}} _{\theta }}
defines the axis of rotation,
e
^
r
{\displaystyle \scriptstyle \mathbf {\hat {e}} _{r}}
= unit vector in direction of r ,
e
^
θ しーた
{\displaystyle \scriptstyle \mathbf {\hat {e}} _{\theta }}
= unit vector tangential to the angle.
Translation
Rotation
Velocity
Average:
v
a
v
e
r
a
g
e
=
Δ でるた
r
Δ でるた
t
{\displaystyle \mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}}
Instantaneous:
v
=
d
r
d
t
{\displaystyle \mathbf {v} ={d\mathbf {r} \over dt}}
Angular velocity
ω おめが
=
n
^
d
θ しーた
d
t
{\displaystyle {\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {{\rm {d}}\theta }{{\rm {d}}t}}}
Rotating rigid body :
v
=
ω おめが
×
r
{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} }
Acceleration
Average:
a
a
v
e
r
a
g
e
=
Δ でるた
v
Δ でるた
t
{\displaystyle \mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}}
Instantaneous:
a
=
d
v
d
t
=
d
2
r
d
t
2
{\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}}
Angular acceleration
α あるふぁ
=
d
ω おめが
d
t
=
n
^
d
2
θ しーた
d
t
2
{\displaystyle {\boldsymbol {\alpha }}={\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}}
Rotating rigid body:
a
=
α あるふぁ
×
r
+
ω おめが
×
v
{\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} }
Jerk
Average:
j
a
v
e
r
a
g
e
=
Δ でるた
a
Δ でるた
t
{\displaystyle \mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}}
Instantaneous:
j
=
d
a
d
t
=
d
2
v
d
t
2
=
d
3
r
d
t
3
{\displaystyle \mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {r} }{dt^{3}}}}
Angular jerk
ζ ぜーた
=
d
α あるふぁ
d
t
=
n
^
d
2
ω おめが
d
t
2
=
n
^
d
3
θ しーた
d
t
3
{\displaystyle {\boldsymbol {\zeta }}={\frac {{\rm {d}}{\boldsymbol {\alpha }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\omega }{{\rm {d}}t^{2}}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{3}\theta }{{\rm {d}}t^{3}}}}
Rotating rigid body:
j
=
ζ ぜーた
×
r
+
α あるふぁ
×
a
{\displaystyle \mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a} }
Translation
Rotation
Momentum
Momentum is the "amount of translation"
p
=
m
v
{\displaystyle \mathbf {p} =m\mathbf {v} }
For a rotating rigid body:
p
=
ω おめが
×
m
{\displaystyle \mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} }
Angular momentum
Angular momentum is the "amount of rotation":
L
=
r
×
p
=
I
⋅
ω おめが
{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}}
and the cross-product is a pseudovector i.e. if r and p are reversed in direction (negative), L is not.
In general I is an order-2 tensor , see above for its components. The dot · indicates tensor contraction .
Force and Newton's 2nd law
Resultant force acts on a system at the center of mass, equal to the rate of change of momentum:
F
=
d
p
d
t
=
d
(
m
v
)
d
t
=
m
a
+
v
d
m
d
t
{\displaystyle {\begin{aligned}\mathbf {F} &={\frac {d\mathbf {p} }{dt}}={\frac {d(m\mathbf {v} )}{dt}}\\&=m\mathbf {a} +\mathbf {v} {\frac {{\rm {d}}m}{{\rm {d}}t}}\\\end{aligned}}}
For a number of particles, the equation of motion for one particle i is:[7]
d
p
i
d
t
=
F
E
+
∑
i
≠
j
F
i
j
{\displaystyle {\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}}
where p i = momentum of particle i , F ij = force on particle i by particle j , and F E = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself.
Torque
Torque τ たう is also called moment of a force, because it is the rotational analogue to force:[8]
τ たう
=
d
L
d
t
=
r
×
F
=
d
(
I
⋅
ω おめが
)
d
t
{\displaystyle {\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}=\mathbf {r} \times \mathbf {F} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}}
For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation:
τ たう
=
d
L
d
t
=
d
(
I
⋅
ω おめが
)
d
t
=
d
I
d
t
⋅
ω おめが
+
I
⋅
α あるふぁ
{\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}\\&={\frac {{\rm {d}}\mathbf {I} }{{\rm {d}}t}}\cdot {\boldsymbol {\omega }}+\mathbf {I} \cdot {\boldsymbol {\alpha }}\\\end{aligned}}}
Likewise, for a number of particles, the equation of motion for one particle i is:[9]
d
L
i
d
t
=
τ たう
E
+
∑
i
≠
j
τ たう
i
j
{\displaystyle {\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}}
Yank
Yank is rate of change of force:
Y
=
d
F
d
t
=
d
2
p
d
t
2
=
d
2
(
m
v
)
d
t
2
=
m
j
+
2
a
d
m
d
t
+
v
d
2
m
d
t
2
{\displaystyle {\begin{aligned}\mathbf {Y} &={\frac {d\mathbf {F} }{dt}}={\frac {d^{2}\mathbf {p} }{dt^{2}}}={\frac {d^{2}(m\mathbf {v} )}{dt^{2}}}\\[1ex]&=m\mathbf {j} +\mathbf {2a} {\frac {{\rm {d}}m}{{\rm {d}}t}}+\mathbf {v} {\frac {{\rm {d^{2}}}m}{{\rm {d}}t^{2}}}\end{aligned}}}
For constant mass, it becomes;
Y
=
m
j
{\displaystyle \mathbf {Y} =m\mathbf {j} }
Rotatum
Rotatum Ρ ろー is also called moment of a Yank, because it is the rotational analogue to yank:
P
=
d
τ たう
d
t
=
r
×
Y
=
d
(
I
⋅
α あるふぁ
)
d
t
{\displaystyle {\boldsymbol {\mathrm {P} }}={\frac {{\rm {d}}{\boldsymbol {\tau }}}{{\rm {d}}t}}=\mathbf {r} \times \mathbf {Y} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\alpha }})}{{\rm {d}}t}}}
Impulse
Impulse is the change in momentum:
Δ でるた
p
=
∫
F
d
t
{\displaystyle \Delta \mathbf {p} =\int \mathbf {F} \,dt}
For constant force F :
Δ でるた
p
=
F
Δ でるた
t
{\displaystyle \Delta \mathbf {p} =\mathbf {F} \Delta t}
Twirl/angular impulse is the change in angular momentum:
Δ でるた
L
=
∫
τ たう
d
t
{\displaystyle \Delta \mathbf {L} =\int {\boldsymbol {\tau }}\,dt}
For constant torque τ たう :
Δ でるた
L
=
τ たう
Δ でるた
t
{\displaystyle \Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t}
The precession angular speed of a spinning top is given by:
Ω おめが
=
w
r
I
ω おめが
{\displaystyle {\boldsymbol {\Omega }}={\frac {wr}{I{\boldsymbol {\omega }}}}}
where w is the weight of the spinning flywheel.
The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:
The work done W by an external agent which exerts a force F (at r ) and torque τ たう on an object along a curved path C is:
W
=
Δ でるた
T
=
∫
C
(
F
⋅
d
r
+
τ たう
⋅
n
d
θ しーた
)
{\displaystyle W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\boldsymbol {\tau }}\cdot \mathbf {n} \,{\mathrm {d} \theta }\right)}
where θ しーた is the angle of rotation about an axis defined by a unit vector n .
The change in kinetic energy for an object initially traveling at speed
v
0
{\displaystyle v_{0}}
and later at speed
v
{\displaystyle v}
is:
Δ でるた
E
k
=
W
=
1
2
m
(
v
2
−
v
0
2
)
{\displaystyle \Delta E_{k}=W={\frac {1}{2}}m(v^{2}-{v_{0}}^{2})}
Elastic potential energy [ edit ]
For a stretched spring fixed at one end obeying Hooke's law , the elastic potential energy is
Δ でるた
E
p
=
1
2
k
(
r
2
−
r
1
)
2
{\displaystyle \Delta E_{p}={\frac {1}{2}}k(r_{2}-r_{1})^{2}}
where r 2 and r 1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.
Euler's equations for rigid body dynamics[ edit ]
Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion . These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:[10]
I
⋅
α あるふぁ
+
ω おめが
×
(
I
⋅
ω おめが
)
=
τ たう
{\displaystyle \mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}}
where I is the moment of inertia tensor .
General planar motion [ edit ]
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,
r
=
r
(
t
)
=
r
r
^
{\displaystyle \mathbf {r} =\mathbf {r} (t)=r{\hat {\mathbf {r} }}}
the following general results apply to the particle.
Kinematics
Dynamics
Position
r
=
r
(
r
,
θ しーた
,
t
)
=
r
r
^
{\displaystyle \mathbf {r} =\mathbf {r} \left(r,\theta ,t\right)=r{\hat {\mathbf {r} }}}
Velocity
v
=
r
^
d
r
d
t
+
r
ω おめが
θ しーた
^
{\displaystyle \mathbf {v} ={\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}}
Momentum
p
=
m
(
r
^
d
r
d
t
+
r
ω おめが
θ しーた
^
)
{\displaystyle \mathbf {p} =m\left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}
Angular momenta
L
=
m
r
×
(
r
^
d
r
d
t
+
r
ω おめが
θ しーた
^
)
{\displaystyle \mathbf {L} =m\mathbf {r} \times \left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)}
Acceleration
a
=
(
d
2
r
d
t
2
−
r
ω おめが
2
)
r
^
+
(
r
α あるふぁ
+
2
ω おめが
d
r
d
t
)
θ しーた
^
{\displaystyle \mathbf {a} =\left({\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\omega ^{2}\right){\hat {\mathbf {r} }}+\left(r\alpha +2\omega {\frac {\mathrm {d} r}{{\rm {d}}t}}\right){\hat {\mathbf {\theta } }}}
The centripetal force is
F
⊥
=
−
m
ω おめが
2
R
r
^
=
−
ω おめが
2
m
{\displaystyle \mathbf {F} _{\bot }=-m\omega ^{2}R{\hat {\mathbf {r} }}=-\omega ^{2}\mathbf {m} }
where again m is the mass moment, and the Coriolis force is
F
c
=
2
ω おめが
m
d
r
d
t
θ しーた
^
=
2
ω おめが
m
v
θ しーた
^
{\displaystyle \mathbf {F} _{c}=2\omega m{\frac {{\rm {d}}r}{{\rm {d}}t}}{\hat {\mathbf {\theta } }}=2\omega mv{\hat {\mathbf {\theta } }}}
The Coriolis acceleration and force can also be written:
F
c
=
m
a
c
=
−
2
m
ω おめが
×
v
{\displaystyle \mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}}
Central force motion [ edit ]
For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:
d
2
d
θ しーた
2
(
1
r
)
+
1
r
=
−
μ みゅー
r
2
l
2
F
(
r
)
{\displaystyle {\frac {d^{2}}{d\theta ^{2}}}\left({\frac {1}{\mathbf {r} }}\right)+{\frac {1}{\mathbf {r} }}=-{\frac {\mu \mathbf {r} ^{2}}{\mathbf {l} ^{2}}}\mathbf {F} (\mathbf {r} )}
Equations of motion (constant acceleration)[ edit ]
These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).
Linear motion
Angular motion
v
−
v
0
=
a
t
{\displaystyle \mathbf {v-v_{0}} =\mathbf {a} t}
ω おめが
−
ω おめが
0
=
α あるふぁ
t
{\displaystyle {\boldsymbol {\omega -\omega _{0}}}={\boldsymbol {\alpha }}t}
x
−
x
0
=
1
2
(
v
0
+
v
)
t
{\displaystyle \mathbf {x-x_{0}} ={\tfrac {1}{2}}(\mathbf {v_{0}+v} )t}
θ しーた
−
θ しーた
0
=
1
2
(
ω おめが
0
+
ω おめが
)
t
{\displaystyle {\boldsymbol {\theta -\theta _{0}}}={\tfrac {1}{2}}({\boldsymbol {\omega _{0}+\omega }})t}
x
−
x
0
=
v
0
t
+
1
2
a
t
2
{\displaystyle \mathbf {x-x_{0}} =\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}}
θ しーた
−
θ しーた
0
=
ω おめが
0
t
+
1
2
α あるふぁ
t
2
{\displaystyle {\boldsymbol {\theta -\theta _{0}}}={\boldsymbol {\omega }}_{0}t+{\tfrac {1}{2}}{\boldsymbol {\alpha }}t^{2}}
x
n
t
h
=
v
0
+
a
(
n
−
1
2
)
{\displaystyle \mathbf {x} _{n^{th}}=\mathbf {v} _{0}+\mathbf {a} (n-{\tfrac {1}{2}})}
θ しーた
n
t
h
=
ω おめが
0
+
α あるふぁ
(
n
−
1
2
)
{\displaystyle {\boldsymbol {\theta }}_{n^{th}}={\boldsymbol {\omega }}_{0}+{\boldsymbol {\alpha }}(n-{\tfrac {1}{2}})}
v
2
−
v
0
2
=
2
a
(
x
−
x
0
)
{\displaystyle v^{2}-v_{0}^{2}=2\mathbf {a(x-x_{0})} }
ω おめが
2
−
ω おめが
0
2
=
2
α あるふぁ
(
θ しーた
−
θ しーた
0
)
{\displaystyle \omega ^{2}-\omega _{0}^{2}=2{\boldsymbol {\alpha (\theta -\theta _{0})}}}
For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.
Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω おめが relative to F. Conversely F moves at velocity (—V or —Ω おめが ) relative to F'. The situation is similar for relative accelerations.
Motion of entities
Inertial frames
Accelerating frames
Translation
V = Constant relative velocity between two inertial frames F and F'.
A = (Variable) relative acceleration between two accelerating frames F and F'.
Relative position
r
′
=
r
+
V
t
{\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t}
Relative velocity
v
′
=
v
+
V
{\displaystyle \mathbf {v} '=\mathbf {v} +\mathbf {V} }
Equivalent accelerations
a
′
=
a
{\displaystyle \mathbf {a} '=\mathbf {a} }
Relative accelerations
a
′
=
a
+
A
{\displaystyle \mathbf {a} '=\mathbf {a} +\mathbf {A} }
Apparent/fictitious forces
F
′
=
F
−
F
a
p
p
{\displaystyle \mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }}
Rotation
Ω おめが = Constant relative angular velocity between two frames F and F'.
Λ らむだ = (Variable) relative angular acceleration between two accelerating frames F and F'.
Relative angular position
θ しーた
′
=
θ しーた
+
Ω おめが
t
{\displaystyle \theta '=\theta +\Omega t}
Relative velocity
ω おめが
′
=
ω おめが
+
Ω おめが
{\displaystyle {\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}}
Equivalent accelerations
α あるふぁ
′
=
α あるふぁ
{\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}}
Relative accelerations
α あるふぁ
′
=
α あるふぁ
+
Λ らむだ
{\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}}
Apparent/fictitious torques
τ たう
′
=
τ たう
−
τ たう
a
p
p
{\displaystyle {\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }}
Transformation of any vector T to a rotating frame
d
T
′
d
t
=
d
T
d
t
−
Ω おめが
×
T
{\displaystyle {\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T} }
Mechanical oscillators [ edit ]
SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.
Equations of motion
Physical situation
Nomenclature
Translational equations
Angular equations
SHM
x = Transverse displacement
θ しーた = Angular displacement
A = Transverse amplitude
Θ しーた = Angular amplitude
d
2
x
d
t
2
=
−
ω おめが
2
x
{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x}
Solution:
x
=
A
sin
(
ω おめが
t
+
ϕ
)
{\displaystyle x=A\sin \left(\omega t+\phi \right)}
d
2
θ しーた
d
t
2
=
−
ω おめが
2
θ しーた
{\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta }
Solution:
θ しーた
=
Θ しーた
sin
(
ω おめが
t
+
ϕ
)
{\displaystyle \theta =\Theta \sin \left(\omega t+\phi \right)}
Unforced DHM
b = damping constant
κ かっぱ = torsion constant
d
2
x
d
t
2
+
b
d
x
d
t
+
ω おめが
2
x
=
0
{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0}
Solution (see below for ω おめが ' ):
x
=
A
e
−
b
t
/
2
m
cos
(
ω おめが
′
)
{\displaystyle x=Ae^{-bt/2m}\cos \left(\omega '\right)}
Resonant frequency:
ω おめが
r
e
s
=
ω おめが
2
−
(
b
4
m
)
2
{\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}}
Damping rate:
γ がんま
=
b
/
m
{\displaystyle \gamma =b/m}
Expected lifetime of excitation:
τ たう
=
1
/
γ がんま
{\displaystyle \tau =1/\gamma }
d
2
θ しーた
d
t
2
+
b
d
θ しーた
d
t
+
ω おめが
2
θ しーた
=
0
{\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0}
Solution:
θ しーた
=
Θ しーた
e
−
κ かっぱ
t
/
2
m
cos
(
ω おめが
)
{\displaystyle \theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)}
Resonant frequency:
ω おめが
r
e
s
=
ω おめが
2
−
(
κ かっぱ
4
m
)
2
{\displaystyle \omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}}
Damping rate:
γ がんま
=
κ かっぱ
/
m
{\displaystyle \gamma =\kappa /m}
Expected lifetime of excitation:
τ たう
=
1
/
γ がんま
{\displaystyle \tau =1/\gamma }
Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics (2nd ed.), Springer, ISBN 978-0-387-96890-2
Berkshire, Frank H. ; Kibble, T. W. B. (2004), Classical Mechanics (5th ed.), Imperial College Press, ISBN 978-1-86094-435-2
Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001), Structure and Interpretation of Classical Mechanics , MIT Press, ISBN 978-0-262-19455-6
Linear/translational quantities
Angular/rotational quantities
Dimensions
1
L
L2
Dimensions
1
θ しーた
θ しーた 2
T
time : t s
absement : A m s
T
time : t s
1
distance : d , position : r , s , x , displacement m
area : A m2
1
angle : θ しーた , angular displacement : θ しーた rad
solid angle : Ω おめが rad2 , sr
T−1
frequency : f s−1 , Hz
speed : v , velocity : v m s−1
kinematic viscosity : ν にゅー ,specific angular momentum : h m2 s−1
T−1
frequency : f , rotational speed : n , rotational velocity : n s−1 , Hz
angular speed : ω おめが , angular velocity : ω おめが rad s−1
T−2
acceleration : a m s−2
T−2
rotational acceleration s−2
angular acceleration : α あるふぁ rad s−2
T−3
jerk : j m s−3
T−3
angular jerk : ζ ぜーた rad s−3
M
mass : m kg
weighted position: M ⟨x ⟩ = ∑ m x
ML2
moment of inertia : I kg m2
MT−1
Mass flow rate :
m
˙
{\displaystyle {\dot {m}}}
kg s−1
momentum : p , impulse : J kg m s−1 , N s
action : 𝒮 , actergy : ℵ kg m2 s−1 , J s
ML2 T−1
angular momentum : L , angular impulse : Δ でるた L kg m2 s−1
action : 𝒮 , actergy : ℵ kg m2 s−1 , J s
MT−2
force : F , weight : F g kg m s−2 , N
energy : E , work : W , Lagrangian : L kg m2 s−2 , J
ML2 T−2
torque : τ たう , moment : M kg m2 s−2 , N m
energy : E , work : W , Lagrangian : L kg m2 s−2 , J
MT−3
yank : Y kg m s−3 , N s−1
power : P kg m2 s−3 , W
ML2 T−3
rotatum : P kg m2 s−3 , N m s−1
power : P kg m2 s−3 , W