Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Linear, surface, volumetric mass density λ or μ (especially in acoustics , see below) for Linear, σ for surface, ρ for volume. m = ∫ λ d ℓ {displaystyle m=int lambda ,mathrm {d} ell }
m = ∬ σ d S {displaystyle m=iint sigma ,mathrm {d} S}
m = ∭ ρ 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 ( 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 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 Pair of masses = m 1 and m 2 μ = 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 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
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 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 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 Angular jerk ζ ζ = d α d t = n ^ d 3 θ 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
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Momentum p p = m v {displaystyle mathbf {p} =mmathbf {v} }
kg m s−1 M L T−1 Force F F = d p / d t {displaystyle mathbf {F} =mathrm {d} mathbf {p} /mathrm {d} t}
N = kg m s−2 M L T−2 Impulse J , Δp , I J = Δ p = ∫ t 1 t 2 F d t {displaystyle mathbf {J} =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 0 ,L , J , S L = ( r − r 0 ) × p {displaystyle mathbf {L} =left(mathbf {r} -mathbf {r} _{0}right)times mathbf {p} }
Most of the time we can set r 0 = 0 if particles are orbiting about axes intersecting at a common point.
kg m2 s−1 M L2 T−1 Moment of a force about a position point r 0 ,Torque
τ , M τ = ( r − r 0 ) × F = d L d t {displaystyle {boldsymbol {tau }}=left(mathbf {r} -mathbf {r} _{0}right)times mathbf {F} ={frac {mathrm {d} mathbf {L} }{mathrm {d} t}}}
N m = kg m2 s−2 M L2 T−2 Angular impulse ΔL (no common symbol) Δ L = ∫ t 1 t 2 τ d t {displaystyle Delta mathbf {L} =int _{t_{1}}^{t_{2}}{boldsymbol {tau }},mathrm {d} t}
kg m2 s−1 M L2 T−1
General energy definitions
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension Mechanical work due to a Resultant ForceW 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 W O N = − Δ 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 , Φ, U , V , Ep Δ W = − Δ 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. 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)} 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 functionS , 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 , but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector
n ^ = e ^ r × e ^ θ {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}} r
e ^ θ {displaystyle scriptstyle mathbf {hat {e}} _{theta }}
Translation Rotation Velocity Average:
v a v e r a g e = Δ r Δ t {displaystyle mathbf {v} _{mathrm {average} }={Delta mathbf {r} over Delta t}}
v = d r d t {displaystyle mathbf {v} ={dmathbf {r} over dt}}
Angular velocity ω = n ^ d θ d t {displaystyle {boldsymbol {omega }}=mathbf {hat {n}} {frac {{rm {d}}theta }{{rm {d}}t}}} rigid body :
v = ω × r {displaystyle mathbf {v} ={boldsymbol {omega }}times mathbf {r} }
Acceleration Average:
a a v e r a g e = Δ v Δ 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 {dmathbf {v} }{dt}}={frac {d^{2}mathbf {r} }{dt^{2}}}}
Angular acceleration
α = d ω d t = n ^ d 2 θ 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 × r + ω × 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 a Δ 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 {dmathbf {a} }{dt}}={frac {d^{2}mathbf {v} }{dt^{2}}}={frac {d^{3}mathbf {r} }{dt^{3}}}}
Angular jerk
ζ = d α d t = n ^ d 2 ω d t 2 = n ^ d 3 θ 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 × r + α × 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} =mmathbf {v} }
For a rotating rigid body:
p = ω × m {displaystyle mathbf {p} ={boldsymbol {omega }}times mathbf {m} }
Angular momentum Angular momentum is the “amount of rotation”:
L = r × p = I ⋅ ω {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 {dmathbf {p} }{dt}}={frac {d(mmathbf {v} )}{dt}}\&=mmathbf {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 _{ineq j}mathbf {F} _{ij}}
where p i i , F ij on i by j , and F E i does not exert a force on itself.
Torque Torque τ is also called moment of a force, because it is the rotational analogue to force:[ 8]
τ = d L d t = r × F = d ( I ⋅ ω ) 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 = d L d t = d ( I ⋅ ω ) d t = d I d t ⋅ ω + I ⋅ α {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 E + ∑ i ≠ j τ i j {displaystyle {frac {mathrm {d} mathbf {L} _{i}}{mathrm {d} t}}={boldsymbol {tau }}_{E}+sum _{ineq 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 {dmathbf {F} }{dt}}={frac {d^{2}mathbf {p} }{dt^{2}}}={frac {d^{2}(mmathbf {v} )}{dt^{2}}}\[1ex]&=mmathbf {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} =mmathbf {j} }
Rotatum Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank:
P = d τ d t = r × Y = d ( I ⋅ α ) 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 p = ∫ F d t {displaystyle Delta mathbf {p} =int mathbf {F} ,dt}
For constant force F :
Δ p = F Δ t {displaystyle Delta mathbf {p} =mathbf {F} Delta t}
Twirl/angular impulse is the change in angular momentum:
Δ L = ∫ τ d t {displaystyle Delta mathbf {L} =int {boldsymbol {tau }},dt}
For constant torque τ :
Δ L = τ Δ t {displaystyle Delta mathbf {L} ={boldsymbol {tau }}Delta t}
The precession angular speed of a spinning top is given by:
Ω = w r I ω {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 on an object along a curved path C is:
W = Δ T = ∫ C ( F ⋅ d r + τ ⋅ n d θ ) {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 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}}
v {displaystyle v}
Δ 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})}
For a stretched spring fixed at one end obeying Hooke’s law , the elastic potential energy is
Δ 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 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 + ω × ( I ⋅ ω ) = τ {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 .
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 , t ) = r r ^ {displaystyle mathbf {r} =mathbf {r} left(r,theta ,tright)=r{hat {mathbf {r} }}}
Velocity
v = r ^ d r d t + r ω θ ^ {displaystyle mathbf {v} ={hat {mathbf {r} }}{frac {mathrm {d} r}{mathrm {d} t}}+romega {hat {mathbf {theta } }}}
Momentum
p = m ( r ^ d r d t + r ω θ ^ ) {displaystyle mathbf {p} =mleft({hat {mathbf {r} }}{frac {mathrm {d} r}{mathrm {d} t}}+romega {hat {mathbf {theta } }}right)}
Angular momenta
L = m r × ( r ^ d r d t + r ω θ ^ ) {displaystyle mathbf {L} =mmathbf {r} times left({hat {mathbf {r} }}{frac {mathrm {d} r}{mathrm {d} t}}+romega {hat {mathbf {theta } }}right)}
Acceleration
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}}}-romega ^{2}right){hat {mathbf {r} }}+left(ralpha +2omega {frac {mathrm {d} r}{{rm {d}}t}}right){hat {mathbf {theta } }}}
The centripetal force is
F ⊥ = − m ω 2 R r ^ = − ω 2 m {displaystyle mathbf {F} _{bot }=-momega ^{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 m d r d t θ ^ = 2 ω m v θ ^ {displaystyle mathbf {F} _{c}=2omega m{frac {{rm {d}}r}{{rm {d}}t}}{hat {mathbf {theta } }}=2omega mv{hat {mathbf {theta } }}}
The Coriolis acceleration and force can also be written:
F c = m a c = − 2 m ω × v {displaystyle mathbf {F} _{c}=mmathbf {a} _{c}=-2m{boldsymbol {omega times v}}}
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 2 ( 1 r ) + 1 r = − μ r 2 l 2 F ( r ) {displaystyle {frac {d^{2}}{dtheta ^{2}}}left({frac {1}{mathbf {r} }}right)+{frac {1}{mathbf {r} }}=-{frac {mu mathbf {r} ^{2}}{mathbf {l} ^{2}}}mathbf {F} (mathbf {r} )}
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 − ω 0 = α 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 − θ 0 = 1 2 ( ω 0 + ω ) 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 − θ 0 = ω 0 t + 1 2 α 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 n t h = ω 0 + α ( 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}=2mathbf {a(x-x_{0})} }
ω 2 − ω 0 2 = 2 α ( θ − θ 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 relative to F. Conversely F moves at velocity (—V or —Ω ) 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 = Constant relative angular velocity between two frames F and F’.Λ = (Variable) relative angular acceleration between two accelerating frames F and F’.
Relative angular position
θ ′ = θ + Ω t {displaystyle theta ‘=theta +Omega t}
ω ′ = ω + Ω {displaystyle {boldsymbol {omega }}’={boldsymbol {omega }}+{boldsymbol {Omega }}}
Equivalent accelerations
α ′ = α {displaystyle {boldsymbol {alpha }}’={boldsymbol {alpha }}}
Relative accelerations
α ′ = α + Λ {displaystyle {boldsymbol {alpha }}’={boldsymbol {alpha }}+{boldsymbol {Lambda }}}
Apparent/fictitious torques
τ ′ = τ − τ 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 × T {displaystyle {frac {{rm {d}}mathbf {T} ‘}{{rm {d}}t}}={frac {{rm {d}}mathbf {T} }{{rm {d}}t}}-{boldsymbol {Omega }}times mathbf {T} }
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 displacementθ = Angular displacementA = Transverse amplitudeΘ = Angular amplitude d 2 x d t 2 = − ω 2 x {displaystyle {frac {mathrm {d} ^{2}x}{mathrm {d} t^{2}}}=-omega ^{2}x}
Solution:
x = A sin ( ω t + ϕ ) {displaystyle x=Asin left(omega t+phi right)}
d 2 θ d t 2 = − ω 2 θ {displaystyle {frac {mathrm {d} ^{2}theta }{mathrm {d} t^{2}}}=-omega ^{2}theta }
Solution:
θ = Θ sin ( ω t + ϕ ) {displaystyle theta =Theta sin left(omega t+phi right)}
Unforced DHM b = damping constantκ = torsion constant d 2 x d t 2 + b d x d t + ω 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’ ):
x = A e − b t / 2 m cos ( ω ′ ) {displaystyle x=Ae^{-bt/2m}cos left(omega ‘right)}
Resonant frequency:
ω r e s = ω 2 − ( b 4 m ) 2 {displaystyle omega _{mathrm {res} }={sqrt {omega ^{2}-left({frac {b}{4m}}right)^{2}}}}
Damping rate:
γ = b / m {displaystyle gamma =b/m}
Expected lifetime of excitation:
τ = 1 / γ {displaystyle tau =1/gamma }
d 2 θ d t 2 + b d θ d t + ω 2 θ = 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 = Θ e − κ t / 2 m cos ( ω ) {displaystyle theta =Theta e^{-kappa t/2m}cos left(omega right)}
Resonant frequency:
ω r e s = ω 2 − ( κ 4 m ) 2 {displaystyle omega _{mathrm {res} }={sqrt {omega ^{2}-left({frac {kappa }{4m}}right)^{2}}}}
Damping rate:
γ = κ / m {displaystyle gamma =kappa /m}
Expected lifetime of excitation:
τ = 1 / γ {displaystyle tau =1/gamma }
Angular frequencies Physical situation Nomenclature Equations Linear undamped unforced SHO k = spring constantm = mass of oscillating bob ω = k m {displaystyle omega ={sqrt {frac {k}{m}}}}
Linear unforced DHO k = spring constantb = Damping coefficient ω ′ = k m − ( b 2 m ) 2 {displaystyle omega ‘={sqrt {{frac {k}{m}}-left({frac {b}{2m}}right)^{2}}}}
Low amplitude angular SHO I = Moment of inertia about oscillating axisκ = torsion constant ω = κ I {displaystyle omega ={sqrt {frac {kappa }{I}}}}
Low amplitude simple pendulum L = Length of pendulumg = Gravitational accelerationΘ = Angular amplitude Approximate value
ω = g L {displaystyle omega ={sqrt {frac {g}{L}}}}
Exact value can be shown to be:
ω = g L [ 1 + ∑ k = 1 ∞ ∏ n = 1 k ( 2 n − 1 ) ∏ n = 1 m ( 2 n ) sin 2 n Θ ] {displaystyle omega ={sqrt {frac {g}{L}}}left[1+sum _{k=1}^{infty }{frac {prod _{n=1}^{k}left(2n-1right)}{prod _{n=1}^{m}left(2nright)}}sin ^{2n}Theta right]}
Energy in mechanical oscillations Physical situation Nomenclature Equations SHM energy T = kinetic energyU = potential energyE = total energyPotential energy
U = m 2 ( x ) 2 = m ( ω A ) 2 2 cos 2 ( ω t + ϕ ) {displaystyle U={frac {m}{2}}left(xright)^{2}={frac {mleft(omega Aright)^{2}}{2}}cos ^{2}(omega t+phi )} x = A :
U m a x m 2 ( ω A ) 2 {displaystyle U_{mathrm {max} }{frac {m}{2}}left(omega Aright)^{2}}
Kinetic energy
T = ω 2 m 2 ( d x d t ) 2 = m ( ω A ) 2 2 sin 2 ( ω t + ϕ ) {displaystyle T={frac {omega ^{2}m}{2}}left({frac {mathrm {d} x}{mathrm {d} t}}right)^{2}={frac {mleft(omega Aright)^{2}}{2}}sin ^{2}left(omega t+phi right)}
Total energy
E = T + U {displaystyle E=T+U}
DHM energy E = m ( ω A ) 2 2 e − b t / m {displaystyle E={frac {mleft(omega Aright)^{2}}{2}}e^{-bt/m}}
Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics ISBN 978-0-387-96890-2 Berkshire, Frank H. ; Kibble, T. W. B. (2004), Classical Mechanics 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
: 
![[en] | List of equations in classical mechanics](https://en.wikipedia.org/wiki/File:Kinematics.svg)
![[en] | List of equations in classical mechanics](https://en.wikipedia.org/wiki/File:Classical_angular_momentum.svg)
and p = p(t) are vectors of the generalized coords and momenta, as functions of time
= unit vector in direction of 
= unit vector tangential to the angle.
Instantaneous:
Rotating 
![{displaystyle {begin{aligned}mathbf {Y} &={frac {dmathbf {F} }{dt}}={frac {d^{2}mathbf {p} }{dt^{2}}}={frac {d^{2}(mmathbf {v} )}{dt^{2}}}\[1ex]&=mmathbf {j} +mathbf {2a} {frac {{rm {d}}m}{{rm {d}}t}}+mathbf {v} {frac {{rm {d^{2}}}m}{{rm {d}}t^{2}}}end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a298a3dbfef1b35f56011f1b5d941a948e09024)
and later at speed 
is: 
Relative velocity 
![{displaystyle omega ={sqrt {frac {g}{L}}}left[1+sum _{k=1}^{infty }{frac {prod _{n=1}^{k}left(2n-1right)}{prod _{n=1}^{m}left(2nright)}}sin ^{2n}Theta right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d051a3e661a258eec322d4e38c4560c8013746a)
Maximum value at x = A: 
