List of equations in classical mechanics

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. 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. 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.

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. This page 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

Mass and inertia

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.	$m=\int \lambda \mathrm {d} \ell$ $m=\iint \sigma \mathrm {d} S$ $m=\iiint \rho \mathrm {d} V\,\!$	kg m−n, n = 1, 2, 3	[M][L]−n
Moment of mass	m (No common symbol)	Point mass: $\mathbf {m} =\mathbf {r} m\,\!$ Discrete masses about an axis $x_{i}\,\!$: $\mathbf {m} =\sum _{i=1}^{N}\mathbf {r} _{\mathrm {i} }m_{i}\,\!$ Continuum of mass about an axis $x_{i}\,\!$: $\mathbf {m} =\int \rho \left(\mathbf {r} \right)x_{i}\mathrm {d} \mathbf {r} \,\!$	kg m	[M][L]
Centre of mass	rcom (Symbols vary)	ith moment of mass $\mathbf {m} _{\mathrm {i} }=\mathbf {r} _{\mathrm {i} }m_{i}\,\!$ Discrete masses: $\mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{\mathrm {i} }m_{i}={\frac {1}{M}}\sum _{i}\mathbf {m} _{\mathrm {i} }\,\!$ Mass continuum: $\mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\int \mathrm {d} \mathbf {m} ={\frac {1}{M}}\int \mathbf {r} \mathrm {d} m={\frac {1}{M}}\int \mathbf {r} \rho \mathrm {d} V\,\!$	m	[L]
2-Body reduced mass	m12, μ Pair of masses = m1 and m2	$\mu =\left(m_{1}m_{2}\right)/\left(m_{1}+m_{2}\right)\,\!$	kg	[M]
Moment of inertia (MOI)	I	Discrete Masses: $I=\sum _{i}\mathbf {m} _{\mathrm {i} }\cdot \mathbf {r} _{\mathrm {i} }=\sum _{i}\left|\mathbf {r} _{\mathrm {i} }\right|^{2}m\,\!$ Mass continuum: $I=\int \left|\mathbf {r} \right|^{2}\mathrm {d} m=\int \mathbf {r} \cdot \mathrm {d} \mathbf {m} =\int \left|\mathbf {r} \right|^{2}\rho \mathrm {d} V\,\!$	kg m2	[M][L]2

Derived kinematic quantities

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	$\mathbf {v} =\mathrm {d} \mathbf {r} /\mathrm {d} t\,\!$	m s−1	[L][T]−1
Acceleration	a	$\mathbf {a} =\mathrm {d} \mathbf {v} /\mathrm {d} t=\mathrm {d} ^{2}\mathbf {r} /\mathrm {d} t^{2}\,\!$	m s−2	[L][T]−2
Jerk	j	$\mathbf {j} =\mathrm {d} \mathbf {a} /\mathrm {d} t=\mathrm {d} ^{3}\mathbf {r} /\mathrm {d} t^{3}\,\!$	m s−3	[L][T]−3
Angular velocity	ω	${\boldsymbol {\omega }}=\mathbf {\hat {n}} \left(\mathrm {d} \theta /\mathrm {d} t\right)\,\!$	rad s−1	[T]−1
Angular Acceleration	α	${\boldsymbol {\alpha }}=\mathrm {d} {\boldsymbol {\omega }}/\mathrm {d} t=\mathbf {\hat {n}} \left(\mathrm {d} ^{2}\theta /\mathrm {d} t^{2}\right)\,\!$	rad s−2	[T]−2

Derived dynamic quantities

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 (L is not always parallel to ω)

bottom: momentum p and it's radial position r from the axis.

The total angular momentum (spin + orbital) is J.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Momentum	p	$\mathbf {p} =m\mathbf {v} \,\!$	kg m s−1	[M][L][T]−1
Force	F	$\mathbf {F} =\mathrm {d} \mathbf {p} /\mathrm {d} t\,\!$	N = kg m s−2	[M][L][T]−2
Impulse	Δp, I	$\mathbf {I} =\Delta \mathbf {p} =\int _{t_{1}}^{t_{2}}\mathbf {F} \mathrm {d} t\,\!$	kg m s−1	[M][L][T]−1
Angular momentum about a position point r0,	L, J, S	$\mathbf {L} =\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {p} \,\!$ Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point.	kg m2 s−1	[M][L]2[T]−1
Moment of a force about a position point r0, Torque	τ, M	${\boldsymbol {\tau }}=\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {F} =\mathrm {d} \mathbf {L} /\mathrm {d} t\,\!$	N m = kg m2 s−2	[M][L]2[T]−2
Angular impulse	ΔL (no common symbol)	$\Delta \mathbf {L} =\int _{t_{1}}^{t_{2}}{\boldsymbol {\tau }}\mathrm {d} t\,\!$	kg m2 s−1	[M][L]2[T]−1

General energy definitions

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Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Mechanical work due to a Resultant Force	W	$W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r} \,\!$	J = N m = kg m2 s−2	[M][L]2[T]−2
Work done ON mechanical system, Work done BY	WON, WBY	$\Delta W_{\mathrm {ON} }=-\Delta W_{\mathrm {BY} }\,\!$	J = N m = kg m2 s−2	[M][L]2[T]−2
Potential energy	φ, Φ, U, V, Ep	$\Delta W=-\Delta V\,\!$	J = N m = kg m2 s−2	[M][L]2[T]−2
Mechanical power	P	$P=\mathrm {d} E/\mathrm {d} t\,\!$	W = J s−1	[M][L]2[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

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File:Generalized coordinates 1df.svg

Generalized coordinates for one degree of freedom (of a particle moving in a complicated path). Instead of using all three Cartesian coordinates x, y, z (or other standard coordinate systems), only one is needed and is completely arbitrary to define the position. Four possibilities are shown.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Generalized coordinates	q, Q		varies with choice	varies with choice
Generalized velocities	${\dot {q}},{\dot {Q}}\,\!$	${\dot {q}}\equiv \mathrm {d} q/\mathrm {d} t\,\!$	varies with choice	varies with choice
Generalized momenta	p, P	$p=\partial L/\partial {\dot {q}}\,\!$	varies with choice	varies with choice
Lagrangian	L	$L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} )-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,\!$ where $\mathbf {q} =\mathbf {q} (t)\,\!$ and p = p(t) are vectors of the generalized coords and momenta, as functions of time	J	[M][L]2[T]−2
Hamiltonian	H	$H(\mathbf {p} ,\mathbf {q} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,\!$	J	[M][L]2[T]−2
Action, Hamilton's principle function	S, $\scriptstyle {\mathcal {S}}\,\!$	${\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\mathrm {d} t\,\!$	J s	[M][L]2[T]−1

Kinematics

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

${\mathbf {\hat {n}}}={\mathbf {\hat {e}}}_{r}\times {\mathbf {\hat {e}}}_{\theta }\,\!$

defines the axis of rotation, $\scriptstyle {\mathbf {\hat {e}}}_{r}\,\!$ = unit vector in direction of r, $\scriptstyle {\mathbf {\hat {e}}}_{\theta }\,\!$ = unit vector tangential to the angle.

	Translation	Rotation
Velocity	Average: $\mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}$ Instantaneous: $\mathbf {v} ={d\mathbf {r} \over dt}$	Angular velocity ${\boldsymbol {\omega }}={\mathbf {\hat {n}}}{\frac {{\rm {d}}\theta }{{\rm {d}}t}}\,\!$ Rotating rigid body: $\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!$
Acceleration	Average: $\mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}$ Instantaneous: $\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}$	Angular acceleration ${\boldsymbol {\alpha }}={\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}={\mathbf {\hat {n}}}{\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}\,\!$ Rotating rigid body: $\mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} \,\!$
Jerk	Average: $\mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}$ Instantaneous: $\mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {r} }{dt^{3}}}$	Angular jerk ${\boldsymbol {\zeta }}={\frac {{\rm {d}}{\boldsymbol {\alpha }}}{{\rm {d}}t}}={\mathbf {\hat {n}}}{\frac {{\rm {d}}^{2}\omega }{{\rm {d}}t^{2}}}={\mathbf {\hat {n}}}{\frac {{\rm {d}}^{3}\theta }{{\rm {d}}t^{3}}}\,\!$ Rotating rigid body: $\mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a} \,\!$

Dynamics

	Translation	Rotation
Momentum	Momentum is the "amount of translation" $\mathbf {p} =m\mathbf {v}$ For a rotating rigid body: $\mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} \,\!$	Angular momentum Angular momentum is the "amount of rotation": $\mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}$ 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: ${\begin{aligned}\mathbf {F} &={\frac {d\mathbf {p} }{dt}}={\frac {d(m\mathbf {v} )}{dt}}\\&=m\mathbf {a} +\mathbf {v} {\frac {{\rm {d}}m}{{\rm {d}}t}}\\\end{aligned}}\,\!$ For a number of particles, the equation of motion for one particle i is: ${\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!$ where pi = momentum of particle i, Fij = force on particle i by particle j, and FE = resultant external force (due to any agent not part of system). Particle 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: ${\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}=\mathbf {r} \times \mathbf {F} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}\,\!$ For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: ${\begin{aligned}{\boldsymbol {\tau }}&={\frac {{\rm {d}}{\mathbf {L}}}{{\rm {d}}t}}={\frac {{\rm {d}}({\mathbf {I}}\cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}\\&={\frac {{\rm {d}}{\mathbf {I}}}{{\rm {d}}t}}\cdot {\boldsymbol {\omega }}+{\mathbf {I}}\cdot {\boldsymbol {\alpha }}\\\end{aligned}}\,\!$ Likewise, for a number of particles, the equation of motion for one particle i is: ${\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,\!$
Yank	Yank is rate of change of force: ${\begin{aligned}\mathbf {Y} &={\frac {d\mathbf {F} }{dt}}={\frac {d^{2}\mathbf {p} }{dt^{2}}}={\frac {d^{2}(m\mathbf {v} )}{dt^{2}}}\\&=m\mathbf {j} +\mathbf {2a} {\frac {{\rm {d}}m}{{\rm {d}}t}}+\mathbf {v} {\frac {{\rm {d^{2}}}m}{{\rm {d}}t^{2}}}\\\end{aligned}}\,\!$ For constant mass, it becomes; $\mathbf {Y} =m\mathbf {j}$	Rotatum Rotatum Ρ is also called moment of a Yank, becuause it is the rotational analogue to yank: ${\boldsymbol {\mathrm {P} }}={\frac {{\rm {d}}\mathbf {\tau } }{{\rm {d}}t}}=\mathbf {r} \times \mathbf {Y} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\alpha }})}{{\rm {d}}t}}\,\!$
Impulse	Impulse is the change in momentum: $\Delta \mathbf {p} =\int \mathbf {F} dt$ For constant force F: $\Delta \mathbf {p} =\mathbf {F} \Delta t$	Angular impulse is the change in angular momentum: $\Delta \mathbf {L} =\int {\boldsymbol {\tau }}dt$ For constant torque τ: $\Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t$

Precession

The precession angular speed of a spinning top is given by:

${\boldsymbol {\Omega }}={\frac {wr}{I{\boldsymbol {\omega }}}}$

where w is the weight of the spinning flywheel.

Energy

The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)

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:

$W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\boldsymbol {\tau }}\cdot \mathbf {n} {\mathrm {d} \theta }\right)\,\!$

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy

$\Delta E_{k}=W={\frac {1}{2}}m(v^{2}-{v_{0}}^{2})$

Elastic potential energy

For a stretched spring fixed at one end obeying Hooke's law:

$\Delta E_{p}={\frac {1}{2}}k(r_{2}-r_{1})^{2}\,\!$

where r₂ and r₁ 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

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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:

$\mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}\,\!$

where I is the moment of inertia tensor.

General planar motion

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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,

$\mathbf {r} ={\mathbf {r}}(t)=r{\mathbf {\hat {e}}}_{r}\,\!$

the following general results apply to the particle.

Kinematics	Dynamics
Position $\mathbf {r} ={\mathbf {r}}\left(r,\theta ,t\right)=r{\mathbf {\hat {e}}}_{r}$
Velocity $\mathbf {v} ={\mathbf {\hat {e}}}_{r}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\mathbf {\hat {e}}}_{\theta }$	Momentum $\mathbf {p} =m\left({\mathbf {\hat {e}} }_{r}{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}+r\omega {\mathbf {\hat {e}} }_{\theta }\right)$ Angular momenta $\mathbf {L} =m{\mathbf {r}}\times \left({\mathbf {\hat {e}}}_{r}{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}+r\omega {\mathbf {\hat {e}}}_{\theta }\right)$
Acceleration $\mathbf {a} =\left({\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\omega ^{2}\right){\mathbf {\hat {e}}}_{r}+\left(r\alpha +2\omega {\frac {\mathrm {d} r}{{\rm {d}}t}}\right){\mathbf {\hat {e}}}_{\theta }$	The centripetal force is $\mathbf {F} _{\bot }=-m\omega ^{2}R{\mathbf {\hat {e}}}_{r}=-\omega ^{2}\mathbf {m} \,\!$ where again m is the mass moment, and the coriolis force is $\mathbf {F} _{c}=2\omega {\frac {{\rm {d}}r}{{\rm {d}}t}}{\mathbf {\hat {e}}}_{\theta }=2\omega v{\mathbf {\hat {e}}}_{\theta }\,\!$ The Coriolis acceleration and force can also be written: $\mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}$

Central force motion

For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centres of masses of the two objects, the equation of motion is:

${\frac {d^{2}}{d\theta ^{2}}}\left({\frac {1}{\mathbf {r} }}\right)+{\frac {1}{\mathbf {r} }}=-{\frac {\mu \mathbf {r} ^{2}}{\mathbf {l} ^{2}}}\mathbf {F} (\mathbf {r} )$

Equations of motion (constant acceleration)

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
$v=v_{0}+at\,$	$\omega _{1}=\omega _{0}+\alpha t\,$
$s={\frac {1}{2}}(v_{0}+v)t$	$\theta ={\frac {1}{2}}(\omega _{0}+\omega _{1})t$
$s=v_{0}t+{\frac {1}{2}}at^{2}$	$\theta =\omega _{0}t+{\frac {1}{2}}\alpha t^{2}$
$v^{2}=v_{0}^{2}+2as\,$	$\omega _{1}^{2}=\omega _{0}^{2}+2\alpha \theta$
$s=vt-{\frac {1}{2}}at^{2}$	$\theta =\omega _{1}t-{\frac {1}{2}}\alpha t^{2}$

Galilean frame transforms

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 $\mathbf {r} '=\mathbf {r} +\mathbf {V} t\,\!$ Relative velocity $\mathbf {v} '=\mathbf {v} +\mathbf {V} \,\!$ Equivalent accelerations $\mathbf {a} '=\mathbf {a}$	Relative accelerations $\mathbf {a} '=\mathbf {a} +\mathbf {A}$ Apparent/fictitious forces $\mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }$
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 $\theta '=\theta +\Omega t\,\!$ Relative velocity ${\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}\,\!$ Equivalent accelerations ${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}$	Relative accelerations ${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}$ Apparent/fictitious torques ${\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }$
	Transformation of any vector T to a rotating frame ${\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T}$

Mechanical oscillators

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 = Angular displacement A = Transverse amplitude Θ = Angular amplitude	${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x\,\!$ Solution: $x=A\sin \left(\omega t+\phi \right)\,\!$	${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta \,\!$ Solution: $\theta =\Theta \sin \left(\omega t+\phi \right)\,\!$
Unforced DHM	b = damping constant κ = torsion constant	${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0\,\!$ Solution (see below for ω'): $x=Ae^{-bt/2m}\cos \left(\omega '\right)\,\!$ Resonant frequency: $\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}\,\!$ Damping rate: $\gamma =b/m\,\!$ Expected lifetime of excitation: $\tau =1/\gamma \,\!$	${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0\,\!$ Solution: $\theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)\,\!$ Resonant frequency: $\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}\,\!$ Damping rate: $\gamma =\kappa /m\,\!$ Expected lifetime of excitation: $\tau =1/\gamma \,\!$

Angular frequencies

Physical situation	Nomenclature	Equations
Linear undamped unforced SHO	k = spring constant m = mass of oscillating bob	$\omega ={\sqrt {\frac {k}{m}}}\,\!$
Linear unforced DHO	k = spring constant b = Damping coefficient	$\omega '={\sqrt {{\frac {k}{m}}-\left({\frac {b}{2m}}\right)^{2}}}\,\!$
Low amplitude angular SHO	I = Moment of inertia about oscillating axis κ = torsion constant	$\omega ={\sqrt {\frac {I}{\kappa }}}\,\!$
Low amplitude simple pendulum	L = Length of pendulum g = Gravitational acceleration Θ = Angular amplitude	Approximate value $\omega ={\sqrt {\frac {g}{L}}}\,\!$ Exact value can be shown to be: $\omega ={\sqrt {\frac {g}{L}}}\left[1+\sum _{k=1}^{\infty }{\frac {\prod _{n=1}^{k}\left(2n-1\right)}{\prod _{n=1}^{m}\left(2n\right)}}\sin ^{2n}\Theta \right]\,\!$

Energy in mechanical oscillations

Physical situation	Nomenclature	Equations
SHM energy	T = kinetic energy U = potential energy E = total energy	Potential energy $U={\frac {m}{2}}\left(x\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\cos ^{2}(\omega t+\phi )\,\!$ Maximum value at x = A: $U_{\mathrm {max} }{\frac {m}{2}}\left(\omega A\right)^{2}\,\!$ Kinetic energy $T={\frac {\omega ^{2}m}{2}}\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\sin ^{2}\left(\omega t+\phi \right)\,\!$ Total energy $E=T+U\,\!$
DHM energy		$E={\frac {m\left(\omega A\right)^{2}}{2}}e^{-bt/m}\,\!$

See also

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• List of physics formulae
• Defining equation (physics)
• Defining equation (physical chemistry)
• Constitutive equation
• Mechanics
• Optics
• Electromagnetism
• Thermodynamics
• Acoustics

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• List of equations in wave theory
• List of relativistic equations
• List of equations in fluid mechanics
• List of equations in gravitation
• List of electromagnetism equations
• List of photonics equations
• List of equations in quantum mechanics
• List of equations in nuclear and particle physics

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A international individual (any one who isn't a Singapore citizen, Singapore Company, Singapore restricted liability partnership or a Singapore society) will still want approval from the Singapore Land Authority (SLA) to purchase land-titled property such as houses, bungalows and vacant plots of land. Housing Improvement Board Properties SINGAPORE NON-PUBLIC RESIDENTIAL PROPERTIES Singapore firm; Singapore society. We provide invaluable services to expats who are considering relocation to Singapore For every kind of Singapore property related providers, we're there to obtain your name. You are positive to get one of the best Singapore properties with our help. RE/MAX Singapore Singapore Residential Business and Industrial Singapore citizen; Singapore Land Authority Thomson Highway, Singapore

Nevertheless, additionally it is a foul investment decision to go in too early. For instance you buy a property near one of the new MRT station location at the Thomson line. Because it takes 10 years to be accomplished, you might need difficulties renting that place out to pay off your mortgage payments. In addition, 10 years is a long time when something can occur including recession, new properties being introduced, modifications in interest rates etc. All these can have an adversarial impact on your property funding.

First, there are generally more rental transactions than gross sales transactions, to permit AV to be determined for every property based on comparable properties. Second, actions in sale costs are extra unstable than rentals. Therefore, using rental transactions to derive the AV helps to maintain property tax extra steady for property owners. If you are shopping for or trying to hire a property. It's tiring to call up individual property agent, prepare appointments, coordinate timing and to go for individual property viewing. What most individuals do is to have a property agent representing them who will organize and coordinate the viewings for all of the properties out there based mostly in your requirements & most well-liked timing. Rent Property District 12 Lease Property District thirteen

The brokers say that many Chinese language patrons are also investing abroad so they can personal property close to major instructional establishments. Some are buying houses close to top faculties — despite the fact that their youngsters are so little they can not walk but. Greater than 80 % of rich Chinese need to ship their children overseas to school, based on the Hurun Report, a Shanghai-based mostly publication. Chinese language patrons sometimes used to pick up properties within the $1 to $5 million vary in New York, typically shopping for two and three at a time for funding purposes, the brokers mentioned. Monika Tu, a dealer at high-end real estate agency Black Diamondz Property Concierge in Sydney, says that over the previous yr mainland Chinese language have develop into 80 % of her firm's business. WCEGA Plaza & Tower

An insurance coverage that covers the reinstatement worth or outstanding loan, whichever is decrease, within the event that the property search singapore (view site…) it insures is destroyed by fire. The coverage does not embody dwelling renovations, moveable household contents and personal belongings. Mortgage Insurance Credit bureau checks (e.g., Good payment data for bank cards / previous or existing loans, no previous blemishes corresponding to discharged bankrupts) A sign by the financial institution of the quantity of mortgage that you are eligible for. Nonetheless, an AIP doesn't constitute a binding loan offer. Additional checks and situations may be imposed by the financial institution, and the phrases of the formal supply will even rely upon the property that you simply intend to buy. Freehold / leasehold (999 years / 99 years)

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• Lectures on classical mechanics
• Biography of Isaac Newton, a key contributor to classical mechanics