List of equations in classical mechanics
   HOME

TheInfoList



OR:

Classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
is the branch of
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
used to describe the motion of
macroscopic The macroscopic scale is the length scale on which objects or phenomena are large enough to be visible with the naked eye, without magnifying optical instruments. It is the opposite of microscopic. Overview When applied to physical phenome ...
objects. It is the most familiar of the theories of physics. The concepts it covers, such as
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
,
acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
, and
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
, are commonly used and known. The subject is based upon a
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
s—as well as other
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
concepts—which relate various physical quantities to one another. These include differential equations,
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s,
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s, and
ergodic theory Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
. This article gives a summary of the most important of these. This article lists equations from
Newtonian mechanics Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body r ...
, see
analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...
for the more general formulation of classical mechanics (which includes Lagrangian and
Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
).


Classical mechanics


Mass and inertia


Derived kinematic quantities


Derived dynamic quantities


General energy definitions

Every
conservative force In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work don ...
has a
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
. 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


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 = \mathbf_r\times\mathbf_\theta defines the axis of rotation, \scriptstyle \mathbf_r = unit vector in direction of , \scriptstyle \mathbf_\theta = unit vector tangential to the angle.


Dynamics


Precession

The precession angular speed of a
spinning top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be rotation, spun on its vertical Axis of rotation, axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will ...
is given by: \boldsymbol = \frac 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 In science, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stren ...
(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 \cdot \mathrm \mathbf + \boldsymbol \cdot \mathbf \, \right ) where θ is the angle of rotation about an axis defined by a
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
n.


Kinetic energy

The change in
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
for an object initially traveling at speed v_0 and later at speed v is: \Delta E_k = W = \frac m(v^2 - ^2)


Elastic potential energy

For a stretched spring fixed at one end obeying
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
, the
elastic potential energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, s ...
is \Delta E_p = \frac 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

Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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:"Relativity, J.R. Forshaw 2009" \mathbf \cdot \boldsymbol + \boldsymbol \times \left ( \mathbf \cdot \boldsymbol \right ) = \boldsymbol where I is the
moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relatively to a rotational axis. It is the ratio between ...
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
.


General planar motion

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 = \mathbf(t) = r\hat\mathbf r the following general results apply to the particle.


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 centers of masses of the two objects, the equation of motion is: \frac\left(\frac\right) + \frac = -\frac\mathbf(\mathbf)


Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then the general
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).


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.


Mechanical oscillators

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.


See also

* List of physics formulae *
Defining equation (physical chemistry) A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definit ...
*
Constitutive equation In physics and engineering, a constitutive equation or constitutive relation is a relation between two or more physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance o ...
*
Mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
*
Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...
*
Electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
*
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
*
Acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
*
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
* 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 This article summarizes equations used in optics, including geometric optics, physical optics, radiometry, diffraction, and interferometry. Definitions Geometric optics (luminal rays) General fundamental quantities Physical optics (EM lum ...
* List of equations in quantum mechanics * List of equations in nuclear and particle physics


Notes


References

* * * {{DEFAULTSORT:Equations In Classical Mechanics Classical mechanics
Classical Mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...