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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 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).
m = \iint \sigma \mathrm{d} S
m = \iiint \rho \mathrm{d} V \,\!
\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} \,\!
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 \,\!
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 \,\!
Torque
to a Resultant Force
system, Work done BY
Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:
where \mathbf{q}=\mathbf{q}(t) \,\! and p = p(t) are vectors of the generalized coords and momenta, as functions of time
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
defines the axis of rotation, \scriptstyle \bold{\hat{e}}_r \,\! = unit vector in direction of r, \scriptstyle \bold{\hat{e}}_\theta \,\! = unit vector tangential to the angle.
Instantaneous:
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:
where θ is the angle of rotation about an axis defined by a unit vector n.
For a stretched spring fixed at one end obeying Hooke's law:
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]}
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,
the following general results apply to the particle.
\mathbf{r} =\bold{r}\left ( r,\theta, t \right ) = r \bold{\hat{e}}_r
Angular momenta \mathbf{L} = m \bold{r}\times \left(\bold{\hat{e}}_r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \bold{\hat{e}}_\theta \right)
where again m is the mass moment, and the coriolis force is
The Coriolis acceleration and force can also be written:
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:
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).
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.
V = Constant relative velocity between two inertial frames F and F'. A = (Variable) relative acceleration between two accelerating frames F and F'.
Relative velocity \mathbf{v}' = \mathbf{v} + \mathbf{V} \,\! Equivalent accelerations \mathbf{a}' = \mathbf{a}
Apparent/fictitious forces \mathbf{F}' = \mathbf{F} - \mathbf{F}_\mathrm{app}
Ω = Constant relative angular velocity between two frames F and F'. Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.
Relative velocity \boldsymbol{\omega}' = \boldsymbol{\omega} + \boldsymbol{\Omega} \,\! Equivalent accelerations \boldsymbol{\alpha}' = \boldsymbol{\alpha}
Apparent/fictitious torques \boldsymbol{\tau}' = \boldsymbol{\tau} - \boldsymbol{\tau}_\mathrm{app}
\frac \,\!
\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 ]\,\!
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 \,\!
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