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Line 27: − + + + + + − '''2. In an inertial reference frame, the acceleration of an object is equal to the force applied on it, divided by the object mass.''' Mathematically:+ + + − <math>\Sigma \vec F = m \vec a</math> where+ − * <math>\Sigma \vec F</math> is the net [[force]] acting on the object+ − * <math>m</math> is the mass of the object+ − * <math>\vec a</math> is the acceleration of the object+ + + Line 72:
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Classical mechanics (view source)
Revision as of 15:15, December 17, 2016
, 15:15, December 17, 2016Tidy up in terms of maths formatting and added section on quantum mechanics
'''Classical mechanics''' refers to a branch of [[physics]] dealing with every-day laws of [[mechanics]] (as opposed to [[quantum mechanics]]). The term is sometimes interchangeable with "Newtonian physics", as it was [[Sir Isaac Newton]] who first proposed the first laws of motion. However, there are other formulations of classical mechanics that are not due to Newton, namely [[Lagrangian dynamics]] and [[Hamiltonian dynamics]]. (These are simply different formulations of the same principles; they are precisely equivalent to Newtonian mechanics.) Classical mechanics is sufficient for explaining most observable phenomena.
'''Classical mechanics''' refers to a branch of [[physics]] dealing with every-day laws of [[mechanics]] (as opposed to [[quantum mechanics]]). The term is sometimes interchangeable with "Newtonian physics", as it was [[Sir Isaac Newton]] who first proposed the first laws of motion. However, there are other formulations of classical mechanics that are not due to Newton, namely [[Lagrangian Dynamics|Lagrangian dynamics]] and [[Hamiltonian Dynamics|Hamiltonian dynamics]]. (These are simply different formulations of the same principles; they are precisely equivalent to Newtonian mechanics.) Classical mechanics is sufficient for explaining most observable phenomena.
==Kinematics==
==Kinematics==
Kinematics is the subfield of mechanics that describes motion, without taking into account its cause. The relevant concepts in kinematics are position, velocity and acceleration.
Kinematics is the subfield of mechanics that describes motion, without taking into account its cause. The relevant concepts in kinematics are position, [[velocity]] and [[acceleration]].
The position of an object is usually described by a [[vector]], with respect to some reference point. That position is described as a vector means that we need several numbers to specify it. For example, we can say that object A is three meters to the right and four meters to the front from object B. Note that a single number is not enough to specify its position. We could say that object A is 5 meters from object B, but in this case we need to specify its direction, so, in any case, we need two numbers.
The position of an object is usually described by a [[vector]], with respect to some reference point. That position is described as a vector means that we need several numbers to specify it. For example, we can say that object A is three meters to the right and four meters to the front from object B. Note that a single number is not enough to specify its position. We could say that object A is 5 meters from object B, but in this case we need to specify its direction, so, in any case, we need two numbers.
Velocity is measures how fast position changes. As such, it is the change in position with respect to time. Velocity is also a vector. Mathematically, it is the [[derivative]] of position with respect to [[time]]:
Velocity is measures how fast position changes. As such, it is the change in position with respect to time. Velocity is also a vector. Mathematically, it is the [[derivative]] of position with respect to [[time]]:
: <math>v=\frac{dx}{dt}</math>
: <math>v=\frac{dx}{dt}</math>
If the velocity is constant, the distance traveled in a time t is:
If the velocity is constant, the distance traveled in a time t is:
:<math> x=vt</math>.
:<math> x = vt</math>.
Similarly, [[acceleration]] measures how fast the velocity changes. It is then mathematically defined as the derivative of velocity with respect to time:
Similarly, [[acceleration]] measures how fast the velocity changes. It is then mathematically defined as the derivative of velocity with respect to time:
: <math>a=\frac{dv}{dt}</math>
: <math>a=\frac{dv}{dt}</math>
If the acceleration is constant, the velocity at a time t is:
If the acceleration is constant, the velocity at a time t is:
: <math>v=v_0+at</math>.
: <math>v=v_0+at</math>.
Where v0 is the velocity at time t = 0.
Where <math>v_0</math> is the velocity at time t = 0.
[[Integral|Integrating]] this equation gives the distance traveled by an object at constant acceleration:
[[Integral|Integrating]] this equation gives the distance traveled by an object at constant acceleration:
: <math>\Delta x=v_0t+\frac{1}{2}at^2</math>.
: <math>\Delta x=v_0t+\frac{1}{2}at^2</math>.
'''1. An object subject to no net external force moves with constant velocity relative to an inertial reference frame.'''
'''1. An object subject to no net external force moves with constant velocity relative to an inertial reference frame.'''
An inertial reference frame is defined as one where Newton's laws are valid. Intuitively, an inertial reference frame is a frame that moves in a straight line at a constant velocity. Since the Earth is rotating, it is not a perfect reference frame; however, it is a very good approximation for everyday purposes.
An inertial reference frame is defined as one where Newton's laws are valid. Intuitively, an [[inertial reference frame]] is a frame that moves in a straight line at a constant velocity. Since the Earth is rotating, it is not a perfect reference frame; however, it is a very good approximation for everyday purposes.
'''2. In an inertial reference frame, the net force acting upon an object is equal to the rate of change of its momentum.''' Mathematically:
<math>\Sigma \vec{F} = \frac{\text{d} \vec{p}}{\text{d}t}</math>
where
:<math>\Sigma \vec F</math> is the net [[force]] acting on the object
:<math>\vec{p}</math> is the [[momentum (physics)|momentum]] of the object
If the mass of the object in question is constant, Newton's second law may be rewritten in the more familiar form of:
<math>\Sigma \vec F = m \vec a</math>
where
:<math>m</math> is the mass of the object
:<math>\vec a</math> is the acceleration of the object
This law also applies to a system of objects.
This law also applies to a system of objects.
Speeds necessary to get a significant Lorentz factor are simply not found in real-life examples of visible objects. For example, at 5 miles per second, (18,000 miles per hour, the speed of a satellite in Earth orbit), the Lorentz factor has a value of 1.000000000360219.
Speeds necessary to get a significant Lorentz factor are simply not found in real-life examples of visible objects. For example, at 5 miles per second, (18,000 miles per hour, the speed of a satellite in Earth orbit), the Lorentz factor has a value of 1.000000000360219.
===Quantum Mechanics===
[[Quantum mechanics]] describes the behaviour of [[matter]] at small distances, such as those of [[atoms]] or [[subatomic particle]]s. Given enough information about a classical system, we can use classical mechanics to predict how it will evolve with certainty. This is not true in quantum mechanics; in quantum mechanics, we can only calculate the probability of say the measurement of the position of a particle. This results from the wave-like nature of matter, referred to as [[wave-particle duality]]. It states that particles (such as [[electron]]s) can sometimes behave like waves and waves (such as light) can sometimes behave like particles. The reason we don't observe quantum effects on macroscopic scales can be seen by considering the equation for the [[wavelength]], <math>\lambda</math> of a particle:
<math>\lambda = \frac{h}{p}</math>
where <math>h</math> is [[Planck's constant]] and <math>p</math> is the [[momentum (physics)]] of the particle.
At macroscopic scales, <math>\lambda</math> is very small as Planck's constant is such a small number, <math>6.63 \times 10^{-34} \; \text{J} \, \text{s}</math>. Therefore the wave-like nature of matter has little effect on macroscopic scales.
The wave-like nature of matter leads to the [[Heisenberg uncertainty principle]], which states that there are certain quantities, such as position and momentum, that cannot be measured to an arbitrary precision simultaneously.
Particles are described by a wavefunction, <math>\Psi</math>, which encodes the results of all possible measurements. The wavefunction is the solution to the [[Schrodinger equation]],
<math>-\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi = i \hbar \frac{\partial \Psi}{\partial t}</math>
where <math>\hbar</math> is the reduced Planck's constant, <math>m</math> is the mass of the particle and <math>V</math> is the potential the particle is in.
[[Category:Mechanics]]
[[Category:Mechanics]]