Difference between revisions of "Kinetic Energy"

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'''Kinetic energy''' represents the [[energy]] asociated with the [[motion]] of an object.<ref>Serway and Beichner, ''Physics for Scientists and Engineers'', Fifth Edition</ref> It is defined as:
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'''Kinetic energy''' represents the [[energy]] associated with the [[motion]] of an object.<ref>Serway and Beichner, ''Physics for Scientists and Engineers'', Fifth Edition</ref> It is defined as the work done by a force to accelerate that object from rest to some speed <math> v </math>, in the absence of any other [[force]]s acting upon the object. Kinetic energy is a scalar and has the same units as work (i.e. [[Joule]]).
  
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K ≡ [[mass|m]][[velocity|v]]<sup>2</sup> / 2 for a point mass and <math> K = {1 \over 2}mV^2 + {1 \over 2} I \omega ^2 </math> for a body, where I is the body's moment of inertia and omega is the body's angular velocity.
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==Classical mechanics==
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===Translational kinetic energy===
  
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The change of kinetic energy in an object is equal to the total [[work]] done on it by a [[force]]. In the case of constant force, this can be expressed as:
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In [[classical mechanics]], the translational kinetic energy of a ridid object, <math> K </math>, can be found as:
  
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Σ''W'' = ΔK = mv<sub>f</sub><sup>2</sup> / 2 - mv<sub>i</sub><sup>2</sup> / 2
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<math> K = \frac{1}{2} m v^2</math>
  
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Where v<sub>i</sub> is [[speed]] at t = 0 and v<sub>f</sub> is speed at [[time]] = t.
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Where
  
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Kinetic energy is a scalar and has the same units as work (i.e. [[Joule]]).  
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:<math> m </math> is the [[mass]] of the object
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:<math> v </math> is the [[velocity]] of the object
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===Rotational kinetic energy===
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The rotational kinetic energy of a rigid object is:
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<math> K = {1 \over 2} I \omega ^2 </math>
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Where
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:<math> I </math> is the [[moment of inertia]] of the object
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:<math> \omega </math> is the angular velocity of the object
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===Work-Energy theorem===
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The change of kinetic energy is equal to the total [[work]] done on it by the resultant of all [[force]]s acting on it. For a point mass this can be expressed as:
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<math> \Sigma W = \Delta K = \frac{1}{2} m v_{f}^{2} - \frac{1}{2} m v_{I}^{2} </math>
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Where
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:<math> v_i </math> is  the initial [[speed]]
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:<math> v_f </math> is the final [[speed]]
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Note that if the mass of an object is increased, the increase in kinetic energy increases linearly; if the [[velocity]] of an object is increased, the increase in kinetic energy increases [[quadratic equation|quadratically]]. For example, doubling the mass of an object doubles its kinetic energy; doubling its velocity quadruples its kinetic energy.
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===Derivation of translational kinetic energy===
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The [[work]] done by a force accelerating an object from rest, which is the kinetic energy is:
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<math> W = K = \int F dx</math>
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From Newton's second law, the [[force]], <math> F</math>, is <math> F =\frac{dp}{dt} </math>. Hence we can make the substitution and use the chain rule
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<math> K = \int \frac{dp}{dt} dx = \int \frac{dp}{dx} \frac{dx}{dt} dx  </math>
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This is the same as
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<math> K = \int v dp </math>
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In [[classical mechanics]], [[momentum]] is given by <math> p = mv </math>. Differentiating and substituting into the above equation results in
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<math> K = \int^{u}_{0} m v dv  </math>
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We want to integrate between 0 and the speed of the object, <math> u </math> as this defines kinetic energy. Performing the integration reveals that the kinetic energy is, as expected, the following:
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<math> K = \frac{1}{2} mv^2 </math>
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A similar method may be used to derive the formula for rotational kinetic energy.
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== Kinetic Energy in Relativity ==
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In [[relativity]], kinetic energy can be expressed as:
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<math> K = (\gamma - 1) m_0 c^2 </math>
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where
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:<math>\gamma</math> is the [[Lorentz factor]]
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:<math>m_0</math> is the [[rest mass]]
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:<math>c</math> is the [[speed of light]]
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This can be shown to be equivalent to the classical equation for kinetic energy, <math>K=\frac{1}{2} m_0 v^2</math>, by performing a binomial expansion on it. Using the result:
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<math>
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(1+x)^n = 1 + nx+ \frac{n(n-1)x^2}{2} + ...
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</math>
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Expanding the [[Lorentz factor]] in this way, we see:
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<math>
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K=(1 + \frac{1}{2} \frac{v^2}{c^2}+ \frac{3}{8} \frac{v^4}{c^4}+...-1) m_0 c^2
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</math>
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This simplifies to
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<math>
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K=\frac{1}{2} m_0 v^2 + \frac{3}{8} \frac{m_0 v^4}{c^2} +...
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</math>
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For speeds encountered everyday, which are a lot less than that of [[speed of light|light]] (such that <math>v<<c</math>), all terms apart from the first are very small and can be neglected. Hence, the formula reduces to:
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<math>K \approx \frac{1}{2} m_0 v^2</math>
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which is the classical formula.
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===Derivation===
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The kinetic energy is the [[work|work done]] accelerating a particle from rest to some speed <math>v</math>. Suppose the particle is at rest at <math>x_1</math> and speed <math>v</math> at position <math>x_2</math>. Hence:
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<math>K = \int^{x_2}_{x_1} \vec{F} \dot d \vec{x}</math>
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Since <math>\vec{F} = \vec{F}_{\perp} + \vec{F}_{\parallel}</math>, and a perpendicular force does no work, we can ignore the perpendicular component and write:
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<math>K = \int^{x_2}_{x_1} F_{\parallel} dx = \int^{x_2}_{x_1}  \gamma^3 m_0 a dx</math>
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Since <math>a \, dx = \frac{dv_{\parallel}}{dt} dx = \frac{dx}{dt} dv</math>, we find the integral can be rewritten as:
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<math>K = \int^{v_2}_{v_1} \frac{mv}{(1- \frac{v^2}{c^2})} dv</math>
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where <math>v_1</math> is the initial speed and hence 0 by definition and <math>v_2</math> is the final speed.
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Performing the integration reveals the kinetic energy as:
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<math>K=(\gamma -1)m_0 c^2</math>
  
 
==References==
 
==References==
 
<references/>
 
<references/>
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[[category:physics]]
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[[Category:mechanics]]
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[[Category:Mechanics]]

Latest revision as of 17:21, April 7, 2017

Kinetic energy represents the energy associated with the motion of an object.[1] It is defined as the work done by a force to accelerate that object from rest to some speed , in the absence of any other forces acting upon the object. Kinetic energy is a scalar and has the same units as work (i.e. Joule).

Classical mechanics

Translational kinetic energy

In classical mechanics, the translational kinetic energy of a ridid object, , can be found as:

Where

is the mass of the object
is the velocity of the object

Rotational kinetic energy

The rotational kinetic energy of a rigid object is:

Where

is the moment of inertia of the object
is the angular velocity of the object

Work-Energy theorem

The change of kinetic energy is equal to the total work done on it by the resultant of all forces acting on it. For a point mass this can be expressed as:

Where

is the initial speed
is the final speed

Note that if the mass of an object is increased, the increase in kinetic energy increases linearly; if the velocity of an object is increased, the increase in kinetic energy increases quadratically. For example, doubling the mass of an object doubles its kinetic energy; doubling its velocity quadruples its kinetic energy.

Derivation of translational kinetic energy

The work done by a force accelerating an object from rest, which is the kinetic energy is:

From Newton's second law, the force, , is . Hence we can make the substitution and use the chain rule

This is the same as

In classical mechanics, momentum is given by . Differentiating and substituting into the above equation results in

We want to integrate between 0 and the speed of the object, as this defines kinetic energy. Performing the integration reveals that the kinetic energy is, as expected, the following:

A similar method may be used to derive the formula for rotational kinetic energy.

Kinetic Energy in Relativity

In relativity, kinetic energy can be expressed as:

where

is the Lorentz factor
is the rest mass
is the speed of light

This can be shown to be equivalent to the classical equation for kinetic energy, , by performing a binomial expansion on it. Using the result:

Expanding the Lorentz factor in this way, we see:

This simplifies to

For speeds encountered everyday, which are a lot less than that of light (such that ), all terms apart from the first are very small and can be neglected. Hence, the formula reduces to:

which is the classical formula.

Derivation

The kinetic energy is the work done accelerating a particle from rest to some speed . Suppose the particle is at rest at and speed at position . Hence:

Since , and a perpendicular force does no work, we can ignore the perpendicular component and write:

Since , we find the integral can be rewritten as:

where is the initial speed and hence 0 by definition and is the final speed.

Performing the integration reveals the kinetic energy as:

References

  1. ↑ Serway and Beichner, Physics for Scientists and Engineers, Fifth Edition
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