Difference between revisions of "Conservation of Angular Momentum"
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| − | ''' | + | The '''conservation of angular momentum''' is a fundamental concept of physics along with other [[conservation law]]s such as those of energy and [[momentum (physics)|linear momentum]]. It states that the angular momentum of a system remains constant unless changed through an action of external [[force]]s. |
| − | + | In [[Newtonian mechanics]], the angular momentum of a point mass about a point is defined as <math>\vec H = \vec r \times \vec p</math> where <math>\vec{r}</math> is the position [[vector quantity|vector]] of the point mass with respect to the point of reference and <math>\vec{p}</math> is the [[momentum (physics)|linear momentum]] vector of the point mass. | |
The principle of angular momentum can be applied to a system of particles by summing the angular momentum of each particle about the same point. This can be represented as: | The principle of angular momentum can be applied to a system of particles by summing the angular momentum of each particle about the same point. This can be represented as: | ||
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:<math>\vec{H}_{sys}</math> is the total angular momentum of the system | :<math>\vec{H}_{sys}</math> is the total angular momentum of the system | ||
:<math>\vec{H}_i</math> is the angular momentum of the i<sup>th</sup> particle | :<math>\vec{H}_i</math> is the angular momentum of the i<sup>th</sup> particle | ||
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| + | ==Proof of conservation== | ||
The [[derivative]] of angular momentum with respect to time is equal to the sum of the external moments (or torque <math>\vec {\tau}</math>) applied to the system. Differentiating angular momentum gives: | The [[derivative]] of angular momentum with respect to time is equal to the sum of the external moments (or torque <math>\vec {\tau}</math>) applied to the system. Differentiating angular momentum gives: | ||
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<math>\vec {\tau} = \vec r \times \vec F + \vec{\dot{r}} \times p</math> | <math>\vec {\tau} = \vec r \times \vec F + \vec{\dot{r}} \times p</math> | ||
| − | For a constant radius, the second term is zero. Hence <math>\vec {\tau}= \vec r \times \vec F</math> From this, it can be concluded that in the absence of an external moment, angular momentum must be conserved. | + | For a constant [[radius]], the second term is zero. Hence <math>\vec {\tau}= \vec r \times \vec F</math> From this, it can be concluded that in the absence of an external moment, angular momentum must be conserved. |
[[Category:Physics]] | [[Category:Physics]] | ||
[[Category:Mechanics]] | [[Category:Mechanics]] | ||
Revision as of 20:47, November 22, 2016
The conservation of angular momentum is a fundamental concept of physics along with other conservation laws such as those of energy and linear momentum. It states that the angular momentum of a system remains constant unless changed through an action of external forces.
In Newtonian mechanics, the angular momentum of a point mass about a point is defined as <math>\vec H = \vec r \times \vec p</math> where <math>\vec{r}</math> is the position vector of the point mass with respect to the point of reference and <math>\vec{p}</math> is the linear momentum vector of the point mass.
The principle of angular momentum can be applied to a system of particles by summing the angular momentum of each particle about the same point. This can be represented as:
<math> \vec{H}_{sys} = \sum_i \vec{H}_i </math>
where
- <math>\vec{H}_{sys}</math> is the total angular momentum of the system
- <math>\vec{H}_i</math> is the angular momentum of the ith particle
Proof of conservation
The derivative of angular momentum with respect to time is equal to the sum of the external moments (or torque <math>\vec {\tau}</math>) applied to the system. Differentiating angular momentum gives:
<math>\vec {\tau} = \vec r \times \vec F + \vec{\dot{r}} \times p</math>
For a constant radius, the second term is zero. Hence <math>\vec {\tau}= \vec r \times \vec F</math> From this, it can be concluded that in the absence of an external moment, angular momentum must be conserved.