The shape of the surface of a rotating mass of water as a variational problem
Variational methods have a long and remarkable role in theoretical physics. Few of our students when first exposed to them fail to admire their elegance and efficacy in the formulation and solution of physical problems. In this paper we apply the variational approach that leads to the Euler-Lagrange equations to the determination of the shape of the surface of a mass of water that partially fills a cylindrical bucket that rotates with constant angular velocity (Newton's bucket). Here this approach will lead us to the principle of minimization of the effective potential energy associated with the system. The effect of an external pressure on the equilibrium shape is also taken into account and two models, the constant pressure model and the linear model are discussed. The level of the discussion is kept accessible to undergraduates taking an intermediate level course in classical mechanics.
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Sociedade Brasileira de Física
2017
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oai:scielo:S1806-111720170002004012017-09-29The shape of the surface of a rotating mass of water as a variational problemSantos,F.C.Tort,A.C. analytical mechanics variational calculus rotating bucket Variational methods have a long and remarkable role in theoretical physics. Few of our students when first exposed to them fail to admire their elegance and efficacy in the formulation and solution of physical problems. In this paper we apply the variational approach that leads to the Euler-Lagrange equations to the determination of the shape of the surface of a mass of water that partially fills a cylindrical bucket that rotates with constant angular velocity (Newton's bucket). Here this approach will lead us to the principle of minimization of the effective potential energy associated with the system. The effect of an external pressure on the equilibrium shape is also taken into account and two models, the constant pressure model and the linear model are discussed. The level of the discussion is kept accessible to undergraduates taking an intermediate level course in classical mechanics.info:eu-repo/semantics/openAccessSociedade Brasileira de FísicaRevista Brasileira de Ensino de Física v.39 n.2 20172017-01-01info:eu-repo/semantics/articletext/htmlhttp://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172017000200401en10.1590/1806-9126-rbef-2016-0204 |
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Santos,F.C. Tort,A.C. |
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Santos,F.C. Tort,A.C. The shape of the surface of a rotating mass of water as a variational problem |
| author_facet |
Santos,F.C. Tort,A.C. |
| author_sort |
Santos,F.C. |
| title |
The shape of the surface of a rotating mass of water as a variational problem |
| title_short |
The shape of the surface of a rotating mass of water as a variational problem |
| title_full |
The shape of the surface of a rotating mass of water as a variational problem |
| title_fullStr |
The shape of the surface of a rotating mass of water as a variational problem |
| title_full_unstemmed |
The shape of the surface of a rotating mass of water as a variational problem |
| title_sort |
shape of the surface of a rotating mass of water as a variational problem |
| description |
Variational methods have a long and remarkable role in theoretical physics. Few of our students when first exposed to them fail to admire their elegance and efficacy in the formulation and solution of physical problems. In this paper we apply the variational approach that leads to the Euler-Lagrange equations to the determination of the shape of the surface of a mass of water that partially fills a cylindrical bucket that rotates with constant angular velocity (Newton's bucket). Here this approach will lead us to the principle of minimization of the effective potential energy associated with the system. The effect of an external pressure on the equilibrium shape is also taken into account and two models, the constant pressure model and the linear model are discussed. The level of the discussion is kept accessible to undergraduates taking an intermediate level course in classical mechanics. |
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Sociedade Brasileira de Física |
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2017 |
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http://old.scielo.br/scielo.php?script=sci_arttext&pid=S1806-11172017000200401 |
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1756430717876699136 |