{"id":3794,"date":"2016-09-30T00:00:42","date_gmt":"2016-09-30T00:00:42","guid":{"rendered":"https:\/\/assignment.essayshark.com\/blog\/?p=3794"},"modified":"2023-01-02T11:45:09","modified_gmt":"2023-01-02T11:45:09","slug":"solved-problems-in-physics-classical-mechanics","status":"publish","type":"post","link":"https:\/\/assignmentshark.com\/blog\/solved-problems-in-physics-classical-mechanics\/","title":{"rendered":"Solved Problems in Physics: Classical Mechanics"},"content":{"rendered":"

Physics assignment<\/h2>\n

Classical mechanics<\/h3>\n

Problem statement<\/strong><\/p>\n

Particles with two different masses m and M are located along a linear harmonic chain
\nof infinite length. The chain has a force constant k (see the picture below). The
\ndistance between two particles with the same mass is equilibrium and equals to a.
\nqj<\/sub> and rj<\/sub> are the deviations of particles mj<\/sub> and Mj<\/sub> from their equilibrium positions respectively.<\/p>\n

    \n
  1. \n
      \n
    1. Find both potential and kinetic energies of the system and the Lagrangian of the
      \nsystem. Write down the equations of motion for qi<\/sub> and ri<\/sub>.<\/li>\n
    2. If the equations \"q_{j}=Q_{j}\exp{[i{\omega}t]}\" and \"r_{j}=R_{j}\exp{[i{\omega}t]}\" are applied, what are the equations for the amplitudes Qj<\/sub> and Rj<\/sub>?<\/li>\n
    3. Let \"Q(k)=\sum_{j=-\infty}^{j=+\infty}{Q_{i}\exp[i(jka)]}\" and \"R(k)=\sum_{j=-\infty}^{j=+\infty}{R_{j}\exp[i(jka)]}\", where \"i=\sqrt(-1)\" and the sum on j is over all particles of mass m for Q(k) or over all particles of mass M for R(k). Perform the sum over all of the amplitudes in part 2) above and determine the equations for Q(k) and R(k) using the above definitions for Q(k) and R(k).<\/li>\n
    4. Determine the normal mode frequencies \"\omega (k)\" for the system, where \"k\in[-\pi/(2a),+\pi/(2a)]\".<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

      \"phisics\"<\/a><\/p>\n

      Solution<\/strong><\/p>\n

        \n
      1. The kinetic energy of the system can be determine from the equation below:
        \n\"t0-5\"
        \nThe potential energy of the system can be found from the equation below:
        \n\"v\"
        \nThe Lagrangian can be found as follows:
        \n\"l\"Where T is the kinetic energy and V is the potential energy, determined above.
        \nThe equations of motion for jth<\/sup> points can be found as the partial differentials from the
        \nLagrangian:
        \n\"d\"<\/li>\n
      2. Let \"q_{j}=Q_{j}\exp{[i({\omega}t})]\" with \"Q(k)=\sum_{j=-\infty}^{j=+\infty}{Q_{i}\exp[ijka]}\" and \"r_{j}=R_{j}\exp{[i({\omega}t)]}\" with \"R(k)=\sum_{j=-\infty}^{j=+\infty}{R_{j}\exp[ijka]}\" .Then \"\"{q}_{j}=-\omega^2Q_{j}\exp[i(\omega{t})]\" and \"\"{r}_{j}=-\omega^2R_{j}\exp[i(\omega{t})]\" .
        \nTherefore, \"k\"<\/li>\n
      3. In order to find the solution for part 3, multiplication by exp[i(jka)] should be
        \nperformed.
        \nTherefore, \"result\"Applying the differentiation for Q(k), R(k) enables to sum over j:\"result\"<\/li>\n
      4. For non-trivial solutions Q(k) and R(k), the determinant, as shown below, should
        \nbe equal to zero.
        \n\"result\"
        \n\"result\"<\/li>\n<\/ol>\n

        Physics Assistance from the Experts<\/h2>\n

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        Here is one more physics sample paper<\/a> that can be helpful in your studies.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

        Physics assignment Classical mechanics Problem statement Particles with two different masses m and M are located along a linear harmonic chain of infinite length. The chain has a force constant k (see the picture below). The distance between two particles with the same mass is equilibrium and equals to a. qj and rj are the […]<\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[59,35],"tags":[],"class_list":["post-3794","post","type-post","status-publish","format-standard","hentry","category-physics","category-samples"],"_links":{"self":[{"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/posts\/3794","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/comments?post=3794"}],"version-history":[{"count":49,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/posts\/3794\/revisions"}],"predecessor-version":[{"id":13195,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/posts\/3794\/revisions\/13195"}],"wp:attachment":[{"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/media?parent=3794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/categories?post=3794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/assignmentshark.com\/blog\/wp-json\/wp\/v2\/tags?post=3794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}