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Easy

Question

Four massless springs whose force constants are 2 k comma 2 k comma k and 2 k respectively are attached to a mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of oscillation of the system is 

  1. fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k over denominator 4 M end fraction end root    
  2. fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 4 k over denominator M end fraction end root    
  3. fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator k over denominator 7 M end fraction end root    
  4. fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 7 k over denominator M end fraction end root    

The correct answer is: fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 4 k over denominator M end fraction end root


    The two spring on left side having spring constant of 2 k each are in series, equivalent constant is fraction numerator 1 over denominator open parentheses fraction numerator 1 over denominator 2 k end fraction plus fraction numerator 1 over denominator 2 k end fraction close parentheses end fraction equals k. The two springs on right hand side of mass M are in parallel. Their effective spring constant is open parentheses k plus 2 k close parentheses equals 3 k
    Equivalent spring constants of value k and 3 k are in parallel and their net value of spring constant of all the four springs is k plus 3 k equals 4 k
    therefore Frequency of mass isn equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 4 k over denominator M end fraction end root

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