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Physics-

General

Easy

Question

Four massless springs whose force constants are 2k, 2k, k and 2k 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

$\frac{1}{2 π} \sqrt{\frac{k}{4 M}}$
$\frac{1}{2 π} \sqrt{\frac{4 k}{M}}$
$\frac{1}{2 π} \sqrt{\frac{k}{7 M}}$
$\frac{1}{2 π} \sqrt{\frac{7 k}{M}}$

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 springs on left side having spring constant of 2k 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 $left parenthesis k plus 2 k right parenthesis equals 3 k$ .
Equivalent spring constants of value k and 3k 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 is $n equals fraction numerator 1 over denominator 2 pi end fraction square root of fraction numerator 4 k over denominator M end fraction end root$

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