First slide

Simple hormonic motion

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

Moderate

A

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

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

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

$\frac{1}{2 π} \sqrt{\frac{7 k}{M}}$

Solution

The two springs on left side having spring constant of 2k each are in series, equivalent constant is $\frac{1}{(\frac{1}{2 k} + \frac{1}{2 k})} = k .$ The two springs on right hand side of mass M are in parallel. Their effective spring constant is (k + 2k) = 3k.
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 + 3k = 4k
$∴ Frequency of mass is n = \frac{1}{2 π} \sqrt{\frac{4 k}{M}}$

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