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 the figure. If the mass M is displaced in the horizontal direction, then the frequency of the system is

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 the figure. If the mass M is displaced in the horizontal direction, then the frequency of the system is (A) (1/2π) √(k/4M) (B) (1/2π) √(4k/M) (C) (1/2π) √(k/7M) (D) (1/2π) √(7k/M)

Tardigrade Admin · Accepted Answer

Two springs on the L.H.S. of mass M are in series and two springs on the R.H.S. of mass M are in parallel. These combinations of springs will be considered in parallel to mass M. Thus effective spring constant,

K = \frac{2 k \times 2 k}{2 κ + 2 k} + (k + 2 k) = 4 k

∴ F re q u e n cy υ = \frac{1}{2 π} \frac{K}{M} = \frac{1}{2 π} \frac{4 k}{M}

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

$\frac{1}{2 π} \frac{k}{4 M}$

$\frac{1}{2 π} \frac{4 k}{M}$

$\frac{1}{2 π} \frac{k}{7 M}$

$\frac{1}{2 π} \frac{7 k}{M}$

Q. 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 the figure. If the mass M is displaced in the horizontal direction, then the frequency of the system is

2π1​4Mk​​

2π1​M4k​​

2π1​7Mk​​

2π1​M7k​​

Two springs on the L.H.S. of mass M are in series and two springs on the R.H.S. of mass M are in parallel. These combinations of springs will be considered in parallel to mass M. Thus effective spring constant, K=2κ+2k2k×2k​+(k+2k)=4k ∴Frequencyυ=2π1​MK​​=2π1​M4k​​

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

$\frac{1}{2 π} \frac{k}{4 M}$

$\frac{1}{2 π} \frac{4 k}{M}$

$\frac{1}{2 π} \frac{k}{7 M}$

$\frac{1}{2 π} \frac{7 k}{M}$