A system of springs with their spring constants are as shown in figure. What is the frequency of oscillations of the mass m ?

Options

A. `(1)/(2pi)sqrt((k_(1)k_(2)(k_(3)+k_(4)))/([(k_(1)+k_(2))+(k_(3)+k_(4))+k_(1)k_(4)]m))`
B. `(1)/(2pi)sqrt((k_(1)k_(2)(k_(3)+k_(4)))/([(k_(1)+k_(2))+(k_(3)+k_(4))+k_(1)k_(2)]m))`
C. `(1)/(2pi)sqrt((k_(1)k_(2)(k_(3)+k_(4)))/([(k_(1)+k_(2))(k_(3)+k_(4))+k_(1)k_(2)]m))`
D. `(1)/(2pi)sqrt(((k_(1)+k_(2))(k_(3)+k_(4))+k_(1)k_(2))/(k_(1)k_(2)(k_(3)+k_(4))m))`

Answer

B

Explanation

Here `k_(3)andk_(4)` are in parallel, their equivalent spring constant, `k=(k_(3)+k_(4))`. Now, `k_(1),k_(2)andk` are in series, their equivalent spring constant k. is
`(1)/(k.)=(1)/(k_(1))+(1)/(k_(2))+(1)/((k_(3)+k_(4)))`
or `(1)/(k.)=(k_(2)(k_(3)+k_(4))+k_(1)(k_(3)+k_(4))+k_(1)k_(2))/(k_(1)k_(2)(k_(3)+k_(4)))`
or `k.(k_(1)k_(2)(k_(3)+k_(4)))/((k_(1)+k_(2))(k_(3)+k_(4))+k_(1)k_(2)) therefore` Frequency, `upsilon=(1)/(2pi)sqrt((k.)/(m))`
`v=(1)/(2pi)sqrt((k_(1)k_(2)(k_(3)+k_(4)))/([(k_(1)+k_(2))(k_(3)+k_(4))+k_(1)k_(2)]m))`