Question

three spring of spring constant K are connected in series and parallel with the mass M calculate the ratio of frequency in series and parallel.​

Answer 1

Answer:

Keq in Series

Keq = K/n

n = 3

Keq = K/3

Keq in Parallel

Keq = nK

n = 3

Keq = 3K

f = 1/2π √K/M

f is directly proportional to √K

f1/f2 = √K/3 × 1/3K

f1/f2 = 1/3

Thanks for Reading

Answer 2

Explanation:

We Have :-

3 springs of spring constant K

To Find :-

Ratio of srequency

Formula Used :-

$frequency = \dfrac{1}{2\pi} \sqrt{ \dfrac{k}{m} }$

Solution :-

In series

$\dfrac{1}{k_{s}} = \dfrac{1}{k} + \dfrac{1}{k} + \dfrac{1}{k} \\ \\ \\ \dfrac{1}{k_{s}} = \dfrac{3}{k} \\ \\ \\ k _{s} = \dfrac{k}{3}$

In Parallel

$k_{p} = k + k + k \\ \\ \\ k_{p} = 3k$

Frequency is Series

$frequency = \dfrac{1}{2\pi} \sqrt{ \dfrac{k}{3m} }$

Frequency in Parallel

$frequency = \dfrac{1}{2\pi} \sqrt{ \dfrac{3k}{m} }$

Ratio of Frequency in Series to Parallel

$\dfrac{f_{s} }{f_{p} } = \dfrac{ \sqrt{ \dfrac{k}{3m} } }{ \sqrt{ \dfrac{3k}{m} } } \\ \\ \\ \dfrac{f_{s} }{f_{p} } = \sqrt{ \dfrac{k}{3m} } \times \sqrt{ \dfrac{m}{3k} } \\ \\ \\ \dfrac{f_{s} }{f_{p} } = \sqrt{ \dfrac{k \times m}{3m \times 3k} } \\ \\ \\ \dfrac{f_{s} }{f_{p} } = \sqrt{ \dfrac{1}{9} }$