Two identical springs are connected in series and parallel as shown in the figure. If fs and fp are frequency of series and parallel arrangement what is (fs/fp) ?

Question

Two identical springs are connected in series and parallel as shown in the figure. If fs and fp are frequency of series and parallel arrangement what is (fs/fp) ? (A) 1: 2 (B) 2: 1 (C) 1: 3 (D) 3: 1

Tardigrade Admin · Accepted Answer

In first case, springs are connected in parallel, so their equivalent spring constant

k_{p} = k_{1} + k_{2}

So, frequency of this spring block system is given by

f_{p} = \frac{1}{2 π} \frac{k _{p}}{m}

or

f_{p} = \frac{1}{2 π} \frac{k _{1} + k _{2}}{m}

but

k_{1} = k_{2}

∴ f_{p} = \frac{1}{2 π} \frac{2 k}{m}

... (i)
Now, in second case, springs are connected in series,
so their equivalent spring constant

k = \frac{k _{1} k _{2}}{k _{1} + k _{2}}

Hence, frequency of this arrangement is given by

f_{s} = \frac{1}{2 π} \frac{k _{1} k _{2}}{( k _{1} + k _{2} )}

or

f_{s} = \frac{1}{2 π} \frac{k}{2 m}

... (ii)
Dividing Eq. (ii) by Eq. (i), we get

\frac{f _{s}}{f _{p}} = \frac{\frac{1}{2 π} \frac{k}{2 m}}{\frac{1}{2 π} \frac{2 k}{m}}

= \frac{1}{4} = \frac{1}{2}

Q. Two identical springs are connected in series and parallel as shown in the figure. If $f_{s}$ and $f_{p}$ are frequency of series and parallel arrangement what is $\frac{f _{s}}{f _{p}}$ ?

1 : 2

2 : 1

1 : 3

3 : 1

Q. Two identical springs are connected in series and parallel as shown in the figure. If fs​ and fp​ are frequency of series and parallel arrangement what is fp​fs​​ ?

1 : 2

2 : 1

1 : 3

3 : 1

Q. Two identical springs are connected in series and parallel as shown in the figure. If $f_{s}$ and $f_{p}$ are frequency of series and parallel arrangement what is $\frac{f _{s}}{f _{p}}$ ?