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

Question

Two identical springs are connected in series and parallel as shown in the figure. If fs and fp are frequencies of series and parallel arrangements, what is (fs/fP) ? <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/e8409143b2575ab2610b714d9f932862-.png" /> (A)  1: 2  (B)  2: 1  (C)  1: 3  (D)  3: 1

Tardigrade Admin · Accepted Answer

In first arrangement springs are in parallel and in second arrangement springs are in series.
In first case, springs are connected in parallel, so their equivqlent spring constant

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

So, frequency of this spring-block system is

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} = k

∴ 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} ) m}

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}

or

\frac{f _{s}}{f _{p}} = \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 frequencies of series and parallel arrangements, 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 frequencies of series and parallel arrangements, 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 frequencies of series and parallel arrangements, what is $\frac{f _{s}}{f _{P}}$ ?

$1 : 2$

$2 : 1$

$1 : 3$

$3 : 1$