Question

Choose the correct option:

Two identical springs of constant k are connected, first in series and then in parallel. A metal block of mass m is suspended from their combination. The ratio of their frequencies of vertical oscillations will be in a ratio

पर्याय

• 1:4

• 1:2

• 2:1

• 4:1

Answer 1

1:2

Explanation:

Take the case of a series combination.

Express the relation of equivalent spring constant.

`1/k_(eq) = 1/k_1 + 1/k_2`  .....(i)

Here, k₁ and k₂ are the spring constants of the two springs.

`k_(eq)` is the equivalent spring constant.

For both the springs, spring constant k₁ = k and k₂ = k.

Substitute k for k₁ and k for k₂ in equation (i).

`1/k_(eq) = 1/k + 1/k`

`1/k_(eq) = 2/k`

`k_(eq) = k/2`

 Hence equivalent spring constant for series combination is `k/2`.

Take the case of the parallel combination.

Express the relation of equivalent spring constant.

`k_(eq) = k_1 + k_2`  .....(ii)

Here, k₁ and k₂ are the spring constants of the two springs.

`k_(eq)` is the equivalent spring constant.

For both the springs, spring constant k₁ = k and k₂ = k.

Substitute k for k₁ and k for k₂ in equation (ii).

`k_(eq) = k + k`

= 2k

Hence equivalent spring constant for parallel combination is 2k.

Express the relation of frequency f of oscillation.

`f = 1/(2pi) sqrt(k_(eq)/m)`

Here, `k_(eq)` is the equivalent spring constant.

m is the mass of the block.

For series combination substitute `k/2` for `k_(eq)`.

`f_1 = 1/(2pi) sqrt(k/(2m)`  ......(iii)

For parallel combination substitute 2k for `k_(eq)`.

`f_2 = 1/(2pi) sqrt((2k)/m)`  .....(iv)

Now find the ratio of the two frequencies `f_1/f_2` from equation (iii) and equation (iv).

`f_1/f_2 - (1/(2pi) sqrt(k/(2m)))/(1/(2pi) sqrt((2k)/m))`

`f_1/f_2 = 1/2`

Hence the ratio of frequencies for vertical oscillations is 1:2.