Correct Answer - Option 3 : 4 ∶ 3

**Concept:**

**Spring Force**

- When a spring is stretched by x distance from its natural length, then the restoring force is proportional to the x.
- The force is given as

F = k x

- The potential energy gained by spring when stretched by distance x is given as

\(P = \frac{1}{2} k x^2\)

- The potential energy is done by doing work against the restoring force.

**Calculation:**

Given, two springs have **spring constant in ratio 9: 16**

\(\frac{k_1}{k_2} = \frac{9}{16}\) -- (1)

If the same work is done, they will be having the **same potential energy. **

Let the elongation of the first spring be x_{1}, and the second spring be x_{2}, then

\(\frac{1}{2} k_1 (x_1)^2 = \frac{1}{2} k _2(x_2)^2\)

\(\implies (\frac{x_1}{x_2} )^2 = \frac{k_2}{k_1}\)

\(\implies (\frac{x_1}{x_2} ) = \sqrt{\frac{k_2}{k_1}} \) -- (2)

Putting (1) in (2) we get

\(\implies (\frac{x_1}{x_2} ) = \sqrt{\frac{16}{9}} = \frac{4}{3}\)

**So, the required ratio is 4 : 3.**