# The ratio of the force constant of two springs is 9 ∶ 16. If same work is done on them, then the ratio of elongation will be -

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The ratio of the force constant of two springs is 9 ∶ 16. If same work is done on them, then the ratio of elongation will be -
1. 9 ∶ 16
2. 16 ∶ 9
3. 4 ∶ 3
4. 3 ∶ 4

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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 x1, and the second spring be x2, 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.