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.