Question

A spring when stretched by $ 2\;mm\; $ its potential energy becomes $ 4\;J $ . If it is stretched by $ 10\;mm $ , its potential energy is equal to
A) $ 4{\text{ }}J $
B) $ 54{\text{ }}J $
C) $ 415{\text{ }}J $
D) $ 100{\text{ }}J $

Answer 1

Hint: According to the question when a spring is stretched by $ 2\;mm\; $ its potential energy becomes $ 4\;J $ . If it is stretched by $ 10\;mm $ , then we have to calculate its potential energy now. Here the formula which will be used to solve this question is $ U = \dfrac{1}{2}k{x^2} $ , where k is the spring constant.

Complete step by step answer:
A spring is an elastic object that stores mechanical energy. Springs are typically made of spring steel. There are many spring designs. In everyday use, the term often refers to coil springs. In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Elastic potential energy is Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done to stretch the spring, which depends upon the spring constant k as well as the distance stretched. According to Hooke's law the force required to stretch the spring will be directly proportional to the amount of stretch.
Since the force has the form, $ F = - kx $
Then the work done to stretch the spring a distance x is $ Work = \Delta PE = \dfrac{1}{2}k{x^2} $
Let extension produced in the spring be x initially in the stretched condition. Spring will have potential energy, $ U = \dfrac{1}{2}k{x^2} $ , where k is the spring constant.
Further we know that, $ {x_1} = 2cm $ and $ {x_2} = 10cm $
Now if x becomes 5 times then energy will become 25 times, i.e., $ 4 \times 25 = 100J $ ( as there is square relation ).
Hence the final answer is D) $ 100J $ .

Note:
Be careful while applying the conditions given in the question. While solving the question, keep in mind the formula of potential energy. Calculation mistakes are possible, so try to avoid them. Another thing to keep in mind is that if x becomes $ 5 $ times then energy will become $ 25 $ times as there is a square relation.