A spring with force constant k is initially stretched by \[{{x}_{1}}\]. If it further stretched by \[{{x}_{2}}\] , then increase in its potential energy is:
A. \[\dfrac{1}{2}k{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}\]
B. \[\dfrac{1}{2}k{{x}_{2}}\left( {{x}_{2}}+2{{x}_{_{1}}} \right)\]
C. \[\dfrac{1}{2}kx_{1}^{2}+\dfrac{1}{2}kx_{2}^{2}\]
D. \[\dfrac{1}{2}k{{\left( {{x}_{1}}+{{x}_{2}} \right)}^{2}}\]
E. \[\dfrac{1}{2}k{{\left( {{x}_{1}}+{{x}_{2}} \right)}^{3}}\]

Hint: To solve this question we will be using the formula of potential energy for the spring system. We will first find the potential energy for first instance i.e. when the spring is stretched by \[{{x}_{1}}\] and later the potential energy when the string is further stretched by \[{{x}_{2}}\]. We shall then calculate the change in energy by using these values.
Formula used:
\[U=\dfrac{1}{2}k{{x}^{2}}\]
Where,
U = potential energy
k = the spring constant
x = the displacement of the string

Complete answer:
We have been given two different instances in the question. The first instance is when the spring is stretched by \[{{x}_{1}}\] as shown in the figure. Which means that x = \[{{x}_{1}}\].

Therefore, substituting this value for x in equation
\[U=\dfrac{1}{2}k{{x}^{2}}\] ……………….. (A)
We get,
\[{{U}_{1}}=\dfrac{1}{2}k{{x}_{1}}^{2}\] …………… (1)
Now at the second instance when the spring is further stretched by \[{{x}_{2}}\] as sown in the figure. Which means that,
 x = (\[{{x}_{_{1}}}\] +\[{{x}_{2}}\])
Therefore, substituting this in equation (A)
We get,
\[{{U}_{2}}=\dfrac{1}{2}k{{({{x}_{1}}+{{x}_{2}})}^{2}}\]
On solving above equation
We get,
\[{{U}_{2}}=\dfrac{1}{2}k(x_{1}^{2}+x_{2}^{2}+2{{x}_{1}}{{x}_{2}})\] ………………. (2)
Now, the increase in potential energy will be given by
\[{{U}_{2}}-{{U}_{1}}\]
Therefore, from (1) and (2)
We get,
 \[{{U}_{2}}-{{U}_{1}}\] = \[\dfrac{1}{2}k(x_{1}^{2}+x_{2}^{2}+2{{x}_{1}}{{x}_{2}})-\dfrac{1}{2}k{{x}_{1}}\]
On solving the above equation,
We get,
\[{{U}_{2}}-{{U}_{1}}\] = \[\dfrac{1}{2}k{{x}_{2}}\left( {{x}_{2}}+2{{x}_{_{1}}} \right)\]

So, the correct answer is “Option B”.

Additional Information:
Potential energy is the energy that is stored in an object due to the position of the object. An example of potential energy is the stored water in dams. When any object is high in air it has potential energy which is later converted into kinetic energy, because of which the object starts falling.

Note:
If the string is compressed instead of stretching the value of distance i.e. x will be taken as –x. The k in the equation of potential energy is known as the spring constant. It is the measure of stiffness of the spring. The value for constant k is unique for every spring. This spring constant mainly depends on material used for the spring and the thickness of the coiled wire.