The potential energy in an elastic string of force constant K which has been extended by X metres is expresses as

The potential energy stored in an elastic string that has been stretched by a distance X is given by option (A): \(\frac{1}{2}k \Box ^2\). The force required to stretch an elastic string is proportional to the extension. This proportionality constant is known as the spring constant (K). When a spring is stretched or compressed, it stores potential energy. The amount of energy stored in the spring is directly proportional to the amount of stretching or compression. The formula for the potential energy stored in a spring is given by: Potential energy = 0.5 x Spring constant x (extension)^2 where the extension represents the distance by which the spring is stretched or compressed from its original length. Thus, in this case, the potential energy stored in the elastic string of force constant K, which has been extended by X meters, is given by: Potential energy = 0.5 x K x X^2 This matches option (A): \(\frac{1}{2}k \Box ^2\).