Assume a spring has an un-stretched length Xo. Write an expression for the elastic potential energy of the spring in terms of the position of the end of the spring X and its spring constant K.
To find the expression for the elastic potential energy of the spring, we need to consider Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement. The formula for Hooke's Law is:
F = -KX
Where:
F is the force exerted by the spring
K is the spring constant
X is the displacement from the equilibrium position
To calculate the elastic potential energy of the spring, we can integrate the force equation with respect to displacement. The work done by the spring force in moving a distance X from the equilibrium position is equal to the elastic potential energy (PE):
PE = ∫ F dX
Integrating the force equation gives:
PE = ∫ (-KX) dX
Integrating the above equation, we get:
PE = (1/2)KX^2 + C
Where C is the constant of integration. However, since we are interested in the elastic potential energy of the spring when it is at its equilibrium position (X = 0), the constant of integration disappears, and the equation simplifies to:
PE = (1/2)KX^2
Therefore, the expression for the elastic potential energy of the spring in terms of the position of the end of the spring (X) and its spring constant (K) is:
PE = (1/2)KX^2