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.
Oh, spring physics, the bane of my comedic existence! Well, here's a not-so-funny expression for the elastic potential energy of a spring:
Elastic potential energy (U) = (1/2) * k * (x - Xo)^2
There you have it! Just plug in the spring constant (k) and the position of the end of the spring (x), and you've got yourself some elastic potential energy! Now, if only we could find a way to make springs tell really cheesy jokes...
The elastic potential energy of a spring can be calculated using the equation:
Elastic potential energy = (1/2) * k * (x - Xo)^2
where:
- k is the spring constant,
- x is the position of the end of the spring,
- Xo is the un-stretched length of the spring.
To derive the expression for the elastic potential energy of a spring, we need to consider Hooke's Law, which states that the force exerted by a spring is directly proportional to its displacement.
According to Hooke's Law, the force exerted by the spring can be expressed as:
F = -kx
Where:
F is the force exerted by the spring,
k is the spring constant, and
x is the displacement of the end of the spring from its equilibrium position.
The negative sign indicates that the force exerted by the spring is in the opposite direction to the displacement.
Now, we can find the work done by the spring when it is stretched or compressed from its equilibrium position. The work done is given by the formula:
W = ∫F dx
Where:
W is the work done by the spring, and
∫F dx represents the integral of the force over the displacement.
Since the force applied by the spring is constant, we can simplify the formula:
W = -∫kx dx
W = -k∫x dx
Integrating with respect to x:
W = -k * (1/2)x^2 + C
Where:
C is the constant of integration.
Now, let's determine the value of C. The expression for elastic potential energy should be zero when the spring is at its unstretched length, Xo. Therefore, we can substitute x = Xo and W = 0 into the equation to solve for C:
0 = -k * (1/2)Xo^2 + C
C = k * (1/2)Xo^2
The final expression for the elastic potential energy (PE) of the spring in terms of the position of the end of the spring x and its spring constant k is:
PE = -k * (1/2)x^2 + k * (1/2)Xo^2
Simplifying further:
PE = (1/2)k * (Xo^2 - x^2)
Therefore, the expression for the elastic potential energy of the spring is (1/2)k * (Xo^2 - x^2).