Question

a string of natural lenght L extends to a new lenght L under tensile force F. if hooke's law applies, the work done in stretching the spring is

Answer 1

Task:

A string of natural length $L$ extends to a new length $L_{1}$ under tensile force $F$. If Hooke's Law applies, what is the work done in stretching the spring?

Solution:

A horizontal spring exerts a force $F = (kx, 0, 0)$ that is proportional to its deflection in the $x$ direction. The work of this spring on a body moving along the space curve

$s(t) = (x(t), y(t), z(t))$, is calculated using its velocity, $v = (v_x, v_y, v_z)$, to obtain

$$W = \int_{0}^{t} \boldsymbol{F} \cdot \boldsymbol{v} \, dt = \int_{0}^{t} k x v_x \, dt = \frac{1}{2} k x^2.$$

For convenience, consider contact with the spring occurs at $t = 0$, then the integral of the product of the distance $x$ and the $x$-velocity, $x v_x$, is $(1/2) x^2$.

The function $U(x) = 1/2 k x^2$ is called the potential energy of a linear spring.

Given:

$$x = L_1 - L,$$ $$W = \frac{1}{2} k (L_1 - L)^2$$

Answer:

$$W = \frac{1}{2} k (L_1 - L)^2$$