Spring Force (Hooke's Law) Video Lessons

Concept

# Problem: Consider the force exerted by a spring that obeys Hooke's law. FindU(xf)-U(x0)=∫x0xfFs→·ds→,whereF→s=-kxi^,  ds→=dxi^and the spring constant k is positive.Express your answer in terms of k, x0, and xf.

###### FREE Expert Solution

Evaluating Fs•ds

$\begin{array}{rcl}{\stackrel{\mathbf{⇀}}{\mathbf{F}}}_{\mathbf{s}}\mathbf{·}{\mathbf{d}}_{\mathbf{s}}& \mathbf{=}& \mathbf{-}\mathbf{k}\mathbf{x}\stackrel{\mathbf{^}}{\mathbf{i}}\mathbf{·}\mathbf{dx}\stackrel{\mathbf{^}}{\mathbf{i}}\\ & \mathbf{=}& \mathbf{-}\mathbf{kxdx}\end{array}$

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###### Problem Details

Consider the force exerted by a spring that obeys Hooke's law. Find

$\mathrm{U}\left({\mathrm{x}}_{\mathrm{f}}\right)-\mathrm{U}\left({\mathrm{x}}_{0}\right)={\int }_{{\mathrm{x}}_{0}}^{{\mathrm{x}}_{\mathrm{f}}}\stackrel{\to }{{\mathrm{F}}_{\mathrm{s}}}·d\stackrel{\to }{\mathrm{s}}$,

where

and the spring constant k is positive.