Evaluating F_{s}•ds

$\begin{array}{rcl}{\stackrel{\mathbf{\rightharpoonup}}{\mathbf{F}}}_{\mathbf{s}}\mathbf{\xb7}{\mathbf{d}}_{\mathbf{s}}& \mathbf{=}& \mathbf{-}\mathbf{k}\mathbf{x}\hat{\mathbf{i}}\mathbf{\xb7}\mathbf{dx}\hat{\mathbf{i}}\\ & \mathbf{=}& \mathbf{-}\mathbf{kxdx}\end{array}$

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

$\mathrm{U}\left({\mathrm{x}}_{\mathrm{f}}\right)-\mathrm{U}\left({\mathrm{x}}_{0}\right)={\int}_{{\mathrm{x}}_{0}}^{{\mathrm{x}}_{\mathrm{f}}}\overrightarrow{{\mathrm{F}}_{\mathrm{s}}}\xb7d\overrightarrow{\mathrm{s}}$,

where

${\overrightarrow{\mathrm{F}}}_{\mathrm{s}}=-\mathrm{kx}\hat{\mathrm{i}},\mathrm{d}\overrightarrow{\mathrm{s}}=\mathrm{dx}\hat{\mathrm{i}}$

and the spring constant *k* is positive.

Express your answer in terms of k, x_{0}, and x_{f}.

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Spring Force (Hooke's Law) concept. You can view video lessons to learn Spring Force (Hooke's Law). Or if you need more Spring Force (Hooke's Law) practice, you can also practice Spring Force (Hooke's Law) practice problems.

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Based on our data, we think this problem is relevant for Professor Casimir's class at HU.