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<h2>Conservative and Non-Conservative Forces</h2>
<p>In a previous lesson, you saw that mechanical energy was conserved in the <strong>absence</strong> of non-conservative forces. It is important to know whether a force is an conservative force or an non-conservative force in the system, because this is related to whether the force can change an object's total mechanical energy when it does work on an object.</p>
<p>When the only forces doing work are conservative forces (for example, gravitational and spring forces), energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical energy (\(E_K + E_P\)) is conserved. For example, as an object falls in a gravitational field from a high elevation to a lower elevation, some of the object's potential energy is changed into kinetic energy. However, the sum of the kinetic and potential energies remain constant.</p>
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<h2>Investigation: Non-Conservative Forces</h2>
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<p>We can investigate the effect of non-conservative forces on an object's total mechanical energy by rolling a ball along the floor from point A to point B.</p>
<img alt="f70fc7e8583786ef8c496e4861d8f2b7.png" src="https://data.ulearngo.com/assets/1bc3f282-0fa4-495c-a429-d2b394fb7e65"/>
<p>Find a nice smooth surface (e.g. a highly polished floor), mark off two positions, A and B, and roll the ball between them.</p>
<p>The total mechanical energy of the ball, at each point, is the sum of its kinetic energy (\(E_k\)) and gravitational potential energy (\(E_p\)):</p>

\begin{align*} {E}_{\text{total},\text{A}}&amp; = {E_k}_{,\text{A}}+{E_p}_{,\text{A}}\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2}+mg{h}_{\text{A}}\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2}+mg\left(0\right)\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2} \end{align*}\begin{align*} {E}_{\text{total},\text{B}}&amp; = {E_k}_{\text{B}}+{E_p}_{\text{B}}\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2}+mg{h}_{\text{B}}\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2}+mg\left(0\right)\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2} \end{align*}

<p>In the absence of friction and other non-conservative forces, the ball should slide along the floor and its speed should be <em>the same</em> at positions A and B. Since there are no non-conservative forces acting on the ball, its total mechanical energy at points A and B are equal.</p>

\begin{align*} {v}_{A}&amp; = {v}_{\text{B}}\\ \frac{1}{2}m{v}_{\text{A}}^{2}&amp; = \frac{1}{2}m{v}_{B}^{2}\\ {E}_{\text{total},\text{A}}&amp; = {E}_{\text{total},\text{B}} \end{align*}

<p>Now, let's investigate what happens when there is friction (an <em>non-conservative force</em>) acting on the ball.</p>
<p>Roll the ball along a rough surface or a carpeted floor. What happens to the speed of the ball at point A compared to point B?</p>
<p>If the surface you are rolling the ball along is very rough and provides a large non-conservative frictional force, then the ball should be moving much slower at point B than at point A.</p>
<p>Let's compare the total mechanical energy of the ball at points A and B:</p>

\begin{align*} {E}_{\text{total},\text{A}}&amp; = {\text{EK}}_{\text{A}}+{\text{PE}}_{\text{A}}\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2}+mg{h}_{\text{A}}\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2}+mg\left(0\right)\\ &amp; = \frac{1}{2}m{v}_{\text{A}}^{2} \end{align*}\begin{align*} {E}_{\text{total},\text{B}}&amp; = {\text{EK}}_{\text{B}}+{\text{PE}}_{\text{B}}\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2}+mg{h}_{\text{B}}\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2}+mg\left(0\right)\\ &amp; = \frac{1}{2}m{v}_{\text{B}}^{2} \end{align*}

<p>However, in this case, \({v}_{A}\ne {v}_{B}\) and therefore \({E}_{\text{total},\text{A}}\ne {E}_{\text{total},\text{B}}\). Since</p>

\begin{align*} {v}_{\text{A}}&amp; &gt; {v}_{\text{B}}\\ {E}_{\text{total},\text{A}}&amp; &gt; {E}_{\text{total},\text{B}} \end{align*}

<p>Therefore, the ball has lost mechanical energy as it moves across the carpet.However, although the ball has lost mechanical energy, energy in the larger system has still been conserved. In this case, the missing energy is the work done by the carpet through applying a frictional force on the ball. In this case the carpet is doing negative work on the ball.</p>
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<p>When an non-conservative force (for example friction, air resistance, applied force) does work on an object, the total mechanical energy (\(E_k + E_p\)) of that object changes. If positive work is done, then the object will gain energy. If negative work is done, then the object will lose energy.</p>
<p>When a net <strong>force does work</strong> on an object, then there is <strong>always a change in the kinetic energy</strong> of the object. This is because the object experiences an acceleration and therefore a change in velocity.</p>
<p>This leads us to the work-energy theorem.</p>
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<h2>Definition: Work-Energy Theorem</h2>
<div class="ns-school-defn">
<p>The work-energy theorem states that the work done on an object by the net force is equal to the change in its kinetic energy:</p>

\[W_{\text{net}}=\Delta E_k={E}_{k,f}-{E}_{k,i}\]</div>
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<p>The work-energy theorem is another example of the conservation of energy.</p>

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Work-Energy Theorem

Physics is a science discipline concerned with the nature and properties of matter and energy. Physics topics include mechanics, heat, light and radiation, sound, electricity, magnetism, the structure of atoms, and so on.

Physics

Explore the concepts of mechanical energy, potential and kinetic energy, work, power, and conservation of energy through examples like pendulum, roller coaster, and inclined plane.


Work-Energy Theorem

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