1.4 Force types

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# Engineering Physics - Fall 2024

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$\vec{F}=-G\frac{m_1m_2}{\vert\vec{r}_2-\vec{r}_1\vert^2}\,\hat{e}_r$<br><br>
$r=\frac{a(1-e^2)}{1+e\cos f}$
<br><br>
for $m_1\ll m_2$ and $\vert\vec{r}_2\vert=\vert\vec{r}_1\vert+\epsilon$:
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$F=m_1\,g$   with   $g\approx9.8\frac{m}{s^2}$
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$F_x=-k\,x$
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$x(t)=A\cos\left[\sqrt{\frac{k}{m}}t+\theta\right]$
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$F_x=-m\,\omega^2\,x-\frac{\beta}{m}\,v$
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$x(t)=A\,\exp\left[\frac{t}{2}\left(-\beta\pm\sqrt{\beta^2-4\,\omega^2}\right)\right]$
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$D:=\beta^2-4\,\omega^2$
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<div class="grid-item" style="font-size: 12px;color: tomato;background-color: rgb(200, 200, 200);">Tension<br>
  <span style='color: black;font-size: 10px;'>The force an infinitesimal element of rope, <strong>taut</strong> by forces on its endpoints, would
  experience if it would not be constrained by its neighbouring elements.</span></div>
<div class="grid-item" style="font-size: 12px;color: tomato;background-color: rgb(200, 200, 200);">Normal force:<br>
  <span style='color: black;font-size: 10px;'> When a body presses on a surface, the surface deforms and pushes on the body with a force
 that is perpendicular, i.e., normal to the surface   </span>

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<p style="color: tomato;">Predict the height at which $M$ and $m$ come to rest given:
  <ul   style="list-style-type:square;text-align: left;font-size: 16px;">
    <li> the spring constant <span style="color:dodgerblue;font-size: 150%;">$\alpha$</span>
    <li> the coefficient of friction between $M$ and the incline <span style="color:dodgerblue;font-size: 150%;">$\nu$</span>
      <li> the incline makes an angle <span style="color:dodgerblue;font-size: 150%;">$\theta$</span> with the horizontal
  </ul>
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<p style="color: darkgreen;">Solution recipe:
  <ul   style="list-style-type:square;text-align: left;font-size: 16px;">
    <li> draw the <span style="color:dodgerblue;font-size: 150%;">free-body diagram</span><br>
      <i style="color:lightgray;font-size: 80%;">"Have I checked for forces of friction, gravity, spring, normal, and tension for each object?</i>
    <li> obtain the vector components of all forces in your diagram in <strong>your</strong> <span style="color:dodgerblue;font-size: 100%;">coordinate system</span>.<br>
    <li> write out Newton's 2nd law for the horizontal <strong>and</strong> vertical coordinate: <br>
      <span style="color:dodgerblue;font-size: 100%;">$$ma_{x/y}=\sum F_{x/y}$$</span>
      <li> if stationary, $\vec{a}=0$, and you can solve the equations for the unknown quantity.
  </ul>
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