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Physics - Classical mechanics - Circular dynamic and vertical circle motion examples by drifter1

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Physics - Classical mechanics - Circular dynamic and vertical circle motion examples
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<p><img src="http://www.aplusphysics.com/courses/regents/circmotion/images/rollercoasterbottom.gif"/></p>
<p>Hello it's a me! Today we continue with some more examples in <strong>Physics</strong>.</p>
<p>The last two posts where about examples in:</p>
<p><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-object-equilibrium-and-2nd-law-application-examples">Object equilibrium and 2nd law applications</a></p>
<p><a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-contact-force-and-friction-examples">Contact forces and friction</a></p>
<p>Today's topic will be circular dynamics and vertical circle motion examples.</p>
<p>The theory for that can be found <a href="https://steemit.com/physics/@drifter1/physics-classical-mechanics-dynamics-of-circular-motion">here </a>and I highly suggest you to check it out!</p>
<p>So, without further do, let's get started!</p>
<p><br></p>
<h3>Example 1 (flat curve):</h3>
<p><img src="http://s3.amazonaws.com/answer-board-image/82665ba0-3186-4039-8f29-b98d058019b9.jpeg" width="300" height="165"/></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Suppose we have a flat/horizontal curve on a highway with an radius (r) of 200m. A car takes this corner with a velocity of 20 m/s. Which is the least possible (static) friction factor μs so that the car doesn't slide out?</p>
<p><br></p>
<p>In constant velocity circular motions we know that the acceleration is a = u^2/r.</p>
<p>That way from the second law we have:</p>
<p>ΣF = F = ma = mu^2/r =&gt;</p>
<p>F = m*20^2/200 =&gt;</p>
<p>F = 2m N</p>
<p>The maximum static friction is also equal to:</p>
<p>F = μs*Ν = μmg = 10μm N</p>
<p>By setting those equations equal to each other we get:</p>
<p>2m = 10μsm =&gt;</p>
<p><strong>μs = 0.2</strong></p>
<p><br></p>
<h3>Example 2 (banked curve):</h3>
<p><img src="https://session.masteringphysics.com/problemAsset/1011163/41/MLD_cm_7_a.jpg" width="320" height="260"/></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Suppose that we now have a banked curve on a highway with an radius (r) of 350m that will be constructed with an angle (θ). A car needs to be able to take this corner with a velocity of 25 m/s and the friction is theoretically supposed as zero (non-existent). Which is the least angle (θ) so that this can happen?</p>
<p><br></p>
<p>In my theory post we already found the equations but here a small explanation</p>
<p>The vertical ground force has an angle θ to the gravity and so we find the components:</p>
<p>Nsinθ and Ncosθ (one for each axis)</p>
<p>Using the 1st and 2nd law for each axis respectively we get:</p>
<p>N * sinθ = mu^2 / r &nbsp;&nbsp;&nbsp;(2nd law)</p>
<p>and</p>
<p>N * cosθ = mg&nbsp;&nbsp;&nbsp;(1st law)&nbsp;</p>
<p>By dividing these equations we get:</p>
<p>tanθ = u^2/gr =&gt;&nbsp;</p>
<p>θ = arctan(u^2/gr) =&gt; θ = arctan(25^2/3500) ~= arctan(0.1785) =&gt;</p>
<p><strong>θ ~= 10.12 degrees&nbsp;</strong></p>
<p><br></p>
<h3>Example 3(vertical circle):</h3>
<p>The radius (r) of a theme park wheel is 9m and it does a circle in 12 seconds.</p>
<p>Calculate the phenomenal weight of an 80kg passenger at the highest and lowest point of the wheel.</p>
<p><br></p>
<p>Because the period is T = 12 seconds we can calculate the velocity from:</p>
<p>u = 2πr / T &nbsp;=&gt; u ~= 4.71 m/s&nbsp;</p>
<p>The centripetal force Fc stays the same and so:</p>
<p>Fc = mu^2/r =&gt; Fc != 197 N</p>
<p><br></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;Because the gravity also stays the same and is W = mg = 800 N we know that there is a upward force at the top that is declining the gravity so that:</p>
<p>Fc = W - Ft = mu^2 / r =&gt; Ft = W - Fc =&gt;</p>
<p>Ft = 800 - 197 = 603 N (upwards)</p>
<p>This force is acting like a vertical ground force and so the phenomenal weight the top is 603N!</p>
<p><br></p>
<p>In the same way at the bottom there is a upward force that is also declining gravity so that:</p>
<p>Fc = Fb - W = mu^2 / r =&gt; Fb = W + Fc =&gt;</p>
<p>Fb = 197 + 800 = 997N (upwards)</p>
<p>So, the phenomenal weight at the bottom is 997N!</p>
<p><br></p>
<blockquote>&nbsp;&nbsp;&nbsp;&nbsp;Another interesting topic is how we <strong>generate artificial gravity by rotating a space station</strong> in space!</blockquote>
<p>You can read about that <a href="https://www.school-for-champions.com/science/gravity_artificial_equations.htm#.WpMC-qhuYaY">here</a>.</p>
<p><br></p>
<h3>Image sources:</h3>
<p><a href="http://www.aplusphysics.com/courses/regents/circmotion/images/rollercoasterbottom.gif">http://www.aplusphysics.com/courses/regents/circmotion/images/rollercoasterbottom.gif</a></p>
<p><a href="http://s3.amazonaws.com/answer-board-image/82665ba0-3186-4039-8f29-b98d058019b9.jpeg">http://s3.amazonaws.com/answer-board-image/82665ba0-3186-4039-8f29-b98d058019b9.jpeg</a></p>
<p><a href="https://session.masteringphysics.com/problemAsset/1011163/41/MLD_cm_7_a.jpg">https://session.masteringphysics.com/problemAsset/1011163/41/MLD_cm_7_a.jpg</a></p>
<p><br></p>
<p>And this is actually it and I hope that you learned something!<br>
&nbsp;&nbsp;&nbsp;Next time we will get into more advanced Newton law examples and after that we are finished with everything that I wanted to cover about Classical mechanics (for now)! This means that we will get into a new physics branch!</p>
<p>Bye!</p>
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authordrifter1
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created2018-02-25 18:41:12
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