<html> <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> 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 =></p> <p>F = m*20^2/200 =></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 =></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> 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 (2nd law)</p> <p>and</p> <p>N * cosθ = mg (1st law) </p> <p>By dividing these equations we get:</p> <p>tanθ = u^2/gr => </p> <p>θ = arctan(u^2/gr) => θ = arctan(25^2/3500) ~= arctan(0.1785) =></p> <p><strong>θ ~= 10.12 degrees </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 => u ~= 4.71 m/s </p> <p>The centripetal force Fc stays the same and so:</p> <p>Fc = mu^2/r => Fc != 197 N</p> <p><br></p> <p> 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 => Ft = W - Fc =></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 => Fb = W + Fc =></p> <p>Fb = 197 + 800 = 997N (upwards)</p> <p>So, the phenomenal weight at the bottom is 997N!</p> <p><br></p> <blockquote> 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> 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> </html>
| author | drifter1 |
|---|---|
| permlink | physics-classical-mechanics-circular-dynamic-and-vertical-circle-motion-examples |
| category | physics |
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| created | 2018-02-25 18:41:12 |
| last_update | 2018-02-25 18:41:12 |
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