Loop the Loop Problem (Find Minimum Initial Drop Height and Minimum Velocity at Top of Loop)

YouTube Excerpt: Loop the loop problem and solution with experimental and theoretical values In this video we will be going over how to calculate the minimum velocity at the top of loop the loop and minimum height that is needed to make a toy car do a loop the loop. We will be using 4 equations Initial potential energy equals the potential energy at a given point plus the kinetic energy at a given point Gravitational potential energy equals mass times gravity times height kinetic energy equals one half mass times velocity squared. And finally centripetal acceleration equals velocity squared over the radius Inorder to solve this problem we will need to know what the radius of the loop is. In this case the diameter of the loop is .220 meters. Being the radius is half of the diameter we can take the diameters and divide by 2 to get .110 meters for our radius. Now we can use our centripetal acceleration formula to find what the minum velocity will need to be at the top of the loop the loop. Inorder for the car to stay on the track we must have the centripetal acceleration cancel out the acceleration due to gravity. So the centripetal acceleration must equal the acceleration due to gravity. (ELVATOR EXAMPLE) If it is unclear why this is we can think in terms of elevators. If you are in an elevator accelerating downward you would feel lighter. Now if that elevator was acclerarting just above the acceleration due to gravity you would be pressed up into the top of the elevator So we need the centripetal acceleration to be great enough to just push the car into the track. So we will set our centripetal acceleration equal to the acceleration due to gravity Now we can plug in gravity and rearrange the centripetal acceleration formula to have velocity to one side. After plugging in our numbers we are left with a velocity of 1.038 meters per second. When solving this problem we assume that energy is conserved. This means that the intial potential energy is equal to the potential energy at a given point plus the kinetic energy at a given point. This means that we can trade intial potential energy for more kinetic energy if the height is lower. Note some of the energy will go into friction but that will assume not friction to keep the calculation simple. Now we can plug in the potential energy formula of mass times gravity times height and the kinetic energy formula of one half mass time velocity squared. We notice that each part of the equation is multiplied times mass so it can be canceled out or in other words removed. Now we must recall from the previous slide that velocity at point 2 equals the square root of gravity times the radius. And that the height at point 2 is simply 2 times the radius of the loop the loop. Anywhere there was a h2 in the equation we can plug in 2 times the radius and any where there was a velocity in the original equation we can plug in sqrt of gravity times radius We notice we are taking the square of a square root so this is canceled out. We also notice that all separate values are multiplied times gravity so gravity can also be canceled out or removed. After adding the 2 parts of the equation together we get that h1 equals 2 and one half radi of the loop the loop. We can plug in our numbers to get a height at point one of .275. I tested out this minimum height with my toy car track. At a height of .275 meters the car does not go all the way around the loop the loop. This may be due to friction and that fact that the track is not rigid. (the track moves when the car goes down and around the loop) I tested it out at multiply points and ended up with an experimental height of about .375 meters. This leaves us with an experimental error of 36.3 %. The lesson I am taking away from this is if you are going to do loop the loops in your car make sure it is rigid. Disclaimer These videos are intended for educational purposes only (students trying to pass a class) If you design or build something based off of these videos you do so at your own risk. I am not a professional engineer and this should not be considered engineering advice. Consult an engineer if you feel you may put someone at risk.

Loop the loop problem and solution with experimental and theoretical values In this video we will be going over how to calculate the minimum...

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