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# Converting Between Linear & Rotational

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Sections
Rotational Position & Displacement
More Connect Wheels (Bicycles)
Intro to Connected Wheels
Converting Between Linear & Rotational
Rotational Velocity & Acceleration
Types of Acceleration in Rotation
Rolling Motion (Free Wheels)
Equations of Rotational Motion

Concept #1: Converting Between Linear & Rotational

Transcript

Example #1: Length of string rotating point mass

Transcript

Here we have a small object that rotates at the end of a light string. Here's a string and you got a small object, so IÕm gonna do this and it's spinning like this. Imagine that if you spin a string, it forms a circular path. The object reaches 120 rpm from rest in just 4 seconds. I'm giving you a ton of information here. First you started at rest so w initial is zero. You reach the final rpm of 120 and you did this in just 4 seconds. It also says here that tangential acceleration, tangential acceleration remember is at, after the 4 seconds is 15 m/s^2 and we want to know what is the length of the string. We haven't talked about the length of string yet but I hope you can figure out that the length of the string is this distance here. It's the radius of the circle that forms the circular path you get or it is the radial distance from the center of rotation which is your hand and the edge of rotation which is where the object is. Little r, the distance to the middle is your length. Essentially what we're looking for is little r. Think of it as little r or not as L because there's no LÕs in any of these equations so you're not going to find out. This is a little bit of a mess because we're going to have to use the combination of equations here. If you look through all the equations we've used so far, you might first think about one of the three or four motion equations. You might think of that because I gave you w initial is 0. I gave you _t. I gave you rpm which we can convert to w final. If you do that, you're going to have three out of five variables once you convert. However notice that if you look through all those four equations, there are no rÕs in them. You're not going to be able to solve for r by doing this. If you look a little further, I do have an equation that I gave you recently that links at with r. That equation is at = r_. I know at so all I have to do is find _. r = at which is 15 right there divided by _. What we're going to have to do is find _ and plug in here. How do I find _? _ is one of my five variables of motion so I'm going to be able to use these three guys to find _, and that's what we're going to do now. First I'm going to convert from rpm into w final. Remember that w final is 2¹f and F is rpm/60. 120/60 is 2 therefore w final is 2¹ and instead of f I'm going to put a 2 which is 4¹. I'm going to rewrite this here just to clean it up. w initial = 0, w final is 4¹, _t is 4. We're looking for _ and the ignored variable is the __. Notice how I know three things. The ignored variable is __. The only equation that doesn't have __ is the first one out of the four. w final = w initial + at. If we're looking for _, we just got to move everything out of the way. Initial is zero so _ is w final / t. w final is we found it here, itÕs 4¹ divided by time, time is 4. 4 cancels with 4 and alpha is 3.1415 rad/s^2. All you got to do is plug in this number here and we're good. 15 divided by 3.14 and if you divide the two you get that r is 4.77 meters and that is the final answer. ThatÕs it for this one. It's an interesting question that combines these two equations. The basic idea here is just that old school physics hassle of you got stuck in one and you're going to have to go find the other and just kind of work your way through it. There's not a very clear path. There's a few different ways you could have done this. But the most important thing is try to figure out what's the equation that has my variable and then look for all different ways to find all the letters, all the variables, you have to solve for. That's it for this one. Let me know if you guys have any questions.

Practice: A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc.

Practice: A rock rotates around a light, 4-m long string. The rock is initially at rest, but reaches 150 RPM in 3 seconds. Calculate its tangential acceleration after 3 s.

BONUS: Calculate its tangential speed after 3 s.

Practice: A 4 m long blade initially at rest begins to spin with 3 rad/s2 around its axis, which is located at the middle of the blade. It accelerates for 10 s. Find the tangential speed of a point at the tip of the blade 10 s after it starts rotating.