The turntable of a DJ set is spinning at a constant rate just before it is turned off. If the turntable decelerates at 3 rad/s 2 and goes through an additional 30 rotations before stopping, how fast (in RPM) was the turntable initially spinning? BONUS: How long (in seconds) does the turntable take to stop?

Hi guys. So in this practice problem, we have the turntable of a DJ set and it's spinning at a constant rate. It doesn't tell us what the rate is just before it's turned off. Now in these problems, when you have some sort of disc that's spinning and you'll turn it off, it's usually going to slow down and stop. And it tells us here that the turntable decelerates at 3 radians per second. So let's try a little turntable here. Alpha deceleration or deceleration, that'd be negative 3 because it's decelerating. And it goes through an additional 30 rotations. So that's delta theta equals 30 rotations. I'm going to write 30 rev, before stopping. Before stopping implies that the final velocity will be 0. Now I'm supposed to have we're supposed to have delta theta in radians, not revolutions. Remember, one revolution is just 2 pi radians. So you can just multiply this by 2 pi, you get 60 pi, or you can show the conversion if you want. So I'm going to write here 2 pi radian divided by 1 rev because because 1 rev is 2 pi. These cancel and you get 60 pi radians. Okay? Alright. What else? It's asking how fast was it initially spinning. So it's asking for Omega initial, but notice that it wants Omega initial in RPM. Now these equations, will always spit out Omega in radians per second, but we have to convert from Omega to RPM. Okay. So we're basically asking for RPM after we find Omega. Okay. So there's 4 variables here and the 5th one is time, delta t equals sad face. That's our ignored variable. Because this is our ignored variable, we know to use the second equation. So let's do that real quick. Omega final squared, omega initial squared plus 2 alpha Delta Theta. We're looking for this guy right here and Omega final is 0. Let's move everything to the other side, negative 2 alpha Delta Theta. And I can also take the square root of both sides and it looks like that. We're ready to plug in numbers. Got a negative inside of a square root, but it's okay because the alpha will become a negative 3 is a negative 3. And this is 60 pi. So at this point, you would multiply everything inside, take the square root of it, and you would get and you get 33.6 radians per second. But remember, that's not what we want. We want to go to RPM. Now remember, the 4 variables are RPM, which can go into frequency, which then can go into either period or or omega. Obviously, you can go both ways. These guys are also connected. In this case, we want to go from omega to RPM. Okay. The equation that links these 2 is that frequency equals RPM over 60 and the equation that links these 2 is that omega is 2pi f. If I'm going from if I'm going from omega to RPM, notice that frequency is what they have in common, right? So what you can do is you can combine these two equations into just 1, okay, and I can write w equals 2 pi. Instead of f, I'm going to write RPM over 60. Okay? That's the merged equation there. It allows me to skip straight from one to the other. So I'm going to use this to find RPM. Okay. Right here. RPM, therefore, would be 60 w divided by 2 pi. So 60 times 33.6, that'd be right there, equals, divided by 2 pi. And if you do all this, you get 3, 2, 1, RPM. So the revolutions per minute or your rate, how quickly you're spinning is 321, RPM is your initial rotation or your initial velocity. Cool? Alright, that's it for this one. Let me know if you have any questions.

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