(II) A small rubber wheel is used to drive a large pottery whe...

Question

(II) A small rubber wheel is used to drive a large pottery wheel. The two wheels are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of 6.8 rad/s², and it is in contact with the pottery wheel (radius 27.0 cm) without slipping. Calculate the time it takes the pottery wheel to reach its required speed of 65 rpm.

Accepted Answer

Welcome back. Everyone in this problem. We want to figure out how long it takes for a large milling wheel to achieve a speed of 66 revolutions per minute when rotated by a small driving gear with a radius of eight centimeters accelerating at eight radians per second squared. Given that the gear and wheel make contact without slipping and the wheels radius is 22 centimeters for our answer choices. A says it takes 2.37 seconds. B 3.87 seconds, C 4.97 seconds and the D 7.01 seconds. Let's first make sure we understand what's going on here to help us figure out the time it takes. So we're talking about a small driving gear. Ok. Let's say this is our gear and this smart driving gear is moving a large milling wheel. Ok. So we could say our milling wheel looks like that and we know the radius for our small driving gear. Let's call it R one, the acceleration for a small driving gear. Let's call it alpha one. And for our large milling wheel, we know it's radius, let's call it R two. But we don't know it's acceleration, it's angular acceleration. So let's just label that as alpha two. However, we know that we want to get our large milling wheel to achieve a speed of 66 revolutions per minute. Now, let's ask ourselves, what do we know about speed acceleration and time that can help us or, or that is related here? Well, we know that our final angular velocity, OK, is going to be equal to our initial angular velocity plus the angular velocity of our large milling wheel, which if we think about it, that angular velocity will be equal to alpha I I its angular acceleration multiplied by T the time. OK. No. And we know that our initial angular velocity is equal to zero. So that means then that our final angular velocity is going to be the same as the angular velocity of our milling wheel alpha two T. Therefore, the time that it's going to take is going to be equal to our initial sorry, our final angular velocity divided by alpha two. So we know how to find the time. All we need to do now is to figure out what our angular final velocity is and our angular acceleration of the milling wheel. Let's start by trying to figure out the angular, the final angular velocity. OK. Now we know that we want our milling to achieve a speed of 66 revolutions per minute. If we can take that or we can take that actually to solve for our final angular velocity because now we can turn that 66 revolutions per minute into si units. OK? By multiplying it by two P Harris divided by 60 seconds. OK. Now, when we do that, because we know that this is our, the speed we want to achieve. So its angular velocity is going to be the same. OK. And so when we do that and we multiply those, we get it to be equal to 6.91 radiance per second. Thus, our final angular velocity is 6.91 radiance per second. Now that we have that next, we need to solve for alpha I I the angular acceleration of the large milling wheel. No, if we think back to the information we have from our problem. OK. Recall that we know the gear and wheel make contact without slipping. OK. So in the case of no slip, the product of the radius and the angular acceleration for both a large milling wheel and a small driving gear is constant. In other words, R one, alpha one is equal to R two alpha two, the alpha alpha two. OK. Then we can divide R one alpha one by R two. We know our one, the radius of our uh driving gear which is eight centimeters. So that's eight multiplied by 10 to the negative 2 m multiplied by alpha one, which we know is eight radiance per second squared. And that is going to be divided by R two, the million wheel radius, which is 22 centimeters. So that's 22 multiplied by 10 to the negative 2 m. Now, when we do that, OK, when we do that, put that in our calculator, we should find we get alpha two to be equal to 2.91 radiance per second squared know that we have alpha two and we have omega F or final angular velocity we can solve for our time because no, the time T is going to be equal to 6.91 radius per second divided by 2.91. OK. 2.91 radiance per second squared. Now, if we go ahead and do that, then we get our time to be equal to 2.37 seconds. That's how long it will take for the milling wheel to achieve a speed of 66 revolutions per minute. If we look back at our answer choices. A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Angular Acceleration

Relationship Between Linear and Angular Velocity

Time to Reach a Specific Speed

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