A thread is being pulled off a spool at the rate of 59.4 cm pe...

Question

A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.

Accepted Answer

Welcome back. I am so glad you're here. We are told that a film was taken out of its reel at a linear speed of 20.4 centimeters per second. What is the radius of the wheel? If it had 18 revolutions per minute? Our answer choices are answer choice. A 7.2 centimeters. Answer choice B 9.1 centimeters. Answer choice C 12.7 centimeters and answer choice D 10.8 centimeters. All right. So what do we know? And what are we trying to find out? We're trying to find our, that's our unknown, the radius. We know the linear speed, linear speed is V. Linear speed is given to us in terms of the formula V equals the radius multiplied by Omega are multiplied by Omega. And this is given it's equal to 20.4 centimeters per second. So we have this formula that contains an R. We know what V is. We're trying to find R. Do we know what Omega is? Omega is the angular speed? Well, Omega is given in terms of theta divided by T which is radiance per unit of time and we have something per unit of time. We're given 18 revolutions per minute. But that is not radiance per minute. It's revolutions per minute. Revolutions and radiance aren't the same thing, but we can convert from one to the other. We know that one revolution once around the circle is equal to two pi radiance once around the circle. So using unit conversion, we can get from one to the other and then we'll have our angular speed, our omega. So we'll multiply 18 revolutions per minute by the unit, we'll put two pi radians in the numerator divided by one revolution in the denominator. So that our revolution units cancel out, then multiplying straight across 18, multiplied by two pi is 36 pi and then what's left is our units which are radians per minute. There's our omega 36 pie radians per minute. Now, we can plug that in to our formula V equals R omega and solve for R. So on the left V is 20.4 centimeters per second equals the unknown. R multiplied by our omega which is 36 pie radians per minute. We want to get R by itself. So we want to get rid of our omega on the right hand side because there's a fraction there, we can multiply by the reciprocal. So instead of 36 pi we're multiplying it by one divided by 36 pi and instead of radiance per minute, we're multiplying it by minutes per radium. And what we do to one side, we do to the other as well. One divided by 36 pi minutes per reading and on the right, everything except the R cancels out, which is wonderful on the left. However, there's not so much canceling out and that's because we're dealing with two different units of time. We have minutes and seconds, but we can use unit conversion again to get rid of those. We know that one minute is equal to 60 seconds. And to get both of those units to cancel out, we can put for the unit, we'll put seconds in the numerator, 60 seconds divided by one minute. And so now lots of our units will cancel out the seconds, cancel out the minutes, cancel out and we can start multiplying in the numerator, 60 multiplied by one, multiplied by 20.4 gives us 1224 for the units in the numerator that's centimeters in the denominator. For numbers, we just have 36 pi we have radiance in the denominator. However, we're talking about a radius here which is measured in terms of distance. It's not measured in terms of radiance. So we can disregard the radiance for our answer here and then simplifying this because all of our answer choices are rounded decimals, 1224 divided by 36 pi gives us approximately 10.8 and then our unit is centimeters. So our radius here is 10.8 centimeters. We look at our answer choices and that matches with answer choice d well done. We'll catch you on the next one.

A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.

Key Concepts

Relationship Between Linear and Angular Velocity

Conversion of Units for Angular Velocity

Circumference of a Circle

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