If a compressor pulley has a 5 34 inch diameter and a motor...

If a compressor pulley has a 5 3/4 inch diameter and a motor pulley has a 2 3/4 inch diameter, what is the ratio of the revolutions per minute for the motor pulley to the revolutions per minute for the compressor pulley expressed as a fraction in lowest terms?

Transcript

00:01 Two pulleys of radius 3 feet and 1 .5 feet are connected with the belt if the larger pulley rotates at 37 rpm a question is how fast is the smaller pulley rotating in radiance per minute now the rotation that is specified here is basically the speed so the radiance per minute that they are asked us to find out is the angular speed so there is a relationship between linear speed and angular speed so the linear speed can be said as v angular speed is is denoted by the term omega so v can be said as equal to r times omega where a radius is equal to r and angular speed.

01:04 Now why does it become important in this problem it becomes important because considering the the pulleys are connected by a belt in this problem we can say that the linear speed of both pulleys is the same both the police is the same that's because it's connected by the same belt so obviously the total velocity the linear velocity will be the same for both the bullies but the angular velocity will be so different because it depends on the radius of the pulley now with this we can establish a formula by calculating the individual the angular velocities given the given the information that they have given us the rotation rate now let's write down the given information now considering that we've looked at the theory behind this problem so the given information is we've got the radius of small smaller pulley calling that to be equal to r1 that is given to be 1 .5 feet the radius of larger pulley is given by let's call it r2 which is given by three feet which is given as three feet actually and they've also given us the the rotation rate which would be in this case rate of rotation in exact terms of the notation it says rotates at 37 rpm so we can actually calculate we can directly write a formula by saying the angular speed of small smaller pulley is given by and this is for us to find out this is the question that we are trying to find so let's let's say this is given this is equal to some x or something so it's easier for us to substitute i'm calling it as a variable x now the angular speed of larger pulley is omega 2 and that can be obtained by multiplying the rate of the revolutions here 37 revolutions per minute so you can multiply the 37 revolutions per minute multiplied by 2 pi and that would give you the answer in radiance radiance per minute so this would be in radiance per minute so that is going to be in radiance per minute so let's calculate omega 2 just to complete the problem so that is going to be 74 radiance per minute so omega 2 is 74 radiance per minute this is what we get now there is a relationship we have for which we have already seen we've said that v equals to our omega where v is the linear speed so there therefore, we can use the relationship for smaller pulley and larger pulley.

05:31 Considering that the linear speed is going to be the same for both the cases, we can say that r1 multiplied by omega 1 will be equal to r2 multiplied by omega 2...

Question

If a compressor pulley has a 5 3/4 inch diameter and a motor pulley has a 2 3/4 inch diameter, what is the ratio of the revolutions per minute for the motor pulley to the revolutions per minute for the compressor pulley expressed as a fraction in lowest terms?

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