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Brainteaser


Harry P.

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I think I've posted this before, a long time ago, but it's kind of fun to think about, so here goes...

Imagine you're looking directly down onto the surface of a record on a turntable. A 12" LP, let's say. And imagine there are two dots painted on the surface of the record... one directly next to the spindle hole, and one directly outboard of that, on the outer edge of the record.

Now imagine a straight line outwards from the spindle, through both dots. Let's call that imaginary line the starting line.

We turn on the turntable, and the record begins to rotate around the spindle. The record is rotating at a constant speed (33 rpm). In one revolution around the spindle, from starting line, around, and back to the starting line, both dots arrive back at the starting line at exactly the same time (because they are on the record surface and can't move independently of one another).

Yet it's obvious that the outer dot... the one on the edge of the record... has traveled a far greater distance than the dot at the center of the record next to the spindle. But both dots cross the starting line at the same time!

So how is it possible that the outer dot, traveling a far greater distance per revolution around the spindle, arrives at the starting line at the same time as the inner dot that only traveled a very short distance, if they are both moving at the same speed (33 rpm)? :blink:

 

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I recall this one, explained in some specific detail and concerning angular displacement versus time, distance and speed. Simple math, really, if you understand the terms and concepts.

The dot on the outer rim is moving at a far greater speed than the dot close to the spindle in order to maintain the same angular displacement over a specified time, necessarily constant all along the line between the dots. The one farther from the center has to travel a greater distance because the circumference of the circle it traverses is greater than that of the inner dot. Hence the outer dot has to travel at a greater speed to cover the greater distance in the time available to do so.

As stated above, RPM and speed are two discrete concepts, and are not interchangeable...even though the record player may be marked as "speeds" of 33 1/3, 45, etc., RPM IS NOT THE SAME THING AS SPEED.

Speed is the measure of distance traveled over a specified time. Like miles-per-hour.

RPM is a way of stating angular displacement over a specified time. One RPM...revolutions-per-minute... is 360 degrees of angular displacement in one minute (a circle is 360 degrees, so one full revolution is also 360 degrees).

Edited by Ace-Garageguy
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Ok, how about this...

It takes a certain amount of time for both dots to make the trip around the spindle, right? Isn't speed the ratio of distance traveled over a specific time? As in X miles per hour?

So how can the outer dot be traveling at a higher speed if both dots make the trip around the spindle at the same time?

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Ok, how about this...

It takes a certain amount of time for both dots to make the trip around the spindle, right? Isn't speed the ratio of distance traveled over a specific time? As in X miles per hour?

So how can the outer dot be traveling at a higher speed if both dots make the trip around the spindle at the same time?

The outer dot has to be traveling at a higher speed, because it has farther to go.

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Remember the relationship between the radius of a circle and its circumference?

The inner dot travels a circle defined by its distance from the center, which is the radius (R) of the circle it travels. 

The outer dot also travels in a circle defined by its distance from the center, which is greater than the distance to the inner dot from the center.

Because the radius of the outer circle is larger, the circumference (C) of the outer circle is larger, as determined by the formula any 6th grader should have been required to learn and understand.

Image result for 2pir

Because the circumference of the outer circle is larger, the outer dot has farther DISTANCE to travel than the inner dot, which is traveling on a smaller circle.

BOTH dots travel the same number of DEGREES OF ANGLE in the same time, but because the circumference of the outer dot's circle is greater than the circumference of the inner dot's circle, more DISTANCE is required to be traveled by the outer dot to cover the same number of DEGREES.

That's all there is. No smoke, no mirrors, no magic, no logic disconnect. :D

Edited by Ace-Garageguy
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Remember please...RPM is NOT the same thing as SPEED, and the two are not interchangeable concepts.

A precise understanding of the meaning of the words here is imperative to grasp what's happening.

FORGET the thought that both dots are on the same record for a minute, because I think that's what is throwing you off. 

Assume you have one record, a 33 1/3 RPM record, with a dot on its edge. At 33 1/3 RPM, the dot will travel 33 1/3 times around the record player spindle in one minute.

Now take a smaller record, like a 45, and put a dot on its rim. Spin it at the same 33 1/3 RPM. The dot on the smaller record will STILL travel 33 1/3 times around in one minute, but because the circumference of the circle the dot on the small record travels is smaller than the circle the dot on the big record has to travel, the dot on the small record will travel LESS DISTANCE over the same time...meaning it will travel through space at a SLOWER SPEED, even though it's at exactly the same RPM.

Now glue the little record to the big record with the dots in a line.

See??

IMG_2496-e1409686526889.jpg

Edited by Ace-Garageguy
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As stated above, RPM and speed are two discrete concepts, and are not interchangeable...

Assume a runner on a circular track exactly one mile in circumference. He runs a lap around the track. He covers one mile of distance (one lap around the track) in X amount of time; let's say five minutes. By plugging in the circumference of the track (one mile), and the time it took the runner to cover that distance (five minutes), we can calculate the speed he was running at (speed being the ratio of distance/time)... or 12 mph. And if the runner maintains a constant speed as he circles the track several times, we can then calculate his RPM around the track. 1/5 RPM. Right?

So why is a dot revolving on a record in a circular motion moving at a certain RPM but not moving at a certain speed?

Or look at it this way. Each dot on the record travels a certain distance in one revolution. Imagine that we take the distance traveled and take that round "distance" and make it a straight line. The inner and outer dots, in their revolution around the spindle, have traveled different distances, yet they both arrived at the "finish line" at the same time! :blink:

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I've explained that SPEED is the way we express DISTANCE traveled over TIME.

RPM is not SPEED. RPM is the NUMBER OF REVOLUTIONS OVER A GIVEN TIME. There are NO distance units in RPM.

The dot that's farther out from the center of the disc has to travel a greater DISTANCE IN ONE REVOLUTION due to the greater circumference of the circle it traverses, and that circumference is greater because the RADIUS of the circle is greater.  The dot that's farther out THEREFORE TRAVELS AT A HIGHER SPEED THAN THE INNER DOT.

HIGHER SPEED, SAME RPM.

RPM IS NOT THE SAME AS SPEED

This isn't hard.

EXAMPLE:

In one complete revolution, each dot travels in one complete circle.

The radius of the circle either dot traverses is the distance the dot is from the center. 

To get the circumference of a circle, you multiply its radius by 2π. (radius times 2 times Pi. Pi is roughly 3.14)

It should be obvious that a larger radius (R) will give a larger circle circumference (C) from the formula C = 2πR

For the EXAMPLE, let's say:

A dot 2" (two inches) from the center (that's a 2" radius, R = 2") will travel 2" X 2 π in one revolution. That's 12.56 inches in ONE REVOLUTION.

A dot 4" (four inches) from the center (that's a 4" radius, R = 4") will travel 4" X 2 π in one revolution. That's 25.12 inches in ONE REVOLUTION.

In this particular case, the outer dot travels TWICE AS FAR IN ONE REVOLUTION AS THE INNER DOT.....but the principle is always the same.

So, in this particular case, at the SAME RPM, the outer dot has to travel TWICE AS FAST to arrive at the starting point, which is the line that both dots are on.

Traveling TWICE AS FAST means that the outer dot's SPEED IS DOUBLE the speed of the inner dot, AT THE SAME RPM.

The formula C=2πR will ALWAYS give the circumference of the circle any dot on the record traverses in one revolution, where R is the distance of the dot from the center of the record.

Once you know the circumferences of the circles any two dots have to travel (DISTANCE) in one RPM, to determine the differential in speed between ANY TWO DOTS on the record at the same RPM is a matter of simple arithmetic.

I don't know how to make it any more clear. :mellow:

 

Edited by Ace-Garageguy
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RPM refers to the frequency of the rotation and will be the same for all points regardless of the radius of the point Angular velocity is calculated from the rpm and radius Therefore if one point is 3 inchs from center and another is 6 inchs the 6 inch point has to travel twice as fast So you are right about the speed being different for the two points

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So why is a dot revolving on a record in a circular motion moving at a certain RPM but not moving at a certain speed?

It is. Make no mistake, each dot (depending on it's distance from the axis of rotation), is traveling at a specific speed calculable from, and relative to, the RPM of the disc. Though speed may not necessarily be implied by the term RPM, it can certainly be determined as part of the equation.

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I think Harry is just having some fun with us.  Or if not, I think the above explanations clearly explained the differences and relations of speed and revolutions per minute on a circular track.

If you look at an oval track for runners, the starting (or finish) positions are staggered as not to give unfair advantage to the runners on the inside of the track (that way all the runners cover the same exact distance). :)

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Harry pull out a few of your old LP's . Then find a few song that are the same length give or take a few seconds.  Then look how much room they take on the LP. The songs on the outer edge will take up less room or they will be thinner than those that are on the inside as they are traveling farther around the outside of the LP.  

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