Brainteaser... just for fun

[Harry P.]

Imagine you are looking straight down onto a turntable with a record on it (you remember turntables and records, right? )...

There are two red dots painted onto the record–one right next to the spindle and one at the outer edge of the record. Both dots are lined up along a radius of the record (line A in the diagram below):

Let's call line A the "starting line." Now we turn on the turntable and the record makes one revolution. The red dots (along with the record) will rotate, the red dots beginning at the starting line, rotating once, with the red dots ending back at the starting line at the same exact time. One revolution. With me so far?

Ok. So we can see that the inner dot (the one next to the spindle) traveled a much shorter distance than the outer dot to cover that one revolution. Yet both dots started and finished their one trip around at the same exact time... they both left the "starting line" at the same instant, traveled one revolution, and ended up back at the starting line at the exact same time... yet the outer dot traveled a much greater distance! If the outer dot had to travel a much greater distance to make its one revolution around and get back to the starting line at the same time as the inner dot, it must have been traveling much faster than the inner dot.

But both dots are on the same record, and the record can only rotate at one speed.

So how can it be that the outer dot travels farther and faster to make its one revolution in the same time as the inner dot... yet they are both traveling the same speed (whatever speed the record is turning). How can the outer dot be traveling faster than the inner dot... and at the same speed as the inner dot?

[Quick GMC]

I've had to do this math several times, but for the life of me I can't remember the equation. I have two bridge saws that run 13-16" blades and I have to calculate blade speed vs. arbor speed for certain materials/types of blades. I usually call my cousin, a physics major.

the same reason you should stick with manufacturer recommended tire and wheel sizes, unless you can reprogram for different sizes. Wheel speed is read at the hub, and it can throw your speedometer off quite a bit.

Edited March 3, 2014 by Quick GMC

[martinfan5]

I am not smart enough to figure it out

[Belugawrx]

its all 78s Harry

or 45s

[Fabrux]

I guess I should've paid more attention in first year mechanics! But, given that my area of work deals with immovable objects, I wasn't too concerned with objects that do...

I do remember, though, that angular displacement and velocity are different than their linear equivalents...

[Belugawrx]

33 1/3

[Ace-Garageguy]

The rotational speed of the record is measured in "RPM", or revolutions per minute, BUT, because the outer dot is farther from the center than the inner dot, as you say, it has to travel farther, thus faster, do do one complete revolution.

For instance, IF the inner dot is 1" from the center, one revolution will be the circumference of a circle with a 1" radius. The formula for that circumference is 2X3.1416X1 (2 times pi times the radius), or 6.2832 inches.

If the outer dot is, say, 3" from the center, one revolution will be 2X3.1416X3, or 18.8496 inches. Much greater distance to travel at the same rotational speed, which is actually ANGULAR VELOCITY, not speed per se.

The angular velocity of the two dots is identical, but the speed of the outer dot is much greater.

Edited March 3, 2014 by Ace-Garageguy

[Harry P.]

The rotational speed of the record is measured in "RPM", or revolutions per minute, BUT, because the outer dot is farther from the center than the inner dot, as you say, it has to travel farther, thus faster, do do one complete revolution.

For instance, IF the inner dot is 1" from the center, one revolution will be the circumference of a circle with a 1" radius. The formula for that circumference is 2X3.1416X1 (2 times pi times the radius), or 6.2832 inches.

if the outer dot is, say, 3" from the center, one revolution will be 2X3.1416X3, or 18.8496. Much greater distance to travel at the same rotational speed.

Exactly. The outer dot is traveling faster than the inner dot. But the record is spinning at only one speed. So how can the outer dot be traveling faster than the inner dot when they are both traveling at the same rotational speed?

Let's say the rotational speed of the record is 33 1/3 RPM (the speed of a normal LP). That means both dots make 33 1/3 revolutions per minute... yet the outer dot is covering a much greater distance in that minute... so it must be traveling faster than the inner dot. But if both dots are making 33 1/3 orbits per minute, how can the outer dot be rotating faster than the inner dot?

[Ace-Garageguy]

Because angular or rotational speed and linear speed are not the same thing. You are mixing terms. RPM is a measure of angular displacement over time, called angular velocity., not speed. Speed is a measure of distance over time.

Both dots travel at exactly the same "rotational speed" (angular velocity), but the outer dot has much more linear distance to cover to maintain the same "rotational speed" (angular velocity).

Edited March 3, 2014 by Ace-Garageguy

[Ace-Garageguy]

Here you go...https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/v/relationship-between-angular-velocity-and-speed

[Harry P.]

Because angular or rotational speed and linear speed are not the same thing. You are mixing terms. RPM is a measure of angular displacement over time, called angular velocity., not speed. Speed is a measure of distance over time.

Both dots travel at exactly the same rotational or angular speed, but the outer dot has much more linear distance to cover to maintain the same rotational speed.

Let's say we have two concentric circular running tracks. The inner track is one mile in circumference, the outer track is five miles in circumference.

In order for the runners to make one lap of their respective tracks and get back to the start/finish line at the same time (like the dots on the record), the runner on the outer track has to run five times as fast (he has to cover five miles while the runner on the inner track only has to cover one mile). Right?

So replace the runners on concentric tracks with dots on a record. What changed?

[Ace-Garageguy]

The ONLY thing that changed is the RADIUS of the circle described by the dots or the runners. It's all in the equations relating angular displacement to radius to speed of a point on a line, and is explained fully in the video I linked to.

Am I missing some trick point? I understand the math.

Edited March 3, 2014 by Ace-Garageguy

[Harry P.]

That video made my head hurt!

[Belugawrx]

changed the whole dynamic!!!!

outta here

[Harry P.]

The ONLY thing that changed is the RADIUS of the circle described by the dots or the runners. It's all in the equations relating angular displacement to radius to speed of a point on a line, and is explained fully in the video I linked to.

Am I missing some trick point? I understand the math.

No trick. I myself don't get it.

To the example of the runners on concentric tracks... for a given interval of time, in order to get back to the starting line at the same time as the inner runner, clearly the runner on the outer track has to be running faster to make a lap in the same interval of time (the "rotational speed" of once around the track) as the runner on the inner track.

Both runners are running at the same rotational speed (one around the track in a given amount of time), yet the outer runner is running much faster.

So the runners are running at the same rotational speed, but at different velocities? The time it takes to run one lap (the rotational speed) is different than the velocity required to attain that rotational speed?

My head hurts...

[Ace-Garageguy]

Again, LINEAR velocity has to do with how far a dot or a runner goes in a given time. ANGULAR velocity in this example has only to do with how many rotations (revolutions) occur in a given time. They are not the same thing, and you seem to be trying to see them as being the same.

Two dots or runners can have IDENTICAL angular velocities, but the one farther away from the center (the greater radius) will have a higher LINEAR velocity at the same ANGULAR velocity.

Maybe thinking in the units involved will help. RPM, the layman's term for angular velocity, is "revolutions per minute" obviously, but there is NO DISTANCE UNIT there, no feet, inches, etc. Only how many REVOLUTIONS...one revolution is 360 degrees of course...so both dots or runners travel 360 degrees in the same time. Identical angular velocity.

SPEED (linear velocity, or how FAR a dot or runner travels) on the other hand, has to be expressed in some kind of DISTANCE unit over time...inches per second, feet per minute, miles per hour, etc. The speeds are markedly different for the two dots or runners, yes, because the one farther from the center does cover more ground (linear) for the same number of degrees traveled (angular).

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Maybe another way to look at it to foster understanding: one RPM is one complete revolution per minute, or 360 DEGREES per minute. There are NO DISTANCE UNITS, and RPM is not really a "speed". It's a way to express the number of DEGREES of a circle traveled in a given time.

All parts of the line the two dots are on travel the same number of DEGREES OF THE CIRCLE in a given time (RPM) but because one dot is farther away from the center, it obviously has to travel a much farther DISTANCE while it traverses an identical number of DEGREES.

Degrees are not distance, but just a proportion of whatever the circumference is of a circle described by a traveling dot.

Probably the real problem is that RPM isn't really a "speed" at all, but has come to be used to represent angular displacement over time, which is angular velocity In common usage, everyone knows what you mean when you say "crankshaft speed" for instance, and express it in RPM. It can be confusing, I know.

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Things that turn have both linear velocity and angular velocity, and though one influences the other, they are not the same, so the statement that the dots are "traveling at the same speed" is meaningless unless one specifies ANGULAR velocity. They are necessarily traveling at different LINEAR velocities.

It's absolutely necessary to separate the concepts of ANGULAR velocity (commonly expressed as RPM) and LINEAR velocity (expressed as inches-per-second, feet-per-minute, etc.) which expresses how far a dot will go at a given radius, at a given angular velocity.

Angular velocity (commonly expressed as RPM) and distance from the center (radius, or how far away from the center the dot is) determine linear velocity of the dot, and how much distance it has to travel.

Edited March 3, 2014 by Ace-Garageguy

[Psychographic]

Just because the turntable is set at a certain RPM. doesn't mean the the needle is traveling on the record at the same speed from beginning to end.

[Pete J.]

Harry, I think the issue you are struggling with is equating RPM's to velocity. RPM is not a measure of velocity. Velocity is distance divided by time. A revolution is not a measure of distance because it varies depending on the radius. In this case you have two different circumferences. The inner circumference is much less than the outer, Thus the velocity at the outer edge of the record equals the circumference times the revolutions divided by the time. In this case since we are using revolutions per minute if you multiply the circumference times the revolutions this will give you the velocity in distance per minute. You can forget the division portion since we are measuring velocity in "per minute" or 1 and division by one yields the same as ignoring it.

As a machinist you are trained to think of rpm as the speed of the bit. In reality it is not. You wouldn't run a 3" mill end at the same RPM as a 1/4" end mill to get the same quality of cut because at the point the metal meets the cutting edge of the mill end. At that point the 3" is running much faster than at the edge of the 1/4".

To put it in terms of cars if you had a 14" tire on one side of a fixed axle and a 17" on the other and both were going at a fixed RPM would you be able to go in a straight line?

Edited March 3, 2014 by Pete J.

[Agent G]

Physics.

Same concept with propellers. The rotating object is a rigid structure, therefore the tip of the prop is always in the same postiion to the the hub, barring physical change.

I don't get it, but what the hey........................

G

[Johnag4004]

Aggggghhhhh, my head hurts...

Andy...(off for a lie down now)

[tubbs]

it all comes down to who's album is playing on the turntable? if it's Jimi Hendrix "are you experienced?", it will take you a lot longer to figure it out... too many trips to the fridge, and then you won't remember where you left off.

kidding, this is one of those things that I just say "it works" and that's all I need to know. but it was interesting. I will have to ask my 20 year old college son to try to figure it out....

[Ace-Garageguy]

As a machinist you are trained to think of rpm as the speed of the bit. In reality it is not. You wouldn't run a 3" mill end at the same RPM as a 1/4" end mill to get the same quality of cut because at the point the metal meets the cutting edge of the mill end. At that point the 3" is running much faster than at the edge of the 1/4".

Yes, as a machinist you're also trained to think of speed of the cutting face of the tool relative to the workpiece, and shop math, which many folks who do machine work don't use correctly (if at all) requires you to figure out the speed of the cutting face as a function of the diameter of either the tool or the workpiece (whichever is turning) and the RPM of the tool spindle or the workpiece.

Ignore the math and you get poor cuts, overheated or broken tools, etc.

Edited March 3, 2014 by Ace-Garageguy

[DonW]

If a tiny person stood in the middle of the record as it turned they would not be moving at all, just spinning on their own axis. As they moved out to the edge of the record they would be going faster and faster relative to their surroundings, but they would still be spinning at the same RPM whatever position they were in.

The tips of a Harvard propeller actually went supersonic, which gave that aeroplane it's distinctive sound.

[Pete J.]

Ok, think on this one a bit. Given the rotation of the earth at the equator any object on the equator is traveling at approximately 1,000 mile per hour. Well in excess of the speed of sound at sea level.

[DonW]

Ok, think on this one a bit. Given the rotation of the earth at the equator any object on the equator is traveling at approximately 1,000 mile per hour. Well in excess of the speed of sound at sea level.

But it doesn't make a noise like a Harvard because the atmosphere is travelling at the same speed, subject to winds of course. Just as well really!

Edited March 3, 2014 by DonW