Physics Problem: Linear vs Angular Velocity

[javacontour]

I did electrical engineering, so after freshman engineering physics, I dumped most of my mechanics.

Some came back in control systems, but I just converted the mechanical systems to their electrical analogues.

The ME's converted the electrical systems to the mechanical analogues and we all did well in this cross-listed ME/EE course

 

[sleddriver]

Originally Posted By: Vikas
I have a question about the "rotating led spot". If we were to take the path traveled by the LED and lay it flat, what would be the distance traveled by the led if the car had traveled say one mile? I am having tough time visualizing that path. For some reason, I think that the path (obvious) and its first derivative (not so obvious) both have to be contiguous. Do you have a picture showing it? - Vikas


Circumference of a circle is Pi x diameter. Given a diameter of 2 feet (in my OP), C = 6.28ft. This would be the distance covered by the (outer) LED in one revolution. One mile = 5280 ft so in 1 mile, the tire makes 840.3 revolutions. So the LED travels 840.3 revs x 6.28ft/rev = 5,277ft.

Sorry, no picture. Thanks to all for chiming in. Perspective of the viewer is important. You've all got me thinking in a different perspective!

 

[Vikas]

Sorry but that is not the correct answer. If you actually take a round object (like a glass) and tape a pen to it and roll the whole contraption (we actually did!) you will see the resultant path. The length of that path is *longer* than the distance traveled by the wheel. The shortest distance between any two points is a line. The path taken by the led is anything but a line. Hence the path is longer than the line.

Assuming the wheel is rotating at "w" radians/second angular velocity and the "r" is the radius of the wheel;

At any time "t", the position of LED will be:-

y = r sin(wt)
x = r cos(wt) + rwt

(or something like that

- Vikas

 

[sleddriver]

Tape a pen to the glass where? So it always remains plumb? Now I'm having a hard time visualizing that. If the LED starts at 6 o'clock and I view the wheel from straight above, and make the tire transparent and rotate the wheel at a constant V, I can see where the LED will first move backwards, then forwards, then backwards over one full revolution. From this given perspective. But how will it wobble from side to side if the wheel doesn't?

However, the original problem stated viewing the wheel from the side. And last I checked the circumference of a wheel still indicated its length of travel over one revolution.

Perhaps this is the trick my friend used to stump me with? As the earlier poster mentioned a 2x4 I think?

Sorry, but you're going to have to explain further. Please do!

 

[Vikas]

Tape the pen to the glass in such a way so that the point would be as close to perimeter as possible. I took a paper cup and taped the pen (dry erase marker) inside and then rolled the paper cup along the bottom of a white board. Now you will see the path drawn by the pen. It is hard to visualize without actually observing it. It is arch after arch after arch. I did not believe it myself.

- Vikas

 

[mechtech2]

Originally Posted By: Hokiefyd
Originally Posted By: mechtech2
And there is no zero speed if there is rotation and the car is moving.


There can be no zero speed relative to what?


To the road surface, of course.
We do not need to think of the relative speed to Mars or whatever non relevant position in the universe.

 

[Pete]

Wouldn't the portion of the led's motion, which is in the direction of the car's cancel itself out, except for partial rotations? I'd say you just need to know the number of rotations and the diameter of the rotation.

Expressing the path as a function is another problem...

 

[757guy]

This discussion reminded me of my 10th grade high school science teacher telling the class that "airline pilots have to factor in the earths rotation to get to the destination city at the proper time and place."

His theory: If you were flying North to South like Seattle to LAX, because the earth rotates to the east, the pilot would have to fly slightly east of south to "meet up" with LAX. If you didn't you would arrive west of your intended destination.

I can tell you as an airline pilot, that is total balony. Flight paths are based upon ground based navigation or when over water, intertial units with GPS updates referenced to ground positions. To really complicate this thinking, throw in magnetic variation for each navaid which changes constantly due to changes in the earths magnetic field.

Don't always believe what your high school science teacher tells you. (It did make me think about that for quite some time though!)

757 Guy

 

[Mr Nice]

Is it true some airports are repainting the runway number due to the changing magnetic field ?

I saw one one automotive website the debate on whether a jet airliner (Boeing / Airbus) would be able to take off if it was on a thread-mill.

 

[mechtech2]

Originally Posted By: 757guy
This discussion reminded me of my 10th grade high school science teacher telling the class that "airline pilots have to factor in the earths rotation to get to the destination city at the proper time and place."

His theory: If you were flying North to South like Seattle to LAX, because the earth rotates to the east, the pilot would have to fly slightly east of south to "meet up" with LAX. If you didn't you would arrive west of your intended destination.

I can tell you as an airline pilot, that is total balony.



Flight paths are based upon ground based navigation or when over water, intertial units with GPS updates referenced to ground positions. To really complicate this thinking, throw in magnetic variation for each navaid which changes constantly due to changes in the earths magnetic field.

Don't always believe what your high school science teacher tells you. (It did make me think about that for quite some time though!)

757 Guy





Maybe at that time, in the olden days, the teacher reported what was true, or partially true. Since the atmosphere is generally moving with the earth, I don't see an obvious validity, though.
Since then, our GPS went from satellite systems with built in relativity corrections to the newer integrated ground based systems.
So there has been change.

 

[Shannow]

757guy, so are you agreeing that the bottom of the tyre is "stationary" WRT the ground and the top is moving at 2xV or disagreeing ?

 

[PandaBear]

If I remember correctly, moving something from the north to south in the atmosphere will be skewed as the rotational speed relative to the earth is different than the inertia of the air or ocean it is floating in. The "flight path" is still the same for the shortest path, but the amount of acceleration you have to use to keep in the flight path will be different than if you are just traveling on a stationary soup of fluid that isn't rotating. I was told that it is why hurricane always rotate a certain way in the northern hemisphere and the opposite in the south.

 

[expat]

How about this:
Ant 'A' is sitting in the tread of a tire that has a circumference of 8'
Ant 'B' is at the start of a lineal track 8' long
Both ants start side by side ('A' at the 6 o'clock position of the tire) and 'B' keeps pace with the contact patch of the tire until he meets 'B' after the tire rolls to the end of the track.
Both Ants covered the 8' in the same amount of time.
If you plot 'B's course, it was a straight line.
If you plot 'A's course, it was an arc (as he lifted and fell at his place in the tread of the tire)
Did 'A' travel further and/or faster than 'B'?

 

[Shannow]

It was quite successfully answered when I was in Grade 4, and was put in a tractor tyre and pushed down a hill.

The vertical line between the centre and the ground is quite stationary, and the line from centre to the sky is moving like blazes.

 

[Garak]

Originally Posted By: LT4 Vette
Is it true some airports are repainting the runway number due to the changing magnetic field ?

I saw one one automotive website the debate on whether a jet airliner (Boeing / Airbus) would be able to take off if it was on a thread-mill.


I have no idea on the former, but with the latter, any airplane I can think of will be able to take off on a treadmill. The forward motion is provided by a prop or jet engine, not the wheels.

 

[757guy]

Originally Posted By: LT4 Vette
Is it true some airports are repainting the runway number due to the changing magnetic field ?

I saw one one automotive website the debate on whether a jet airliner (Boeing / Airbus) would be able to take off if it was on a thread-mill.



Yes that is true. Several years ago the MSP airport parallel runways were 29L and 29R. (The movie Airport with Dean Martin features this airport and they make a big point of saying they "needed" two-niner left for landing.)

In 1990, runway 29L had a magnetic heading of 293 degrees, but today it is closer to 299 degrees. Today those runways are numbered 30L and 30R to reflect the current magnetic heading.

Magnetic variation is always changing and if it changes enough, then the runway number has to be changed again to reflect the updated magnetic heading of the runway. The Milwaukee area has some of the bigger changes in magnetic variation in the United States throughout the years.

Now the idea of being able to takeoff on a treadmill is a strange one. It does not matter what the treadmill is doing, all that matters is airflow over the wings, i.e., airspeed.

If the treadmill is going 10 MPH, and the wheels are turning to match that speed, the aircraft would be stationary and therefore would not have any airspeed and would not be able to fly. If the treadmill was going 200 MPH, the aircraft would still be stationary!

The speed at which the wings are moving through the air, i.e., airspeed, is all that matters when you are flying. Groundspeed matters only for arrival calculations, i.e., if you are flying in a core of tailwind moving 150 kts., your groundspeed has increased by 150 knots, conversely, if you are flying into a 150 knot headwind, you're groundspeed is 150 knots slower.

Hope that helps.

757 Guy
Delta Pilot MSP

 

[javacontour]

They did this on Mythbusters. You have to have enough air over the wings to take off. The speed the wheels move really doesn't matter.

So if they put the plane on a treadmill, the wheels simply rotate at a different speed relative to the speed they would rotate without the treadmill.

Either faster or slower depending on the direction the treadmill moves relative to the direction the plane is traveling.

I guess I'm having a hard time understanding how someone would think the speed of the wheels would even matter. How in the world would a float plane take off if it wasn't on wheels?

Good grief.

 

[Hokiefyd]

Originally Posted By: mechtech2
Originally Posted By: Hokiefyd
Originally Posted By: mechtech2
And there is no zero speed if there is rotation and the car is moving.


There can be no zero speed relative to what?


To the road surface, of course.
We do not need to think of the relative speed to Mars or whatever non relevant position in the universe.


Then in that case, I don't understand how there can NOT be zero speed at the ground contact point of the tire, assuming there is no slip between the tire and the ground, as specified by the OP in the beginning of the thread.

 

[757guy]

Originally Posted By: javacontour
They did this on Mythbusters. You have to have enough air over the wings to take off. The speed the wheels move really doesn't matter.

So if they put the plane on a treadmill, the wheels simply rotate at a different speed relative to the speed they would rotate without the treadmill.

Either faster or slower depending on the direction the treadmill moves relative to the direction the plane is traveling.

I guess I'm having a hard time understanding how someone would think the speed of the wheels would even matter. How in the world would a float plane take off if it wasn't on wheels?

Good grief.


One of the limitations we have to consider when taking off is a tire limit speed. On the Boeing 757, the tire limit speed is 196 knots.

Performance calculations are done for each runway using current weather conditions to determine if it's legal to use that runway. A 10 knot tailwind is ok provided there is a long enough runway to reach the proper airspeed for takeoff (and you don't exceed the 196 kt. tire limit speed). There are a multitude of other factors to consider too, but for our discussion it's best to keep it simple.

Maybe more than you wanted to know...

757 Guy
Delta Pilot MSP