[PHYSICS] Challenge Problem

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badwiz
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[PHYSICS] Challenge Problem

Postby badwiz » Sun Oct 17, 2010 6:34 am UTC

I know many people on this forum have various experience in studying physics, and I am looking for challenging problems in classical mechanics; what do I consider a "challenging" problem? How about I tell you what I don't consider "challenging" and then give an example problem I did you can munch on 8)

1. Intractable Problems where approximation methods are necessary. If the problem requires a computer, or cannot be performed in a few hours in a test environment with pen, paper, and a math reference, then I don't want to bother with it. (of course, approximations like a small oscillation approximation are allowed if expected in the problem)

2. Common Problems. I have done enough double pendulum problems, double atwood machines, etc... . I am looking for quirky or unusual problems to sharpen my creative skills.

Tricky Newtonian problems are especially useful if you know any. Problems with constraints are fine too; I know Lagrangian and Hamiltonian forms.

Here's an example problem I did yesterday; If anything, please discuss this problem if few additional problems are posted. Feel free to post your solution; I am always interested in seeing alternative methods. I will give my solution later on when I have more time.

A ladder of length L and uniform mass m rests vertically against a frictionless wall and floor. If the ladder is nudged slightly, so it approximately starts from rest, at what angle does it separate from the wall?
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Re: [PHYSICS] Challenge Problem

Postby Ashton Whithers » Sun Oct 17, 2010 8:27 am UTC

Okay, I've got one for you. A simple enough conceptual problem, but with hairy implications: What is the opposite of relativity? More specifically, relativity theory looks at the greatest extremes of speed relative to other objects also travelling through space. What happens if you look at the other end of that spectrum, and stop an object completely relative to space itself. For a reference point, use the center of the universe, which ought to have a black hole at nearly the exact location of the Big Bang, a black hole which is not moving at all through space (assuming that the point from which the Bang originated was itself stationary).

I've been considering this, and I know that there isn't much thought given to velocity relative to space itself because space is usually thought of as nothing, but humor me. If you achieve lightspeed, time stops passing for you, you gain infinite mass, and you compress to infinitely small size in the direction of travel. So, if you stop completely, would infinite time pass for you, you lose all mass, and you expand in every direction to the size of the universe? What happens then? If so, what would it mean for the rest of physics? Or is this result impossible?

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Re: [PHYSICS] Challenge Problem

Postby jmorgan3 » Sun Oct 17, 2010 8:47 am UTC

A rope of mass m and length l is suspended by one end above a scale, with the lower end just barely brushing the scale. The rope is dropped at time t=0. Find the reading of the scale as a function of time.
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Re: [PHYSICS] Challenge Problem

Postby gmalivuk » Sun Oct 17, 2010 3:37 pm UTC

Ashton Whithers wrote:OFor a reference point, use the center of the universe, which ought to have a black hole at nearly the exact location of the Big Bang
There's no such thing.
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Re: [PHYSICS] Challenge Problem

Postby Tass » Sun Oct 17, 2010 4:02 pm UTC

Ashton Whithers wrote:Okay, I've got one for you. A simple enough conceptual problem, but with hairy implications: What is the opposite of relativity? More specifically, relativity theory looks at the greatest extremes of speed relative to other objects also travelling through space. What happens if you look at the other end of that spectrum, and stop an object completely relative to space itself. For a reference point, use the center of the universe, which ought to have a black hole at nearly the exact location of the Big Bang, a black hole which is not moving at all through space (assuming that the point from which the Bang originated was itself stationary).

I've been considering this, and I know that there isn't much thought given to velocity relative to space itself because space is usually thought of as nothing, but humor me. If you achieve lightspeed, time stops passing for you, you gain infinite mass, and you compress to infinitely small size in the direction of travel. So, if you stop completely, would infinite time pass for you, you lose all mass, and you expand in every direction to the size of the universe? What happens then? If so, what would it mean for the rest of physics? Or is this result impossible?


Read the sticky thread "relativity questions (and other common queries)". You have several misconceptions about relativity, space, and the big bang here, and they have been dealt with several times before.

Besides this thread was for classical newtonian mechanics.

As for the OP: A ring has a mass of m. It is standing vertically on a table. Two point masses M can slide frictionlessly along the ring. If they start from rest at the top point of the ring and slide down simultaneously on each side, what relationship between m and M must be satisfied for the ring to lift from the table?

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Re: [PHYSICS] Challenge Problem

Postby Nlelith » Sun Oct 17, 2010 4:15 pm UTC

From 200 Puzzling Physics Problems by Peter Gnädig, G. Honyek, and KF Riley:

Image

Hint:
Spoiler:
"The time intervals of the motions and the lengths of the paths do not have to be found exactly; only the inequalities relating them need to be determined."

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Re: [PHYSICS] Challenge Problem

Postby Tass » Sun Oct 17, 2010 4:52 pm UTC

Solution to above:

Spoiler:
Obviously A is in motion longer, they travel the same vertical distance, under gravity, starting from rest, A has an upwards force on it, B does not.

A travels pi/2 meters, B lands two meters away, B travels longer.

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Re: [PHYSICS] Challenge Problem

Postby badwiz » Sun Oct 17, 2010 5:14 pm UTC

Ashton, please don't hijack my thread. Follow Tass' advice and read the relativity sticky. Thanks,

jmorgan3:
Interesting problem. I don't seem to get something I can solve analytically. Here are my thoughts so far:
Spoiler:
The normal force on the rope as a function of time will be what the scale reads.
Let:
N[t] = Normal force
y[t] = Length of rope currently at rest on the scale.
[imath]\lambda[/imath] = linear mass density of the rope = M / L

Using Lagrangian Dynamics:
[imath]L = T - U = 1/2 \lambda (L-y)(\dot{y})^2 - \frac{\lambda g (L-y)^2}{2}[/imath]

The Forces will be:
[imath]-\frac{\delta L}{\delta y} = 1/2 \lambda \dot{y}^2 - \lambda g (L-y)[/imath]
So the Normal Force is this above without the mg term:
[imath]N = 1/2 \lambda \dot{y}^2 + \lambda g y[/imath]

I don't see how to proceed from here. The differential equation is not linear and has non-constant coefficients, so finding y[t] is a bit out of the question.
Here is Mathematicas numerical solution. Its still interesting:
Image
Image


Tass,
I will try your problem soon, seems interesting :)

Nlelith,
THank you for provided your book source, I will look into it. Are they all newtonian style problems or are there some lagrangian-style problems too?

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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Sun Oct 17, 2010 5:33 pm UTC

Consider a large collection of molecules inside a cylinder of radius a, length L and the temperature T. The cylinder rotates at angular speed w, and is in a gravitational field g.

A) What is the hamiltonian for a single molecule?
B) Now allow the cylinder's length to fluctuate. You now apply an external pressure P. What is the average value of the cylinder's length?

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Re: [PHYSICS] Challenge Problem

Postby Noetherlite » Sun Oct 17, 2010 5:41 pm UTC

The rope is a question for 1st years at Cambridge. Interesting & can be done with no "advanced" physics

Spoiler:
If I recall correctly, just sum the weight of rope at rest on the scales with the reaction force provided to decelerate falling rope at each moment in time and it all comes out very neatly

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Re: [PHYSICS] Challenge Problem

Postby Charlie! » Sun Oct 17, 2010 6:38 pm UTC

Of course, if we're looking for challenging and unintuitive problems, DDWFTTW has to rank way up there.

viewtopic.php?f=18&t=31905

To make it an actual physics problem, you could perhaps compute the maximum speed as a function of wind speed and blade pitch (am I forgetting anything), then add in both kinetic friction and/or drag to see if you can find a more realistic maximum speed and optimal blade pitch.
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Re: [PHYSICS] Challenge Problem

Postby Noetherlite » Sun Oct 17, 2010 6:56 pm UTC

1) Ice cube with bits of lead in it floats in a glass of water. What happens to the water level when it melts?
2) Is this different if the enriched ice cube sinks rather than floats?

3) Take a floating cube that is half the density of water & place in water. Displace the cube vertically downwards, describe the motion of the cube when you release it.

4) As 3. But use a wedge shaped floating object ie isosceles triangle prism (can also do cone or sphere but the maths gets fiddly for no real benefit)

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Re: [PHYSICS] Challenge Problem

Postby thoughtfully » Sun Oct 17, 2010 7:22 pm UTC

Another awesome book is Thinking Physics, by Lewis Carroll Upstein. I kid you not. Whether his parents are responsible or not, I dunno. In its third edition! 500+ pages of these puzzles, arranged by subject. Quite a few are the sort you're looking for. A lot of them are shorter and test your intuition more than anything else, though. There's a version of the ice cube with something heavy embedded in it in there, for instance. Solutions and explanations are provided.
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Re: [PHYSICS] Challenge Problem

Postby jmorgan3 » Sun Oct 17, 2010 11:48 pm UTC

badwiz wrote:Using Lagrangian Dynamics:
[imath]L = T - U = 1/2 \lambda (L-y)(\dot{y})^2 - \frac{\lambda g (L-y)^2}{2}[/imath]

The Forces will be:
[imath]-\frac{\delta L}{\delta y} = 1/2 \lambda \dot{y}^2 - \lambda g (L-y)[/imath]
So the Normal Force is this above without the mg term:
[imath]N = 1/2 \lambda \dot{y}^2 + \lambda g y[/imath]

I don't see how to proceed from here. The differential equation is not linear and has non-constant coefficients, so finding y[t] is a bit out of the question.

I confess I don't know Lagrangian dynamics, so I can't check your work, but I believe it should be [imath]N = \lambda \dot{y}^2 + \lambda g y[/imath]. Also, are y and [imath]\dot{y}[/imath] dependent on the normal force, or is there a simple way to find them?
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Re: [PHYSICS] Challenge Problem

Postby Dopefish » Tue Oct 19, 2010 12:46 am UTC

Tass wrote:As for the OP: A ring has a mass of m. It is standing vertically on a table. Two point masses M can slide frictionlessly along the ring. If they start from rest at the top point of the ring and slide down simultaneously on each side, what relationship between m and M must be satisfied for the ring to lift from the table?


I had that as a challenge problem a few years back. Managed it at the time I think too, but the fact it has a solution at all still doesn't sit well with intuition. But of course, those are the best problems.

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Re: [PHYSICS] Challenge Problem

Postby Nlelith » Tue Oct 19, 2010 1:39 am UTC

badwiz wrote:Nlelith,
THank you for provided your book source, I will look into it. Are they all newtonian style problems or are there some lagrangian-style problems too?

All the problems can be solved using the usual Newtonian methods, though I suppose you may find some questions easier when using Lagrangians. The problems are meant to challenge your insight and your understanding of the physics rather than your ability to use complex methods*, as seen in Tass' solution.

*not saying anything bad about Lagrangian mechanics, it's a beautiful and often times necessary way to solve more complicated problems.

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Re: [PHYSICS] Challenge Problem

Postby tbone694 » Wed Oct 20, 2010 5:22 pm UTC

Here's one of my favorite mechanics problems of all time:

Show that the motion of a particle traveling in an arbitrary tunnel through the Earth undergoes harmonic motion. Show that the period is independent of the path and find the period. That is to say, pick any two points A and B on the surface of the Earth and drill a hole straight from A to B (not necessarily through the Earth's center) and find the period of motion on that path, which is the same for all A's and B's. This is the idea behind the gravity train: http://en.wikipedia.org/wiki/Gravity_train .

Edit: I forgot to mention to neglect effects from the Earth's rotation for the period to be constant.

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Re: [PHYSICS] Challenge Problem

Postby Jhackulon » Sat Oct 23, 2010 9:04 am UTC

Hey guys, I'm not sure if this is what your looking for but my dad said this was a problem given to someone he knows in a job interview.

A candle is lit in a closed, high, elevator and the cable is cut, will the candle go out before it hits the ground?

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Re: [PHYSICS] Challenge Problem

Postby Dark Knight Bob » Sun Oct 24, 2010 9:59 pm UTC

No. The candle won't go out.

it's mass is ridiculously low for inertia to have an effect

And the air inside the cabin isn't going to be displaced enough to cause an issue.

This all assumes ideal conditions though.

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Re: [PHYSICS] Challenge Problem

Postby ++$_ » Sun Oct 24, 2010 10:17 pm UTC

Jhackulon wrote:Hey guys, I'm not sure if this is what your looking for but my dad said this was a problem given to someone he knows in a job interview.

A candle is lit in a closed, high, elevator and the cable is cut, will the candle go out before it hits the ground?
Spoiler:
It probably depends on the type of flame, the energy of the established convection, and the height of the drop, but in principle, it could. With zero apparent gravity, convection will stop, possibly smothering the candle.

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Re: [PHYSICS] Challenge Problem

Postby sikyon » Mon Oct 25, 2010 5:42 pm UTC

++$_ wrote:
Jhackulon wrote:Hey guys, I'm not sure if this is what your looking for but my dad said this was a problem given to someone he knows in a job interview.

A candle is lit in a closed, high, elevator and the cable is cut, will the candle go out before it hits the ground?
Spoiler:
It probably depends on the type of flame, the energy of the established convection, and the height of the drop, but in principle, it could. With zero apparent gravity, convection will stop, possibly smothering the candle.

Spoiler:
There is still diffusion due to heat and thermodynaics of the Co2 and the Oxygen. You cannot say for sure if the diffusion rate of oxygen to the candle is sufficient or if convection is needed without doing actual experiment or mathematical analysis.

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Re: [PHYSICS] Challenge Problem

Postby gmalivuk » Mon Oct 25, 2010 5:45 pm UTC

Obviously we need to know more about the situation in order to answer that question, but in general a weightless candle would snuff itself out. So the question is whether we're ignoring air resistance and the elevator is in free-fall, and whether the shaft is high enough for the candle to snuff itself out before it hits the bottom.
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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Mon Oct 25, 2010 7:11 pm UTC

Candles function in 0 gravity.

http://blogs.chron.com/sciguy/archives/ ... re_lg.html

The image on the right is a picture from nasa of a candle in orbit.

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Re: [PHYSICS] Challenge Problem

Postby Tass » Mon Oct 25, 2010 8:24 pm UTC

Ignoring the atmosphere, deviations from a sphere, and rotation: What is the quickest travel you can make between two opposite points on the earth, given a rocket of infinite power, but a cargo that can withstand no more than 2g?

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Re: [PHYSICS] Challenge Problem

Postby gmalivuk » Mon Oct 25, 2010 9:08 pm UTC

presumably the rocket is going around the outside of the sphere?
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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Mon Oct 25, 2010 9:49 pm UTC

With infinite thrust, I dont think going through the sphere would be much of a problem

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Re: [PHYSICS] Challenge Problem

Postby JWalker » Mon Oct 25, 2010 9:51 pm UTC

There are a bunch of fairly challenging ones from the University of Illinois PhD qualifying exam. You can find them here: http://physics.illinois.edu/grad/qual-archive.asp The classical mechanics ones start with "CM."

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Re: [PHYSICS] Challenge Problem

Postby Tass » Tue Oct 26, 2010 7:13 am UTC

gmalivuk wrote:presumably the rocket is going around the outside of the sphere?


I assumed that was understood. I was considering specifying it, but got lazy.

BlackSails wrote:With infinite thrust, I dont think going through the sphere would be much of a problem


Infinite thrust (and fuel), not infinite strength hull :p. That is why I said ignore the atmosphere (or start and end at 80km, doesn't make much difference).

I might actually have said that the rocket just have unlimited fuel, but it is incapable of delivering more than 2g acceleration.

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Re: [PHYSICS] Challenge Problem

Postby BlazeOrangeDeer » Sun Nov 07, 2010 2:50 am UTC

Tass wrote:Ignoring the atmosphere, deviations from a sphere, and rotation: What is the quickest travel you can make between two opposite points on the earth, given a rocket of infinite power, but a cargo that can withstand no more than 2g?

Are we starting at rest? Having an initial velocity would allow us to focus only on the centrifugal force felt by the rocket as a result of being confined to the surface. Since Earth pulls at -1g, we can go around the sphere with a maximum centripetal acceleration of 3g and the rocket will only feel 2g. To get the velocity, since a great circle path is shortest, use the fact that centripetal acceleration equals velocity squared times the radius of earth. Solving gives v=sqrt(3gr), then substitute d/t for v and solve for time. My answer is:

t=pi*r/sqrt(3gr)
3.1416 * 6.3781E6 m / sqrt(3 * 9.80665 m/s^2 * 6.3781E6 m) = 1462.8 seconds

or 24.379 minutes.
of course if you wanted to start at rest I don't want to think about that ;)

Charlie! wrote:Of course, if we're looking for challenging and unintuitive problems, DDWFTTW has to rank way up there.


I found that really interesting, I found it was easier to understand once I thought of it this way: at wind speed, there is no wind relative to the cart but the wheels are turning rather quickly, it is the wheels that drive the propeller to force air upwind relative to the cart(that is, downwind slower than the surrounding wind?) and this increases the speed further. It's almost more like a ground-powered rear-prop car, pretty weird.

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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Sun Nov 07, 2010 3:18 am UTC

Tass wrote:
I might actually have said that the rocket just have unlimited fuel, but it is incapable of delivering more than 2g acceleration.


With infinite fuel, its incapable of giving more than 0g acceleration, unless it also has infinite thrust :P

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Re: [PHYSICS] Challenge Problem

Postby Tchebu » Sun Nov 07, 2010 4:10 am UTC

A pendulum is made of a massless rod and a hollow sphere of radius R and mass m (uniformly distributed). The inside is filled with water. Ignoring viscosity and thermal expansion, how does the period change if we freeze the water?
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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Sun Nov 07, 2010 4:20 am UTC

Assuming the sphere is fully filled, then it doesnt change, since the mass distribution remains the same. A full container of liquid water doesnt slosh around, since water is basically incompressible.

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Re: [PHYSICS] Challenge Problem

Postby ++$_ » Sun Nov 07, 2010 4:38 am UTC

BlackSails wrote:Assuming the sphere is fully filled, then it doesnt change, since the mass distribution remains the same. A full container of liquid water doesnt slosh around, since water is basically incompressible.
The water could still rotate, though.

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Re: [PHYSICS] Challenge Problem

Postby BlackSails » Sun Nov 07, 2010 6:05 am UTC

So could the ice ball.

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Re: [PHYSICS] Challenge Problem

Postby nehpest » Sun Nov 07, 2010 6:14 am UTC

++$_ wrote:
BlackSails wrote:Assuming the sphere is fully filled, then it doesnt change, since the mass distribution remains the same. A full container of liquid water doesnt slosh around, since water is basically incompressible.
The water could still rotate, though.

BlackSails wrote:So could the ice ball.

Wouldn't conservation of angular momentum cause the ice ball to rotate in exactly the same way as the water, since we're ignoring viscosity?

I haven't taken a fluid dynamics class, so forgive me if this is off-base.
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Re: [PHYSICS] Challenge Problem

Postby Charlie! » Sun Nov 07, 2010 9:42 am UTC

I think he's talking about
1) the ice ball doesn't rotate inside the ball.
2) the water moves like an approximation and just rotates with the same shape.
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Re: [PHYSICS] Challenge Problem

Postby Tass » Sun Nov 07, 2010 9:44 am UTC

BlackSails wrote:
Tass wrote:
I might actually have said that the rocket just have unlimited fuel, but it is incapable of delivering more than 2g acceleration.


With infinite fuel, its incapable of giving more than 0g acceleration, unless it also has infinite thrust :P


I didn't say infinite, just unlimited. Meaning "enough". It's a photon drive, okay? The expenditure of fuel will be negligible on a journey this short.

And yes, of course, start and stop at rest.

Any more nitpickers? :)

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Re: [PHYSICS] Challenge Problem

Postby LucasBrown » Sun Nov 07, 2010 6:11 pm UTC

Determine the difference between Earth's equatorial and polar radii. A solution can be found here.

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Re: [PHYSICS] Challenge Problem

Postby LaserGuy » Sun Nov 07, 2010 6:51 pm UTC

There's a whole bunch of really good ones here. A few of them do require numerical methods to solve (mainly just finding a zero of a function or something), but most can be solved analytically. Of these, I'd say that #9, 12, 16, 20, 31, 36, 38, and 40 all look particularly interesting. There's solutions here if you get stuck.


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