A few super-fun physics problems! Yay!

[Lobstrosity]

I have no idea whether anybody here is into this sort of thing, but I happen really like uncomplicated conceptual physics problems that make you think a bit. I like to think that the following problems fall into that category. Feel free to attempt any/all/none of the following (though if you've seen one of these explicitly worked out elsewhere, I simply ask that you hold off on offering the solution until others have had the chance or inclination to try). Maybe I'll give a pearl or something to the first person who offers the correct solution (and justification) for each problem.

1)
http://physics.ucsd.edu/~schmidel/icecube.jpg

You place a cube of ice in a glass of water at room temperature such that the ice is freely floating. You return later to find that the ice has totally melted. The initial and final states are depicted above (not necessarily to scale). Assuming enough time has elapsed such that the water is once again at room temperature, will you find that the water level in the glass has risen, stayed the same, or decreased? Why? You may assume that both the water in the liquid and the ice is pure. You may also fully neglect evaporation (i.e. the glass is a closed system).

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2)
http://physics.ucsd.edu/~schmidel/bowlingball.jpg

You're sitting in a little boat that is floating in a small tub of water. In the boat with you is a big bowling ball. You lean over and measure the height of the water in the tub (marked "initial water level" in the diagram above). You then toss the ball overboard into the tub (assume no water splashes out of the tub) and re-measure the height of the water level (marked "final water level" in the diagram above). Keeping in mind that the diagram you see is not necessarily to scale, will you expect to find that the final water level is higher than, equal to, or lower than the initial water level? Why?

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3)
http://physics.ucsd.edu/~schmidel/ballooncar.jpg
You have a car that is accelerating forwards (in the normal driving direction). A helium balloon is floating in the car as pictured above. Which way will the balloon drift relative to the car as the car accelerates (i.e. will it move towards the front windshield, the rear windshield, or stay where it is) and why? You can assume all the car's windows are rolled up. You can also assume that either the car has a frictionless ceiling upon which the balloon rests or that the balloon is tethered to the floor by a long string that would still allow it some horizontal movement.

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4) You have a bouncing ball. It initiall starts from rest at a height of 1 m and falls due to gravity, which you can assume to be 10 m/s². Every time it hits the ground, it bounces back up, losing exactly half of its current energy in the process. The question is this: how much time elapses between the moment the ball is initially released and the moment it comes completely to rest on the floor (i.e. all bouncing has ceased)? You may assume that this occurs in a vacuum (i.e. no air resistance). You may also assume that the time the ball spends in contact with the floor during each bounce is vanishingly small. I assert that the answer is not due to some cheap technicality like non-ideal physics. In fact, this problem is meant to be addressed assuming absolutely ideal classical physics (none of that quantum or statmech stuff).


Problem four is easily the most difficult, but in my opinion it is also by far the most interesting.

[Sher]

/ot are those hotlinked to your website? if so, cool pics!

[Lobstrosity]

Today @ 01:12 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116964#post116964)
SherBear:

/ot are those hotlinked to your website? if so, cool pics!
Not really a website so much as a directory to upload miscellaneous things I wish to link to. What you are witnessing above is magic of MS Paint in all its glory :teeth:

[HRG_new]

Today @ 08:54 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116956#post116956)
Lobstrosity:

I have no idea whether anybody here is into this sort of thing,
[quote]
Me! Me! (But I've seen 1-3 before)


1) to 3) are simple applications of hydrostatics (Archimedes' principle), as it applies not only to external forces like gravity, but also to the (pseudo-) centrifugal force in a non-inertial system.

4) involves a supertask *) and my answer is "twice the time the ball would take to fall and bounce back if there was no energy loss).

*) like the following: I'm switching a lamp on. After 1/2 minute, I'm switching it off. 1/4 minute later, back on, 1/8 minute later, back off etc.

Question: what is the state of the lamp after exactly 2 minutes ?

A problem of mine:

Let's take an iron spring and wind it up closely. It stores potential energy. We throw the spring into concentrated sulfuric acid and dissolve it completely. What happened to the potential energy ?

Corollary: Same with a permanent magnet and the energy of its magnetic field.

Regards,
HRG.

[Lobstrosity]

Today @ 02:17 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116989#post116989)
HRG_new:

4) involves a supertask *) and my answer is "twice the time the ball would take to fall and bounce back if there was no energy loss).
This isn't actually correct, but it's very much on the right track. Recall that the time isn't halving with each bounce, the energy is.

[Lobstrosity]

Today @ 02:17 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116989#post116989)
HRG_new:


Let's take an iron spring and wind it up closely. It stores potential energy. We throw the spring into concentrated sulfuric acid and dissolve it completely. What happened to the potential energy?
Let me see if I can take a quick naive stab at this. The potential stored in a wound spring is in truth stored in the intermolecular bonds that hold the molecules of the spring together. As you wind the spring, you stretch these bonds, decreasing the magnitude of the binding energy. When the acid reacts with the molecules, it is breaking these intermolecular bonds through a chemical reaction. As the bonds are easier to break, the molecules will come off the spring with a higher kinetic energy than they would from an unwound spring. In short, I would guess that the original potential energy stored in the spring would be converted into thermal energy.

Similarly, with a permanent magnet, each tiny dipole (i.e. each atom of the magnet) feels a force proportional to the gradent of the field at its location. When the acid breaks the molecular bonds holding the dipole in place, it will be accelerated away by the non-uniform magnetic field. This will confer kinetic energy to the liberated dipoles and result in increased thermal energy.

Is any of this anywhere close to being right?

[Sher]

Today @ 04:52 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116982#post116982)
Lobstrosity:


Not really a website so much as a directory to upload miscellaneous things I wish to link to. What you are witnessing above is magic of MS Paint in all its glory :teeth:

:lol: I meant the wedding pics! :rofl:

But the MS Paint are good too ...

[Sher]

Today @ 03:54 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116956#post116956)
Lobstrosity:

I simply ask that you hold off on offering the solution until others have had the chance or inclination to try


Today @ 05:17 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116989#post116989)
HRG_new:

[answers]

HRG!

:bonk:

[Archimedes]

Today @ 08:54 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116956#post116956)
Lobstrosity:

4) You have a bouncing ball. It initiall starts from rest at a height of 1 m and falls due to gravity, which you can assume to be 10 m/s². Every time it hits the ground, it bounces back up, losing exactly half of its current energy in the process. The question is this: how much time elapses between the moment the ball is initially released and the moment it comes completely to rest on the floor (i.e. all bouncing has ceased)? You may assume that this occurs in a vacuum (i.e. no air resistance). You may also assume that the time the ball spends in contact with the floor during each bounce is vanishingly small. I assert that the answer is not due to some cheap technicality like non-ideal physics. In fact, this problem is meant to be addressed assuming absolutely ideal classical physics (none of that quantum or statmech stuff).

Hmm, my first guess would be that the ball never stops bouncing because you never get zero kinetic energy when you keep halving it. But you also dubbed this problem "easily the most difficult" and I haven't solved the first two, so there is probably something I'm overlooking.

[DunnySaze]

I think I figured them out, but want to mull it over for a bit longer.

Here's one of mine.

You have 2 eggs identical in size and shape. One is hard boiled, the other raw. Assume both eggs are now at room temperature. Without damaging either of the eggs in any way, how can you tell which is which?

[Sher]

Today @ 09:53 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117184#post117184)
DunnySaze:

I think I figured them out, but want to mull it over for a bit longer.

Here's one of mine.

You have 2 eggs identical in size and shape. One is hard boiled, the other raw. Assume both eggs are now at room temperature. Without damaging either of the eggs in any way, how can you tell which is which?

Now this is one I don't have to be a science geek to get :teeth:

Ahem ...

:: puts on mother hat ::

:: envokes Mutant power "Super Easter Egg" ::

A hard-boiled egg will spin longer (and faster, actually) than a raw one ... :smile:

~Sher, who really hopes she remembered correctly after all this silliness ...

[Archimedes]

Today @ 08:54 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=116956#post116956)
Lobstrosity:

1)

You place a cube of ice in a glass of water at room temperature such that the ice is freely floating. You return later to find that the ice has totally melted. The initial and final states are depicted above (not necessarily to scale). Assuming enough time has elapsed such that the water is once again at room temperature, will you find that the water level in the glass has risen, stayed the same, or decreased? Why? You may assume that both the water in the liquid and the ice is pure. You may also fully neglect evaporation (i.e. the glass is a closed system).

The water level stays the same. The weight of the ice cube is equal to the weight of the water it displaces (and its density is lower than that of water, otherwise it wouldn't even float), hence the volume that is displaced is equal to the volume of the water that we get once the ice cube has melted entirely.


2)

You're sitting in a little boat that is floating in a small tub of water. In the boat with you is a big bowling ball. You lean over and measure the height of the water in the tub (marked "initial water level" in the diagram above). You then toss the ball overboard into the tub (assume no water splashes out of the tub) and re-measure the height of the water level (marked "final water level" in the diagram above). Keeping in mind that the diagram you see is not necessarily to scale, will you expect to find that the final water level is higher than, equal to, or lower than the initial water level? Why?

Water level goes down.

The volume displaced by the boat (including the person on board) when the bowling ball is still in is the weight of the boat plus the bowling ball divided by the density of water (because as before, we know that the weight of the boat and its contents is equal to the weight of the water it displaces). This is the same as the weight of boat divided by the density of water, plus the weight of the bowling ball divided by density of water. On the other hand, once you toss the ball overboard, the total volume that affects the water level is the volume displaced by the empty boat, plus the volume displaced by the bowling ball. The former is equal to the weight of the boat divided by the density of water (i.e. the same as the boat's portion of the displaced volume before), but the latter is equal to the weigth of the bowling ball divided by the density of the bowling ball. Because the density of the bowling ball is greater than that of water, it's volume is less, and the total volume of the water in the tub is less, and the water level decreases.

(Lousy explanations, but I'd feel pretty dumb calling myself "Archimedes" if I couldn't whip up at least something...)


3)

You have a car that is accelerating forwards (in the normal driving direction). A helium balloon is floating in the car as pictured above. Which way will the balloon drift relative to the car as the car accelerates (i.e. will it move towards the front windshield, the rear windshield, or stay where it is) and why? You can assume all the car's windows are rolled up. You can also assume that either the car has a frictionless ceiling upon which the balloon rests or that the balloon is tethered to the floor by a long string that would still allow it some horizontal movement.

I'm skipping this one because I've heard it before.

[DunnySaze]

Today @ 03:00 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117194#post117194)
SherBear:



Now this is one I don't have to be a science geek to get :teeth:

Ahem ...

:: puts on mother hat ::

:: envokes Mutant power "Super Easter Egg" ::

A hard-boiled egg will spin longer (and faster, actually) than a raw one ... :smile:

~Sher, who really hopes she remembered correctly after all this silliness ...


You are correct!

Another thing you can do is when your eggs are spinning, place your fingers on top just long enough to stop the spinning, then immediately remove them. The hard boiled egg will remain motionless and the raw one will start to spin again.

Have a pearl on me!

A big round salt-water one. :smile:

[Lobstrosity]

Today @ 07:02 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117198#post117198)
Archimedes:


(Lousy explanations, but I'd feel pretty dumb calling myself "Archimedes" if I couldn't whip up at least something...).

Those first three problems were just for you :wink:

[Lobstrosity]

Today @ 04:34 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117077#post117077)
SherBear:



:lol: I meant the wedding pics! :rofl:

But the MS Paint are good too ...

Oh ... :dunce: Heh, thanks.

[HRG_new]

Today @ 10:52 AM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117002#post117002)
Lobstrosity:


Let me see if I can take a quick naive stab at this. The potential stored in a wound spring is in truth stored in the intermolecular bonds that hold the molecules of the spring together. As you wind the spring, you stretch these bonds, decreasing the magnitude of the binding energy. When the acid reacts with the molecules, it is breaking these intermolecular bonds through a chemical reaction. As the bonds are easier to break, the molecules will come off the spring with a higher kinetic energy than they would from an unwound spring. In short, I would guess that the original potential energy stored in the spring would be converted into thermal energy.

Similarly, with a permanent magnet, each tiny dipole (i.e. each atom of the magnet) feels a force proportional to the gradent of the field at its location. When the acid breaks the molecular bonds holding the dipole in place, it will be accelerated away by the non-uniform magnetic field. This will confer kinetic energy to the liberated dipoles and result in increased thermal energy.

Is any of this anywhere close to being right?

Yep. It is. In general, if energy has to go "somewhere", in 99% it will be heat.

And thanks for clearing up my oops about halving the energy (= halving the height it bounces back => time divided by sqrt(2) )

Regards,
HRG.

[HRG_new]

Today @ 12:37 PM post located here (http://www.theologyweb.com/forum/showthread.php?s=&postid=117081#post117081)
SherBear:





HRG!

:bonk:

Self-:bonk: !

Apologies. I did not see the passage you quoted.

regards,
HRG.

[Joseph Alward]

JOE ALWARD
I believe the exact answer to Problem 4 is

( 2^.5 +1) / (10^.5 - 5^.5), or approximately 2.61 seconds.

[Jaltus]

Joe,

Is that a proper step function?