Author |
Topic: Balls of Steel. (Read 1368 times) |
|
rloginunix
Uberpuzzler
Posts: 1029
|
 |
Balls of Steel.
« on: Apr 30th, 2014, 4:14pm » |
 Quote  Modify
|
Balls of Steel.
All other conditions being equal which one of two identical solid steel balls is it more difficult to heat to the same temperature - the one resting on a non-heat-conducting plane or the one hanging off of a non-heat-conducting rod?
Justify your answer.
|
|
 IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender: 
Posts: 13730
|
|
Re: Balls of Steel.
« Reply #1 on: Apr 30th, 2014, 10:25pm » |
Quote Modify
|
The one hanging from a rod is easier to get a flame under.
The one on the plane possibly lets less heat radiation escape (depending on whether it's also heat-reflective as well as non-conducting)
All conditions being equal, they already have the same starting temperature, and it's equally easy to heat them to that temperature by doing nothing.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #2 on: May 1st, 2014, 8:28am » |
Quote Modify
|
May be I should've been more explicit, my bad, but "all other things being equal" also means that some abstract source of heat, doesn't matter which one, is applied equally to both steel balls. "Equally" means with equal effect and that the ball's configuration, plane or rod, doesn't make any difference and presents no hindrance to the heating process.
Let's say that the balls are heated with Alien2' favorite device - a laser beam, but, again, it doesn't matter.
Also, this is not a trick question - the balls are getting heated. Let's say their initial temperature is T0. They are both heated to the temperature T1 such that T1 > T0. But don't get carried away with the formulas too much - this is what was called in my day a qualitative problem, similar in spirit to the "Scaredy Cat".
SMQ, elsewhere on this forum, came up with a very good line: "physics gives exact solutions for approximate problems while engineering gives approximate solutions to exact (real life) problems".
The first order of business here is to find the exact physics concept for "more difficult" or, more precisely, "difficult", "difficulty". If you put your physics hat on then In physics terms what does it mean "difficult"?
|
|
IP Logged |
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #3 on: May 1st, 2014, 5:44pm » |
Quote Modify
|
To be completely fair here's the pictorial representation of the initial condition:
The rod is attached to the ceiling. The plane is attached to the floor. We can always level the identical steel balls before we start heating them ...
|
|
IP Logged |
|
|
|
Immanuel_Bonfils
Junior Member
Posts: 114
|
|
Re: Balls of Steel.
« Reply #4 on: May 1st, 2014, 6:56pm » |
Quote Modify
|
the one resting on a plane will be compressed, and heat conduction will be "easier" than in the other one hanging, that will be dilated.
|
|
IP Logged |
|
|
|
rmsgrey
Uberpuzzler
    
Gender:
Posts: 2873
|
|
Re: Balls of Steel.
« Reply #5 on: May 2nd, 2014, 4:27am » |
Quote Modify
|
The plane will re-radiate heat back to the ball more efficiently than the rod.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Balls of Steel.
« Reply #6 on: May 2nd, 2014, 6:07am » |
Quote Modify
|
on May 2nd, 2014, 4:27am, rmsgrey wrote:| The plane will re-radiate heat back to the ball more efficiently than the rod. |
|
That's not necessarily true it might be completely transparent to heat radiation.
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #7 on: May 2nd, 2014, 8:42am » |
Quote Modify
|
Immanuel_Bonfils, good observation but the steel balls aren't big enough for compression/stretching to have a noticeable effect and can be safely neglected.
rmsgrey, also good observation but as towr noted the plane may be heat-transparent and more importantly let us consider the time frame before the steel balls are hot enough (if at all) to radiate any significant amount of heat. In other words let's put the ideas of convection a side for a moment.
There are a bit different forces at play here. We still didn't translate the vague "difficult" into something more concrete and measurable.
Let us consider what happens to a steel ball when it gets heated. This isn't just a physics problem, some of my favorite subject is involved too ...
|
|
IP Logged |
|
|
|
SWF
Uberpuzzler
Posts: 879
|
|
Re: Balls of Steel.
« Reply #8 on: May 2nd, 2014, 4:14pm » |
Quote Modify
|
The one resting on the platform will be easier to heat because the base will act to impede the convection flow induced by the ball being warmer than the air.
But I guess the answer you are looking for is related to work done against gravity because of thermal expansion.
|
| « Last Edit: May 3rd, 2014, 7:34am by SWF » |
IP Logged |
|
|
|
rmsgrey
Uberpuzzler
Gender:
Posts: 2873
|
|
Re: Balls of Steel.
« Reply #9 on: May 3rd, 2014, 7:15am » |
Quote Modify
|
The upper ball will fall apart at a temperature where the lower ball just slumps a bit
|
|
IP Logged |
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #10 on: May 3rd, 2014, 9:21am » |
Quote Modify
|
rmsgrey, your powers to predict the future and read my mind are beginning to scare me. Just yesterday I was thinking about the puzzle where your last post ideas may be employed but I won't say any more to save that future puzzle. Now, in this one the absolute temperature doesn't really matter. We might just as well make both steel balls "very very cold" and heat them up by 1, or 2, or 5 degrees with the original question standing. What really matters here is the change of temperature.
SWF, thank you for properly hiding your answer - it contains 50% of the intended solution.
I've posted the pictorial solution on this forum over here.
Before we look at the solution though let's do the remaining 50% of the puzzle and ponder - exactly what is doing work against gravity and what's happening to the second, rod-hanging, ball?
This was my high school physics question. I guess when it's posted in the proper textbook section you kinda know where it's going. Taken completely out of context it becomes less obvious.
Basically, in physics the words "difficult" are translated into energy, work, force. SWF used work, my choice was energy. So the problem can be reworded as "more energy must be expanded to heat which one of two ...". After that step the thought may run thus: as the steel ball is heated it expands, which means that its radius grows, which means its cross section's circumference (in 2D) grows. In pure geometric terms it means that the center of the circle moves. In physics terms that center of the circle (in 2D or 3D) is the center of mass. The growth is inhibited by the plane and by the rod. In case of the plane the ball's center of mass must be moved "up" - against gravity. In case of the rod the ball's center of mass moves "down" - with gravity. Since the center of mass of the ball on the plane must be moved up more energy must be expanded (or more work done) - its potential energy must be increased - hence it's more "difficult" to heat the ball on the plane.
If you, guys, think this is too specific and too involved - let me know and I won't be posting these anymore. Otherwise I have a few more similar puzzles from my high school years and one from the first year in college. No need for exact formulas, just some common sense, a gut feel and a rather concise idea of what's going on.
Also, thank you for a good discussion. I learn a lot from you in terms of mathematical thinking so I hope this was a useful and entertaining non-mathematical experience.
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: Balls of Steel.
« Reply #11 on: May 3rd, 2014, 1:16pm » |
Quote Modify
|
Considering some other things have been filed under "can be safely neglected", somebody should put some numbers to the various effects.
If the plane was not transparent to radiation, I suspect that effect might be considerably greater.
As might the convection of air (or whatever medium, if not vacuum, the setup is in) (which is probably the hardest effect to take into account, considering the equations involved in the turbulence)
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
rmsgrey
Uberpuzzler
Gender:
Posts: 2873
|
|
Re: Balls of Steel.
« Reply #12 on: May 5th, 2014, 5:39am » |
Quote Modify
|
The obvious problem with this question is that there are two stated differences between the two balls - their orientation (one with a fixed point at the top; the other with a fixed point at the bottom) and the nature of their support (one with a single point of contact to a locally infinite plane; the other with a circle of contact to a rod)
It would make the question fairer if both balls were supported by identical rods, eliminating one set of variables.
Numerically:
Corrected per SWF's post below
Steel expands linearly by about 1 part in 105 per degree at around room temperature, has a density of ~8g/cm3, and a specific heat capacity of .42 J/g/K = 420 J/kg/K
Assuming gravity at 10 m/s2, we get the potential energy change of both spheres as:
10*10-5*E*r/0.42 or 2.4*10-4*E*r where E is the heat energy input (in Joules), and r is the radius of the spheres (in centimeters)
10*10-5*E*r/420 or 2.4*10-7*E*r where E is the heat energy input (in Joules) and r is the radius of the spheres (in meters)
So, with spheres a meter in radius, the change in potential energy is about 2.4% 0.000024% of the change in thermal energy. For spheres with a 1cm radius, it's a hundredth of that.
Changing the strength of gravity will also make a difference - the percentage of the thermal energy represented by the change in potential energy is directly proportional to both r and g. If the temperature is significantly different from room temperature, the thermal expansion coefficient will be different too.
|
| « Last Edit: May 7th, 2014, 4:17am by rmsgrey » |
IP Logged |
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #13 on: May 5th, 2014, 10:36am » |
Quote Modify
|
Tiny typo with sup /sup tags, rmsgrey, for the ten to the minus fourth power.
|
|
IP Logged |
|
|
|
rloginunix
Uberpuzzler
Posts: 1029
|
|
Re: Balls of Steel.
« Reply #14 on: May 5th, 2014, 3:37pm » |
Quote Modify
|
After I posted the problem I realized that the plane bit will cause confusion (and is not really needed) but opted not to mess with it - I still remember how nicely I screwed up the "Let There Be Circular Light" discussion.
I would simplify it to the point where it can't be simplified any more: one ball, one rod. Experiment 1 - the rod is attached to the ceiling. Experiment 2 - the rod is attached to the floor:
From rmsgrey's calculation we see that the effect of center of mass movement is itself pretty negligible, in practice. And if we switch to a more realistic tennis ball size, r = 3.35 cm, or better yet a ping pong ball size, r = 2cm, then I would imagine an average high school physics lab instruments won't be able to detect this effect at all.
So in its latest reincarnation it has mostly theoretical interest and could be (and probably was) used as a rapid fire verbal exam question or some such.
|
|
IP Logged |
|
|
|
SWF
Uberpuzzler
Posts: 879
|
|
Re: Balls of Steel.
« Reply #15 on: May 6th, 2014, 6:01pm » |
Quote Modify
|
rmsgrey, those numbers you came up with seemed remarkably high. I checked the math, and believe you were too high by a factor of 100000.
Looks like you missed a factor of 100 by saying r is in centimeters when everything else was in meters. The only cm you mention in the properties was in the density, but you never even used the density because it was not necessary. There was also another factor of 1000 from the specific heat being per gram while the rest of the units were based on kg.
The dimensionless ratio, W/E is g*a*r/c, where
g: gravity
a: coeff of thermal expansion
r: radius
c: specific heat
|
|
IP Logged |
|
|
|
rmsgrey
Uberpuzzler
Gender:
Posts: 2873
|
|
Re: Balls of Steel.
« Reply #16 on: May 7th, 2014, 4:12am » |
Quote Modify
|
on May 6th, 2014, 6:01pm, SWF wrote:rmsgrey, those numbers you came up with seemed remarkably high. I checked the math, and believe you were too high by a factor of 100000.
Looks like you missed a factor of 100 by saying r is in centimeters when everything else was in meters. The only cm you mention in the properties was in the density, but you never even used the density because it was not necessary. There was also another factor of 1000 from the specific heat being per gram while the rest of the units were based on kg.
The dimensionless ratio, W/E is g*a*r/c, where
g: gravity
a: coeff of thermal expansion
r: radius
c: specific heat |
|
You're right - I should have been more careful about standardising my units (particularly once density cancelled out)
|
|
IP Logged |
|
|
|
UgoLocal02
Newbie
Gender:
Posts: 31
|
|
Re: Balls of Steel.
« Reply #17 on: Jun 11th, 2014, 11:40pm » |
Quote Modify
|
on Apr 30th, 2014, 4:14pm, rloginunix wrote:Balls of Steel.
All other conditions being equal which one of two identical solid steel balls is it more difficult to heat to the same temperature - the one resting on a non-heat-conducting plane or the one hanging off of a non-heat-conducting rod?
Justify your answer. |
|
As you mentioned non heat conducting both are equally hard to heat.
|
| « Last Edit: Jun 11th, 2014, 11:41pm by UgoLocal02 » |
IP Logged |
|
|
|
Annettagiles
Junior Member
Gender:
Posts: 67
|
|
Re: Balls of Steel.
« Reply #18 on: Oct 1st, 2014, 4:46am » |
Quote Modify
|
non-heat conducting rod is very difficult i think so
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print