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Classical Mechanics Level 1

Philip forms three equal portions of leftover mashed potatoes into three shapes: a cube, a sphere, and a cylinder. Each portion is heated in the oven to the same uniform temperature.

When he takes the portions out of the oven and leaves them at room temperature, which shape takes the longest time to cool down?

Details and Assumptions:

Assume that the mashed potatoes are uniform in composition and that each shape is maintained throughout the process.
The mashed potatoes are suspended in the air.

Cube Sphere Cylinder All will cool down at the same rate

12 solutions

Gopala Krishnan
Nov 17, 2017

According to Stefan-Boltzmann Law, power radiated from a body is directly proportional to the surface area. Here, for the same mass, the minimum surface area would be of a sphere. Hence sphere will radiate the slowest.

Moderator note:

For a "regular" closed surface (in math jargon, a "compact strictly convex" one), the sphere has the lowest possible surface area to volume ratio and thus any mashed potato shape of the same volume other than the sphere would cool faster. The proof of this, first found by Liebmann in 1900, is not simple.

For more exotic surfaces (no longer "compact strictly convex") it's still thought the sphere has the lowest surface-area-to-volume ratio but nobody knows for sure! An example of an "exotic surface" is below.

The image above is an attempt to render the "Alexander Horned Sphere" . There are an infinite number of holes so it's not exact.

There are a lot of ways to form the cylinder. If radius<<height, then surface area approaches 0, therefore it would take the longest time to cool down.

Kostas Kokolakis - 3 years, 6 months ago

As the radius approaches zero, the height has to grow for the volume to stay constant. If the radius is halved, the height has to increase by a factor of 4. The surface area of a cylinder is 2 pi r^2 + 2 pi r*h. The second term will increase as r decreases if volume is kept constant. It won't approach zero. The ratio of surface area to volume for a cylinder is proportional to (1/h + 1/r). To minimize this ratio (and thus to minimize the surface area for a fixed volume) you'll want h = r. Long, skinny cylinders and short, fat cylinders will have much higher surface area to volume ratios. The sphere is the shape that minimizes the surface area to volume ratio.

Richard Desper - 3 years, 6 months ago

Side observation - when i was calculating the surface area of the 3 masses, I realized that the cylinder surface area for a given volume varied dramatically. At very small radii and very large radii, for a given volume,the surface area increased dramatically. In the example I used (volume =1000) the surface area minimum occurred at a radius of approx 5.419 units and surface area increased at radii above and below 5.419. So, had the question been which mass cooled fastest, the answer could have been the cylinder or the cube, depending on the radius/height of the cylinder.

Greg Grapsas - 3 years, 6 months ago

Cube has 6 units , sphere has 12.57 units , and cylinder has 12.57 units surface area . Right ? So sphere and cylinder have same surface area. Assuming radius = height = length = width = 1 unit.

Chakravarthy R - 3 years, 6 months ago

But you can't assume that, because if all the dimensions are 1 unit, then the volumes come out to be different and the question says the portions are equal which means the volumes are also equal.

Veides Kasera - 3 years, 6 months ago

All three shapes have the same volume. For a fixed volume, a sphere has the minimum surface area.

Pranshu Gaba - 3 years, 6 months ago

They all have the same volume and mass.

Shkidt Isaac - 3 years, 6 months ago

That's right. But the geometry of the object can affect the rate at which it cools.

Pranshu Gaba - 3 years, 6 months ago

I first thought the sphere would also radiate the slowest, but then I thought about the "table" that these objects are presumably resting on. If the table was a good insulator (I think it's also equivalent to say - if the table had low thermal conductivity), then perhaps the side of the cube resting against the table would essentially radiate no energy or heat. If that were true, then perhaps the surface area of a 5 sided cube is less than the surface are of a sphere, and would thus radiate the slowest. I'm no good at math, though, and so I won't attempt the calculation...

Jeffrey Joe - 3 years, 6 months ago

Jeffrey Joe, I followed a similar thought process..

Yolanda Xu - 3 years, 6 months ago

You have a valid point, thanks for pointing it out. I have updated the problem statement so the shapes can lose heat from all sides.

Pranshu Gaba - 3 years, 6 months ago

Brenda Gaines Hunter
Nov 30, 2017

D is wrong. They definitely will not all cool down at the same time. Why? Because each has a different surface area and heat is lost at the surface. I also know that the more corners, the more surface at least as a ratio to volume. So I am assuming that each has the same volume as well. Therefore, what I know is this: the cube will lose heat the fastest. And the sphere will maintain it the longest. The heat is evenly dispersed throughout the sphere. (I am assuming that it is round when I argue that the heat is evenly dispersed.) Heat is lost very fast at the edges.

The sphere has the smallest surface area, so it will take the longest to cool. As for the fastest, we cannot compare the cube and cylinder unless their dimensions are given. If the cylinder is very long and has small radius, or is almost flat with large radius, then its surface area will be greater than that of cube and it will cool the fastest.

Pranshu Gaba - 3 years, 6 months ago

Can you please elaborate on the relationship between the number of angles, volume and area?

adeniyi Adeosun - 3 years, 6 months ago

What was that???

Asif Imad - 3 years, 6 months ago

Carbon Man
Dec 1, 2017

A sphere have the biggest volume for its surface, therefore the least amount of surface is exposed, therefore takes the longest to cool.

they have same portion of potatoes. So each shape has same volume.

Muhammad Hasan - 3 years, 6 months ago

Yes the sphere have the same volume as all the other shapes, but the least surface compared to the other shapes.

carbon man - 3 years, 6 months ago

Ron Walsh
Nov 28, 2017

Given the same mass for the 3 geometric shapes the sphere presents the smallest surface area, thus the least heat radiating area, making it the longest to cool. The sphere has 52% less surface area than the cube and 67% less area than the cylinder for similar dimensions.

The numbers do not seem right. For the same volume, the sphere has 19.2% less surface area than the cube.

Pranshu Gaba - 3 years, 6 months ago

I thought that the cylinder was a bit taller because the picture wasn't very accurate. I think guess and check works best. Why does the sphere take longest? What is the Alexander Horned Sphere? I would rate this problem a 9 because I don't know much about physics. I felt like it was kind of hard but I didn't want to give up.

Lucia Tiberio - 3 years, 6 months ago

Shkidt Isaac
Nov 27, 2017

Another reason is that the centre point of Cube has shortest distance to its closest surface, which is shorter than the radius of the Sphere, so Cube is faster than Sphere to cool down.

Please note that all three pieces have the same volume and mass.

Well, You can suspend these objects on a thread probably hanging somewhere? this way the whole object is exposed and in this case sphere would have the minimum area I guess (Although it might not make sense to suspend mashed potatoes on a thread though)

Gopala Krishnan - 3 years, 6 months ago

The distance from the center of the cube to one of the vertices is greater than the radius of the sphere. It is the surface area of the object that determines the rate of heat loss. If the surface area is greater, then the heat transfer is quicker. In this case, sphere has the least surface area, so it cools down the slowest.

Pranshu Gaba - 3 years, 6 months ago

You haven't adjusted for insulation which contains the heat. The question is posed incorrectly. My practical thought. One of the 'surfaces' on the Cube is on an insulator. Do you know what an insulator is?

Rick Tollefson - 3 years, 6 months ago

You have a valid point, thanks for pointing it out. I have updated the problem statement so the shapes can lose heat from all sides.

Pranshu Gaba - 3 years, 6 months ago

The cube has one of its 6 sides on the 'ground' which is an insulator. So why wouldn't the cube cool slowest?

Rick Tollefson - 3 years, 6 months ago

It is stated that they are suspended in air though

Lee Ye Chuan - 3 years, 6 months ago

Chandan Gupta
Nov 27, 2017

It absorb heat by large surface area $vice versa

Matt Maloney
Nov 27, 2017

Not a solution as much a comment on other solutions. Regarding surface area, if the 3 objects are placed on an insulating material surface to cool down (as opposed to being magically suspended in the room with their entire surface areas exposed), then I would think the sphere may now have the largest exposed surface area, and the other two objects will cool even less uniformly than their shapes alone might suggest.

You're absolutely right, this really should be stated in the assumptions section. If you no longer count the surface area where cylinder and square touch the insulating material surface (though that's just one way to treat this hypothetical), then the cylinder actually ends up having the lowest exposed surface area (assuming the same radius as the sphere). Square still has the most surface area though.

Andy Greig - 3 years, 6 months ago

I definitively support this solution !

Cédric Poncet-Montange - 3 years, 6 months ago

You have a valid point, thanks for pointing it out. I have updated the problem statement so the shapes can lose heat from all sides.

Pranshu Gaba - 3 years, 6 months ago

Atish Behera
Dec 2, 2017

Heat radiated is directly proportional to the exposed surface area. Hence ,the body with least surface area will take longest. Now, out of cube, Sphere and cylinder. The surface area to volume ratio of sphere is least. Hence takes longer.

Kiichi Takeuchi
Dec 2, 2017

Lol it defended my suspects like “what’s happen if I put it on super chilly tray?”

V S
Nov 30, 2017

Sphere has large surface area in comparison to other shapes so more area is in contact with air which results in its fast cooling.

Just the opposite: the Sphere has the lowest surface area to volume ratio, thus will take the LONGEST to cool as radiant losses to the air are minimized. And if it is resting on a surface, it will lose the least amount to that medium as well due to having only a zero-dimensional point of contact versus the one- or two-dimensional areas in contact with the other two shapes.

Brian Sullivan - 3 years, 6 months ago

Nurfatin Najihah
Nov 30, 2017

Answer: Sphere Because it has greater exposed surface area to the environment.. So it time to lose the heat is shorter ...other than the cylinder and cube..

This is not correct. The sphere has the lowest surface area of the three shapes, so it will take the longest to cool.

Pranshu Gaba - 3 years, 6 months ago

Anand Prakash
Nov 27, 2017

As the question has asked which shape will solidify first. I have used the concept of solidification time of material which is used in casting process from material science of mechanical engineering. Solidification time formula is as follows:

time (t)= k*(volume/surface area)^2

time = t

volume = V

Surface area = A

where k is a constant, value taken as 1.

As their volume is same but surface area are different. On comparing areas we can find time of solidification, hence which have higher surface area its solidification time is lower and vice versa. It clearly shows that higher the solidification time means the material is taking longer time to cool and vice versa.

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