Tag: paradox

Galileo’s Paradox

By Anupum Pant

Here’s an image of a contraption. It is basically a long stick hinged at one end and is free to move about the other. At the end of it rests a ball. Near the ball there’s also a cup fastened to the stick. The big stick is lifted up high and is temporarily supported by a small stick.

galileo paradox

Now, what do you think would happen when the temporary support is removed? Normally, it would be very intuitive to think that the cup and the ball would fall at the same speed. In other words, nothing fascinating would happen. Both would fall and the ball would roll away…no?

However, something very unexpected happens when the support is removed. Something that, in a jiffy demonstrates some very important concepts of physics like centre of mass, torque and acceleration.

The big wooden stick (with the fastened cup) falls and it falls faster than the ball. Actually it falls and also rotates. As a result of the swing, the cup comes under the ball just before ball reaches it and the ball ends up inside it.

Under the influence of the same gravitational force, irrespective of the mass, the cup and the ball must have fallen at the same rate, as predicted by Galileo? What really happens? The video explains…

The Birthday Paradox

By Anupum Pant

Imagine you meet a random person in the street and ask him/her when is their birthday, there’s a huge chance that the person’s birthday will not be the same as your birthday. In fact, the probability of both your birthdays being on the same day is around 0.27%. Fat chance. At the back of our heads, this is something that is very clear to all of us.

Again, if you repeat this by asking about 22 people the same question, the chance of you finding someone having the same birthday as yours is still around 5%. Too less. This is too is a very intuitive piece of information.

But consider this. If I put all of the 22 guys and you in a room, there’s a big chance that 2 people in that room will have the same birthday – a 50% chance. Moreover, if there are 70 people in the room, this chance increases to about 99.99%. This is called the birthday problem or the birthday paradox.

So, what changed when 20 people went into the room? It was just the fact that in the room, we are picking 2 people from a group of 23 people. That is equivalent to this – everyone is asking everyone their birth dates. Everyone doing it simultaneously makes the probability much higher. The probability of two people sharing a birth date among a group of 23 people is far higher than you alone going around and asking all the 22 people, and finding someone having the same birthday as your’s.

Suppose there are 200 people in the room. The probability of 2 people sharing their birthday is massive (and yet not definite). There is in fact a 99.9999999999999999999999999998% chance!

1024px-Birthday_Paradox.svg

Finally, if you had 367 people in a room, at least a pair among these 367 people in the room would definitely have the same birth date. The 99.99% chance shoots up to a definite (100%) probability if there are 367 people in the same room. Think about it for a minute.

The Potato Puzzle

By Anupum Pant

Some call it the Potato paradox, but I prefer calling it a puzzle. It isn’t really a paradox. It’s just that we tend to get confused easily when working with such problems and most times end up with the wrong answer. Here’s the question:

You have 100 kg of potatoes. Assume that they are made up of 99% water. Now, you keep them outside to dry for a while. When measured for water content now, they contain 98% water. What do you think is the weight of these dried potatoes now?

Answer quickly first. Then, try and calculate. It isn’t really tough.

Take 15 minutes, or more if you have to. Use papers, pens and calculators if you have to. But whatever you do, stay honest. Don’t search for the solution on the internet, or don’t read further if you haven’t done it yet. Trust me, your brain won’t like the answer.

Solution – [Wikipedia]

The answer is: 50 kg.

For the explanation, watch this visual explanation…

The Coastline Paradox

By Anupum Pant

The length of Australia’s coastline according to two different sources is as follows:

  1. Year Book of Australia (1978) – 36,735 km
  2. Australian Handbook – 19,320 km

There is a significant difference in the numbers. In fact, one is almost double the other. So, what is really happening here? Which one is the correct data?
Actually, it depends. The correct data can be anyone of them or none of them. It completely depends on the kind of precision you decide to use while measuring the coastline. This is the coastline paradox.

The coastline paradox

The coastline paradox is the counter-intuitive observation that the coastline of a landmass does not have a well-defined length. – Wikipedia

The length of the coastline depends, in simple terms, on the length of scale you use to measure. For example, if you use a scale that is several kilometers long, you will get a total length which is much less than what you’d get when you would use a smaller scale. The longer scale, as explained neatly in this picture, will skip the details of the coastline.

This is exactly what happened when the two different sources measured the coastline of Australia. The first, Year Book of Ausralia, used a much longer scale than the one, Australian Handbook used. Ultimately, the great disparity in the result had to do with the precision of measurement. Had they used a scale just 1 mm in length, the result would have been a whooping 132,000 km.

This effect is similar to the mathematical fractal, Koch’s flake. Koch’s snowflake is a figure with finite area but infinite perimeter. Veritasium explains it better in this video:

Another factor is to take into account the estuaries to measure the length. Then,what about those little islands near the coast? and the little rocks that protrude out of the water surface? Which ones do you include to come out with the data?  And the majestic Bunda cliffs? Probably this article from the 1970’s clarifies what was included and what was not during the time the results were published.

So, the next time someone decides to test your general knowledge and asks you the length of certain country’s coastline, your answer should be – “It depends.”