(Probability) A Thanksgiving Puzzle

Happy Gobble Gobble Day!

It’s Thanksgiving Day and you’re hosting dinner for your large extended family of 30 people. In the interest of organization, you’ve assigned seats by setting name-tags for each person around the table. But when everybody goes to sit down at the table, there’s a problem: Great Uncle Cornelius, who is ancient, has already sat down. Being a bit senile, he has forgotten his own name and just picked a seat at random!

Not wanting to disturb him, everybody decides to just let him stay there. Each person, as they come to the table, will sit at their own seat if it is available, but if it is not, they will just pick a seat at random from the ones that are left. You, being the host, are the last one to sit down.

What is the probability that you’ll sit at your correct place?

(Fluids) I Can't Take The Surface Tension!

A Primer on Surface Tension

Surface tension in a liquid is brought about by the attractive forces between the molecules of the liquid. It is what enables a liquid to minimize its surface area, and it is one of the things that affects the evaporation rate of a liquid.

What’s keeping this spider from sinking? Surface tension! The weight of the spider is not enough to break the tension of the surface of the water, and consequently the surface provides an upward force that is equal to the spider’s weight. Note: Don’t try water walking at home. This spider is a professional.

If a force F acts parallel to the surface of a liquid over a distance L, then the surface tension is defined as:

As can be inferred from the above, the SI unit of surface tension is N/m. Take note: “surface tension” is not a force, but rather a force per unit length. But that’s not all! Remember that a N m is also a Joule, the SI unit of energy. So 1 N/m = 1 Nm/m^{2} = 1 J/m^{2}  . So you can also think of surface tension in terms of energy per area.

It helps to think of the liquid molecules as having springs between them. Surface tension describes a tension force per unit length or energy per area that the film can experience before the "springs" break and the film scatters into multiple bodies of liquid. (Or imagine the springs breaking and causing a just barely too heavy bug to sink. Too much dinner.)

Suppose a thin liquid film is stretched over a wire frame, as shown below.

The slider can be moved horizontally (force F moves it to the right of the page), which changes (increases) the area of the liquid film. The surface of a thin film such as this has two sides: one side facing you, and a second side behind it. Consequently the film has some some small, yet nonzero thickness.

For the thin film, the tension on each side of the surface contributes equally to the total force so the surface tension per side is:

Suppose now that a thin circular ring of radius r, called a Du Noüy ring, is slowly drawn out of a liquid with a force F, as shown below.

This time the length L of the film is the circumference of the circular ring: 2πr. And just like with the wire frame, the thin film has two sides, although this time the sides of the surface are not flat. Each side contributes equally to the surface tension. So the surface tension per side is given by:


A rectangular wire frame with slider length L has a thin liquid film stretched across it, and a force of 0.5 N is used to pull the slider to right. A surface tension σ1 develops in the film until the force exerted by the film is exactly equal and opposite to the 0.5 N pulling force.

A Du Noüy ring of radius r is slowly drawn from a sample of the same liquid using a force of 0.5 N. A surface tension σ2 develops in the film, again until the force exerted by the film is exactly equal and opposite to the 0.5 N pulling force.

How does the surface tension σ1 compare to σ2 ?

(Physics) Pushing Against a Wall

Consider a real system of a person on skates. The nervous skater starts by pushing off of a nearby wall that they've been clinging to in desperation. The work done by the contact force of the wall on the person as the person moves is...

(Thermo) Where'd the Energy Go?

A process involving 30 J of work is done by a closed system on its surroundings. What is the heat transfer (and in what direction), if the internal energy of the system decreases by 50 J. ?

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(Changes Made: Single-Topic 2-Week-Series, Heat Focus, Progressive Questions, Deeper Solutions, Hints, Equation-Oriented Problems etc.)

The Decay of Radioactive Isotopes

Cesium-137 is one of the most problematic radioactive isotopes found in many nuclear weapon tests and nuclear accidents such as the Chernobyl & Fukushima disasters. Exposure to it increases chances of cancer, while exposure to high concentrations can lead to serious burns, and even death.

The isotope has a rate of radioactive decay described by the function:


  • y = amount of material left
  • k = the decay constant

The site of a nuclear accident has an abnormally high concentration of Cesium-137. If the decay constant for Cesium-137 is 0.023, how much time does it take for some initial amount of Cesium-137 to decay to half its original amount?

      5 years
      30 years
      400 years
      9000 years
      I don't know...