WMJ Tutoring

Happy New Year! 🎉

From Will to all Students/Parents.

I saw this question and thought it would be a good opportunity to discuss density and states of matter.

Even though this question has an intuitive answer, it's actually more complicated than it seems at first sight.

Physics principles that apply to changes of state (Solid->Liquid) (Melting)
1) When a substance is solid, particles are packed closely together with little movement (due to a lack of kinetic energy)
2) When a substance is liquid, particles are still closed but they have enough energy to move and occupy slightly more space.
3) Because a liquid occupies slightly more volume than a solid of the same substance, the density of said liquid will be lower.

So when a solid melts, its density reduces and the volume increases.

By this logic, when ice melts, its volume increases, thus the water level should rise.

However...

Water is quite an exceptional molecule. It exhibits many unique properties which similar molecules don't. One of these is that the density of water is actually higher than the density of ice. This is why ice cubes float.

The reason why water is more dense than ice is due to the way water molecules interact. At low temperatures ( at under 0°), water molecules try to pack together. Due to electrostatic forces between the polar parts of the water molecules (Hydrogen bonding), the water molecules organise into a hexagonal crystal structure. The arrangement the water molecules form is not as tightly packed as the more casual arrangements found in liquid water. This results in lots of empty space between the molecules, and therefore a lower density.

However...

Ice floats, and so not all of the ice is submerged. We need to think about volume, displacement and buoyancy.

Theoretically, if in a glass of water, 100% of the ice floats, then if the ice melts, the melted ice will contribute to the volume of the water. To calculate the volume added to the glass of water, we would use the mass of ice divided by the density of ice.

Okay so for this specific example, 5 ice cubes are fully submerged and 2 are partially submerged.
The 5 ice cubes and 2(1/2) cubes will decrease the volume and this the water level. The 2(1/2) cubes that are floating will increase the water level.

Overall, due to the reduced displacement of water compared to ice, and the fact that the majority of ice cubes are fully submerged, the water level would in fact decrease.

Let me know if you agree/disagree and why! 😊

Who wants a pi cookie?!
Came across this pi-shaped cookie cutter while out shopping — couldn’t resist taking a photo!
A nice little reminder that maths really is everywhere… even in the baking aisle. 🥧

For those who might not of seen pi in a while here are some fun facts

1) Pi is a Greek letter. We use many greek letters in maths to represent constants, variables and even functions!

2) Pi represents the circumference of a circle divided by it's radius. What this means is that the circumference of any circle is pi X it's radius. If you don't believe me then try it for yourself! 😁

3) Pi features in many different equations and identities. Ranging from simple geometric ratios all the way to formula including imaginary numbers!

4) The value of Pi is approximately 3.14159.... but the numbers carry on infinitely. We can use computers to discover new digits of pi. We currently know the first 105 trillion digits. Maybe you could try finding the digit after? 😉 Even tho we know so many digits, NASA only uses the first 16!

5) Pi is defines as a transcendental number. This means that it fits these criteria:
a) The number cannot be written algebraically (using square roots or fractions). (Unlike √2 for example)
b) It lacks a repeating pattern in decimal form. (Unlike 1/3 for example which is 0.3333333...)

If anyone has any more fun facts about pi, please put them in the comments! 😁

Will

Congratulations to all of my students who received their results today!
100% of them passed!
All of your hard work and commitment has paid off and now you are ready to start the next chapter.

I am incredibly proud and wish you all the best in your future endeavours.

Will
WMJ Tutoring

Have a go at this question!