Parallelogram 25 • Level 4 • 22 Feb 2024Pendulum Patterns
Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/
- a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.
- Tackle each Parallelogram in one go. Don’t get distracted.
- Finish by midnight on Sunday if your whole class is doing parallelograms.
- Your score & answer sheet will appear immediately after you hit SUBMIT.
- Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.
1. Harmonic Pendula
Take a look at this video of a collection of pendula (the plural of pendulum), which behave in a beautiful and surprising way. It’s called a wave pendulum or harmonic pendulum. It lasts three minutes, and it is worth watching right through to the end to see the amazing effect.
(If you have problems watching the video, right click to open it in a new window)Tom Rowett, who was in the video, has a huge and wonderful collection of odd objects,
and we will meet more of them in later Parallelograms. In this case, each pendulum
swings at a different rate, because it has a different length. In fact, there is
a formula that determines the relationship between the length of the pendulum and
the time it takes to swing from left to right and back to the left:
where
- T is the time of a complete swing in seconds.
- π is 3.14... (Wow! Always nice to see π.)
- L is the length of the pendulum in meters.
- g is the acceleration due to gravity (9.8 m/s²). Don’t worry about this – it is a bit of physics that describes how an object falls due to the Earth’s gravity.
If you had a pendulum that was as tall as a human, say 2 metres, how long would it take to swing? If L = 2 meters, then:
T =
T =
So a 2 metre pendulum takes 2.837, or nearly 3 seconds, to swing back and forth.
1.1 If one of the pendula in the video is 0.15 meters long, how long is the period of the swing?
- 0.10 seconds
- 0.15 seconds
- 0.39 seconds
- 0.78 seconds
- 1.00 second
- (Not answered)
1.2 If you wanted a pendulum to have a swing time of 1 second (T = 1 second), how long should the pendulum be? At the moment the equation gives you the time (T) depending on the length (L), so it might be useful to rearrange the equation so that you have the length depend on the time.
- 0.18 metres
- 0.248 metres
- 0.99 metres
- 1.16 metres
- 1.8 metres
- (Not answered)
We want to know L for a T of 1 second, so:
2. Intermediate Maths Challenge Problem (UKMT)
2.1 The northern wheatear is a small bird weighing less than an ounce. Some northern wheatears migrate from sub-Saharan Africa to their Arctic breeding grounds, travelling almost 15 000 km. The journey takes just over 7 weeks. Roughly how far do they travel each day, on average?
- 1 km
- 9 km
- 30 km
- 90 km
- 300 km
- (Not answered)
There are
Now
3. Can you light a match with water?
This is a surprising video from Steve Mould, who shows that you can light a match using water, or at least the gaseous phase of water.
(If you have problems watching the video, right click to open it in a new window)3.1 Why is it possible to light a match with water?
- Because the steam is super-heated.
- Because the steam is super-dry.
- Because the steam is super-fueled.
- Because the steam is super-oxygentated.
- Because the steam has super powers.
- (Not answered)
4. Intermediate Maths Challenge Problem (UKMT)
4.1. The diagram shows four equal discs and a square. Each disc touches its two neighbouring discs. Each corner of the square is positioned at the centre of a disc. The side length of the square is
3 4 3 π 2 6 2 π - (Not answered)
The radius of each circle is half the length of the square’s sides. If the square has sides of
The perimeter of the shape is made up
The radius of each disc is
The perimeter of the figure is made up of three-quarters of each of the four discs. Hence the length of the perimeter is
5. Intermediate Maths Challenge Problem (UKMT)
5.1. What is the area of the figure in question 4.1?
- 1.16
- 1.36
- 1.56
- 1.76
- 1.96
- (Not answered)
The radius of each circle is half the length of the square’s sides. If the square has sides of
The radius of each circle is half the length of the square’s sides. If the square has sides of
The area of the shape is made up 4 x ¾ circles plus the area of the square. So the area of the shape is
Before you hit the SUBMIT button, here are some quick reminders:
- You will receive your score immediately, and collect your reward points.
- You might earn a new badge... if not, then maybe next week.
- Make sure you go through the solution sheet – it is massively important.
- A score of less than 50% is ok – it means you can learn lots from your mistakes.
- The next Parallelogram is next week, at 3pm on Thursday.
- Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.
Cheerio, Simon.