Parallelogram 21 Year 9 5 Mar 2020Pendulum Patterns

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux 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.

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: T=2πLg

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 = 2×3.14×29.8=6.28×0.204=6.28×0.4518

T = 2.837 seconds.

So a 2 metre pendulum takes 2.837, or nearly 3 seconds, to swing back and forth.

2 marks

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
3 marks

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
Show Hint (–2 mark)
–2 mark

We want to know L for a T of 1 second, so:

L=gT÷2π2

L=9.8×1÷6.282

L=0.248 metres

2. Intermediate Maths Challenge Problem (UKMT)

3 marks

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

There are 7×7=49 days in 7 weeks. So we will get a good estimate of the average distance travelled by a wheatear in 7 weeks by dividing the total distance travelled by 50.

Now 1500050=15005=300. So the average distance travelled is roughly 300 km.

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.

2 marks

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.

4. Intermediate Maths Challenge Problem (UKMT)

5 marks

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 2π. What is the length of the perimeter of the figure?

  • 3
  • 4
  • 3π2
  • 6
  • 2π
Show Hint (–1 mark)
–1 mark

The radius of each circle is half the length of the square’s sides. If the square has sides of 2π, then the radius of each circle is 1π.

Show Hint (–1 mark)
–1 mark

The perimeter of the shape is made up 4×34 circles. So the perimeter of the shape is 4×34× the circumference of one full circle.

The radius of each disc is 1π, and hence its circumference is 2π×1π=2.

The perimeter of the figure is made up of three-quarters of each of the four discs. Hence the length of the perimeter is 4×34×2=6.

5. Intermediate Maths Challenge Problem (UKMT)

3 marks

5.1. What is the area of the figure in question 4.1?

  • 1.16
  • 1.36
  • 1.56
  • 1.76
  • 1.96
Show Hint (–1 mark)
–1 mark

The radius of each circle is half the length of the square’s sides. If the square has sides of 2π, then the radius of each circle is 1π.

The radius of each circle is half the length of the square’s sides. If the square has sides of 2π, then the radius of each circle is 1π.

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 4×34×π×1π2 + 2π2 = 1.36.

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.