Technology 6: Mangonels

Today we had a quick brainstorm about siege weaponry: what it was for and what types of seige weapons we could think of. Then we looked at some of the physics of catapults, and mangonels in particular. Here is some of the information we shared:

From http://www.real-world-physics-problems.com/catapult-physics.html

Catapult physics is basically the use of stored energy to hurl a projectile (the payload), without the use of an explosive. The three primary energy storage mechanisms are tension, torsion, and gravity. The catapult has proven to be a very effective weapon during ancient times, capable of inflicting great damage. The main types of catapults used were the trebuchet, mangonel, onager, and ballista.

The Mangonel:

The mangonel consists of an arm with a bowl-shaped bucket attached to the end. In this bucket a payload is placed. Upon release, the arm rotates at a high speed and throws the payload out of the bucket, towards the target. The launch velocity of the payload is equal to the velocity of the arm at the bucket end. The launch angle of the payload is controlled by stopping the arm using a crossbar. This crossbar is positioned so as to stop the arm at the desired angle which results in the payload being launched out of the bucket at the desired launch angle. This crossbar can be padded to cushion the impact.

The mangonel was best suited for launching projectiles at lower angles to the horizontal, which was useful for destroying walls, as opposed to the trebuchet which was well suited for launching projectiles over walls.

However, the mangonel is not as energy efficient as the trebuchet for the main reason that the arm reaches a high speed during the launch. This means that a large percentage of the stored energy goes into accelerating the arm, which is energy wasted. This is unavoidable however, since the payload can only be launched at high speed if the arm is rotating at high speed. So the only way to waste as little energy as possible is to make the arm and bucket as light as possible, while still being strong enough to resist the forces experienced during launch.

The physics behind a mangonel is basically the use of an energy storage mechanism to rotate the arm. Unlike a trebuchet, this mechanism is more direct. It consists of either a tension device or a torsion device which is directly connected to the arm.

The figure below illustrates a mangonel in which the energy source is a bent cantilever, which is a form of tension device. This can consist of a flexible bow-shaped material, made of wood for example.

The point P in the figure is the pivot axle, attached to the frame, about which the arm rotates.

 

In groups of three or four, we researched how we could construct a working model of a mangonel using ice block sticks.

After construction, we took the time to test our creations, recording their capabilities in terms of distance of the projectile and height.

 



Technology 2

Today's challenge was to build a structure which could launch (and steer) a marble towards a target 3 metres away.
The teams only had sheets of newsprint paper to use to create the ramp, plus sellotape. It had to be portable and free standing (they couldn't support it themselves).
All the teams managed the task really well, so accuracy was the testing point. They got 5 points for shooting past the drink bottle (3 metres away), and 10 points for actually hitting it.
There was a tie for first place (with 25 points from three goes):
The team of Danielle, Alena and Stacey, plus the team of Isaac, Aleks and Millie.

Technology 1

Today's challenge was to create a launcher for a table tennis ball with a selection of materials... 2 rubber bands, 6 sheets of paper, 4 straws, 4 ice block stocks, and sellotape. Not all the materials had to be used.
Not all the groups managed to finish, the task, but it was a great activity for practising our team work and for focusing on time management.

Pixar in a Box 2

Today we continued our study of Pixar, and found out more about how this amazing company is able to create such fun and cool new virtual worlds for us every time they release a new movie.
Lesson 2: Crowds
When the designers were planning Wall-e, they wanted him to go from a world where there was only one robot - himself on Earth - to a place where there were hundreds of different robots, all with their own jobs and purposes.
The designers were tasked with creating all these different robots, but instead of inventing a thousand different robots form scratch, they simply used the branch of mathematics known as "combinatorics" to create a few different heads, a few bodies, a few arms, etc, and mixed and matched them.
We also had a design task to carry out or ourselves. The challenge was to create 1000 different dinosaurs by using a limited number of body parts (head, tail, arms, legs and body) - preferably fewer than 25. In the end we worked out that 21 or 24 different body parts is the answer, and we designed what they might look like.
Here's the maths:
2 x 5 x 10 x 5 x 2 = 1000 (24 body parts)
or
4 x 5 x 5 x 2 x 5 = 1000 (21 body parts)

Pixar in a Box 1

The Khan Academy have joined forces with Pixar (the awesome company that brought us films like Toy Story, Brave and Inside Out) to create a series of lessons which combine maths and animation. Through these lessons we are able to learn about some of the techniques the animators use in order to make their films. Plus we're also able to catch a glimpse of what goes on in an enormous company such as Pixar, where so many employees with distinct skills are working together towards a common goal.
Lesson 1: Environment Modelling
This series looked at Brave. One of the challenges when the animators made Brave was creating so much grass which moved in natural ways. We learned how parabolic arcs helped the designers to make single blades of grass, and how they were animated in order to move naturally.

Maths Investigation 3

The Locker Problem
Imagine this: there are 1000 students at a school, each with a locker, which is shut. At the start of the day, the first student comes into school and opens all of the lockers.
Then the second student comes in and touches locker numbers 2, 4, 6, 8 etc... if it's shut, he opens it, if it's open he shuts it.
The third student comes in and touches numbers 3, 6, 9, 12... etc. (If it's shut she opens it, if it's open she shuts it.)
It carries on all day. The students come in, and starting with their own locker, they touch the ones which are multiples of their number.

The challenge is figuring out what state the lockers are in by the end of the day.

PHEW!

There are a couple of things to think about when tackling this problem...
1. Downsize the issue. 1000 is a big number. Try it out with 20 lockers and see if a pattern starts emerging which you can extrapolate.
2. Think about what happens to a specific locker. How many times will locker number 1 be touched? How many times will locker number 21 be touched?

It wasn't long before a few of the students in the group realised that this problem has a lot to do with square numbers - such as 1, 4, 9, 16, and so on. When you write down the factors of each locker number you will realise why.

Maths Investigation 2

Today we looked at a bit of Geometry.
Firstly Polygons: How many diagonal lines are there inside polyhedral shapes? And is there a general rule? Can the number be predicted?
We found that for a four-sided shape the total is 2, five-sided it's 5, six-sided it's 9, seven-sided it's 14, and eight-sided it's 20. So obviously the number of diagonals increases. To find a rule, we looked carefully at each corner, and we found it useful to notice, for every shape, how many diagonals came from each corner.
In the end we did come up with a rule, and used it to predict that the number of diagonals in a 100-sided shape is 4850 (phew!).
Here's our rule, where "s" means the number of sides on the shape:
(s-3) x s / 2. We checked it and it works for all the smaller shapes, so we think it will work for the larger ones too.

Then we had some time to work with Polyhedra. We looked at three nets and noticed that although they are all called pyramids, they were quite different. We folded and made the shapes, then designed our own nets which were made from at least three different 2D shapes.