Maths Coursework - Cables: For this assignment I have been requested to study a particular design of steel cables used by engineering companies in construction work.

GCSE Maths Coursework – Cables

Introduction

For this assignment I have been requested to study a particular design of steel cables used by engineering companies in construction work. The design consist of four circular cross-sectioned strands (radius 1cm) bonded together with plastic and wrapped around with a fine steel thread to assemble a ‘square’ (i.e. 2 strands by 2 strands) like so:

Within my assignment I have investigated thread lengths and amounts of plastic fill for different sized and ‘squares’, which I throughout exposed various useful patterns that could be used in the industry.

Thread Lengths

To commence this branch of my assignment I have worked out the thread length for the modular cable with four strands shown in diagrams 1 and 2:

X = arc touching the thread

Y= the space between two strands

From looking at the above diagram we can perceive that the length of the thread going all around the cable =

This is because if we look carefully we can see that we have four arcs touching the thread (4x) and four sides each with one space between two strands (4y) and together these make the length of thread going around the cable (4x +4y = thread length).

Knowing that the radius is 1 cm we can work out the value of (y):

Below is a close up diagram of the length (y). The two parts of strands shown are both exactly half a strand with an arrow to show the radius (1cm) from the middle of the strand to the edge of the strand. We can see that the length (y) is equivalent to twice this radius i.e.

We can substitute this value into our previous equation for the value of the thread length:

Now we need to know the value of (x) to conclude. We can do this using this formula:

Arc Length = (angle/360) x (total circumference, π x diameter)

Because we know that (x) is an arc.

We will call the diameter (d)
So the circumference is πd
The angle can be found like this:

First we join the centres of the circles …

…Then we join the ends of each arc (x) to the radius of its circle to show us the angle we want to find …

We can see how four rectangles have automatically appeared. The interior angles of the square in the middle are 90º. So now using the theory ‘angles round a point add up to 360º’ we can work out the value of our angle, since it is one out of four angles round a point. We know that if we add together the two 90º angles, the 90º angle and our angle we are going to get a total of 360º, i.e.

After we have worked out the value of (x) we can now substitute into our equation:

A cable with nine strands follows generally the same rules:

We can spot from the diagram beneath that the thread length =

We already know the values of (x) and (y) because they have not changed, so we only have to substitute the values into the equation:

To conclude this part of my assignment I have worked out the thread lengths for several other ‘square’ sizes. (By the word ‘square’ the company means a cable containing a number of strands which is a square number e.g. the modular 2 strands x 2 strands and the enlarged 3 strands x 3 strands cables we did previously). The results for my workings are shown in the table below:

From the table we can make out that as we go further along the square number sequence there is a difference of 8 between the thread lengths at each step. We can find the nth term:

This nth term is exceedingly useful; throughout it the company can work out exactly how much thread they need for any ‘square’ size by simply replacing (n) with the number of the number they want in the square sequence.

From the above ...