How to find the Centre of Mass of a framework

Centres of masses in Mechanics is an important topic which when using this simple method will be very easily understood.

A framework is an object where it can be described as 'hollow' where there is no inside to the object and that there is only 'wire' on the outskirts of the object.

Here is a simple method to finding the centre of masses:

1) Identify the object and split it into individual wires.

2) Identify your origin point for the shape and work out what way your axis will go (the coordinates of the origin must be decided depending on the type of framework).

3) Work out the centres of masses of the individual frames.

4) Set up a table including the mass of the frame, the centre of mass coordinates and totals for these.

5) Use the formula to calculate the centre of mass.

This will all be shown in examples below.

The formula mentioned above is rsin(a)/a. This is used to find the coordinate of the Centre of Mass Point of a semicircle. This can also be done with (2/3)(rsin(a)/a) which will find the Centre of Mass Point of a semicircular lamina.

Types of Infinity

Infinity, cannot be treated as a number, but rather a concept in the sense that it is without any limit. Georg Cantor a famous Mathematician used sets to show how some infinities are larger than others. He did this by showing how a bijection cannot be shown therefore the cardinality (size of the set) must be different between them.

Given two sets A and B, a bijection is said to be true if:

1)There is an Injection between set A and set B (One to one mapping).
2)There is a Surjection between set A and set B (All elements of set A are mapped to an element of set B).

Therefore by proving a bijection between two sets we can conclude that they have the same cardinality.

For example:

Set A (1,2,3,....) and set B (2,4,6,....) can be proved to have the same cardinality. This is because using the function (f(x)=2x), we can create a bijection between set A and set B.

There is a one to one mapping because 2x only has one value.
All elements of set A are mapped to an element of set B because the elements of B are double that of A!

Therefore we have shown that the size of the set 1,2,3,4,5,..... is the same size as the set of 2,4,6,8,10,.....

These are known as countable infinities, because hypothetically speaking, the only thing that stops you from not being able to count all of the numbers 1,2,3,4,... is the time restraint. It is countable because if you counted forever, you could not have missed any numbers between 1 and n, which makes it listable/countable.

However you cannot list the uncountable infinity, because there is no order in which numbers can be arranged, that once you had counted forever, you would have missed no numbers. And for that reason this type of infinity is of different size to the cardinality of the set of natural numbers.

Finding the inverse of a function

A function is  a special mapping where every element from set A (the domain) is mapped exactly to one element in set B (the range).

Domain refers to the x values that the function occupies, and range referring to the y values. For example: A function of 1/x does not have any values of x=0 or y=0 because these are the asymptotes of the equation.

Therefore... for every value of A to map to set B, the domain and range must specify this i.e:
 Domain:   x ∈ R, x ≠0
 Range:       y ∈ R, y ≠0

 Now every element from set A maps to elements in set B and the equation f(x) = 1/x {x ∈ R, x ≠0} and is now a function.

It is regarded as a one-to-one function because every element of the range comes exactly from one element in the domain.

Finding the inverse of a function:

The example function will be: 2x²-7

1.Set the function equal to y:

y=2x²-7

2.Rearrange to make x the subject:

y=2x²-7

y+7=2x²

0.5(y+7)=x²

x=√(0.5(y+7))

3.Replace (x=) with (f^-1(x)=) and replace (y) with (x)

x=√(0.5(y+7))

f^-1(x)=√(0.5(y+7))

f^-1(x)=√(0.5(x+7))

The inverse of a function graphically, is literally just f(x) reflected in the line y=x.

For example:

"https://en.wikipedia.org/wiki/Inverse_function#/media/File:Inverse_Function_Graph.png"

As shown above, the function just gets reflected in the line y=x to give the inverse function.

Exams Approaching - Revision Tips

1. Don't overdo it;
The best way is to plan your time efficiently and go for a mix between quality and quantity.

2. Everyone revises different
You need to find the way that revision best suits you; whether it be mind maps, or spider diagrams. Don't worry if you are different to the way that your mates study, different things work for different people.

3. Take breaks
It's proven that studying in short 45 minute periods is much more beneficial than hours of solid revision, so make sure you plan your rest time too!

4.Make a timetable
Doing so will allow you to maximise the time in your day and will actually give you more free time. If you continue to muck about with choosing what to study, you are wasting time that collectively adds up to a lot of unused, wasted time.

5. Food, Water, Exercise
Three components that are absolutely crucial to exam success. Without these the body cannot function as well as it could have done, and puts you to a disadvantage. Keep hydrated, full of energy and ready to take on the world!

6. Sleep
Having enough sleep will help you concentrate better and keep your focus while studying.

7. Balanced Life
The more you study the better you'll be. However remember to go out and treat yourself for the hard work you've been doing.

8. Start studying early
The earlier you start, the more time you have to learn all your content meaning, the earlier you start, the better you'll do!

Good Luck!

Checking Validity of Functions and Inverse Functions

Sometimes it can be hard to tell whether or not a graph or equation is a valid function or not. Here are some simple steps to checking this:

Horizontal line test: If a straight line can be drawn horizontally that crosses the graph in two places, then this function is called a many to one function. This is only true if a values of x are plotted. However, when finding the inverse of this function, the graph will be reflected in y=x and will no longer abide by the Horizontal line test, but the Vertical line test.

Vertical Line test: If a straight, vertical line can be drawn that crosses the graph in two places, then this is 'one to many' which is not a function. 

1) Vertical line test true = not a function.

2) Horizontal line test true = many to one function

    Horizontal line test true = inverse will not be a function.

Long Division VS Remainder Theorem

Both methods may be used to divide polynomials by a factor such as (x-3), however take different steps in reaching the end goal.


Long Division

As shown, the factor (x-3) is multiplied by a term to create a new expression that can be subtracted from the original polynomial. This is repeated until the polynomial reaches 0 in which case there is no remainder, or it cannot be divided further, in which case there is a remainder.
The terms are then collected and the remainder is stated as above: 29/(x-3) where the remainder is divided by the divisor.

The Remainder Theorem

This method in my opinion takes longer. The idea is that a polynomial (Ax^3 + Bx^2 + Cx + D) is created and using substitution, these values of ABCD can be calculated. First assume there is a remainder i.e. +D.
We then let x =3. This allows us to work out the value of D.
Then we let x =0. This allows us to work out the value of C.
We then compare coefficients i.e. the coefficient (LHS) of x^3 is 1. Therefore the coefficients of all the x^3's on the RHS must also equal to 1. Therefore A=1.
With x^2 LHS equals 1. Therefore all of the x^2 terms i.e. -3Ax^2 + Bx^2 must also equal 1. This can be used to work out the remaining values of ABCD and the remainder stated, divided by the divisor i.e. 29/(x-3).

Mathematical Proof By Induction

Proof by induction is just one type of mathematical proof that follows a main method:

1) BASIS:
Prove the general statement is true for n = 1

2) ASSUMPTION:
Assume the general statement is true for n = k

3) INDUCTIVE:
Show that the general statement is then true for n = k + 1

4) CONCLUSION:
The general statement is then true for all positive integers, n.


And an example question:

(BASE STEP)
Prove by the method of mathematical induction that for n is a set of positive natural numbers:

Sum from r=1 to n; (2r-1) = n^2

n=1; LHS = 2(1)-1 = 1
        RHS = 1^2 = 1
Therefore true for n = 1.

(ASSUMPTION)
Assume that the summation formula is true for n = k;

Sum from r=1 to k (2r-1) = k^2

With n = k + 1, terms the summation formula becomes:

(INDUCTIVE STEP)
Sum from r = 1 to (k+1) of (2r-1) = 1+3+...+(2k-1)+(2k+1)
                                                   
= k^2                 + (2k+1)
=k^2 + 2k + 1
=(k+1)^2

Therefore summation formula is true when n = k + 1

(CONCLUSION)
If the summation formula is true for n = k then it is shown to be true for n = k + 1. As the result is true for n = 1, it is now also true for all n  ≥ 1 and n is a set of positive natural numbers by mathematical induction.

TIPS
When trying to prove the inductive step, a very good idea is to write the formulae out that you have derived^^ i.e. (k^2 + 2k + 1) and then write out the other part of the formula (k^2) but replacing k with the summation i.e. k+1; therefore you get (k+1)^2.
Then it is much easier to prove the formulae as you have (k^2 + 2k + 1) and now it is just a matter of rearranging formulae to get (k+1)^2; which in this case is very easy, but helps a lot when the questions become harder.