Tutorial 3 - Relationships between Quantities and Transposition of Formulae

The transposition (or rearrangement) of formulae is a skill that is essential for successful study of Physics.  A wrong transposition of a formula will lead to a physics error in the exam and you will lose all the marks available in that part of the question.  (However, if you use your incorrect answer correctly in subsequent parts, your error will be carried forward and you will gain the credit.)

 

Some students find rearrangement very difficult and it hampers their progress and enjoyment of the subject.  They try to get round it by learning all the variants of a formula, which is a waste of brain power.

 

Before we go on to look at rearranging formulae, let us look at how quantities are related in physics.  Mostly they are equal, but sometimes they are less than, or greater than.  One quantity may vary as another.  You need to understand the relationships, otherwise physics will not make sense.

 

Relationships between quantities

Physics studies how different physical quantities are related to each other.  The most obvious relationship is when one variable is equal to the product of a number of different variables.  We call this an equation.  For example:

 

 

Direct Proportionality

Equations are the result of proportionalities, where one quantity varies in a predictable way with another.  In a resistor we know that if the voltage doubles the current doubles.  So we can say that voltage is directly proportional to current.

 

We can write:

The symbol µ means “varies as”.

 

To make an equation out of this we have to use a constant of proportionality, which is usually written as k.

 

 

Actually we know that in this case, the constant of proportionality is resistance, R, so we write:

 

 

Or more conventionally:

 

 

If two quantities are directly proportional, then the graph of one plotted against the other is a straight line of positive gradient which passes through the origin.  We will look at this later.

 

Not every straight line results in direct proportionality.

 

A straight line graph that does not pass through the origin tells us that the two quantities are not directly proportional.

 

 

Question 1

In the equation   the power is proportional to the square of the current. 

 

What is the constant of proportionality?

Answer

 
Inverse Proportionality

Not every relationship shows direct proportionality.  If the relationship is of inverse proportionality, we will see:

 

 

The graph looks like this:

If we plot a against 1/b, we get a straight line:

 

 

The gradient gives us the constant of proportionality, k.

 

So we can bring in our constant of proportionality:

 

 

One example of this is the relationship between pressure and volume in a gas.  Double the pressure, the volume will go to half.

 

Question 2

The equation

 

suggests an inverse proportionality. Write down the inverse proportionality using the varies as symbol, . What is the constant of proportionality?

Answer

 
Inequalities

It is possible that some quantities are not equal to the product or sum of the other variables. 

 

We denote that with these symbols, with which you are familiar from GCSE Maths (aren’t you?).

 

 
 
Approximations

Sometimes it is not possible to get an exact answer.  For example the circular number p is what mathematicians call an irrational number.  This means that however many significant figures you write it to, you will never get to the exact value.  We can get an approximation (close to) to the value of p by writing:

 

p ≈ 3;

p ≈ 3.14;

p ≈ 3.1415;

p22

       7

 

Question 3

Use your calculator to show how close an approximation the value 22/7 is to the calculator value of .

Answer

 

Sometimes correct calculations in Physics will give answers that are only approximations to the true answer.  This is because the equations used may not take into account variables that will affect the answer in real life.  For example, if you ignore air resistance when calculating the trajectory of a falling object, you will get an approximation to its path, but in reality, the object will not follow that exact path.

 

This leads to the important concept of uncertainty, in which measured values may not be the true values.  We will learn how to handle uncertainties later.

 
Change in

Quantities can change.  For example, acceleration is defined as the change in velocity per unit time.  The change in a quantity is denoted by the symbol D, (Delta, a Greek capital letter ‘D’).

 

D value = value at end – value at the start

 

 

Question 4

Use the delta notation to write an equation for acceleration.

Answer

 

 

Transposing Formulae

It is far better to get into the habit of rearranging formulae from the start.  The best thing to do is to practise.

 

Key Points:

 

 

Transposing Simple Formulae

Simple formulae are those that consist of three quantities, taking the form A = BC.  A typical example is V = IR

 

A simple trick is to use the formula triangle.  Some physics teachers sneer at this method.  I don’t, as long as you are aware that it only works for three term equations.

 

 

You put your finger over the term you want to be subject of the formula (what you want to find) and then the rest follows:

 

B = A/C

 

However it is better that you follow a more orthodox method.  Suppose we are using the equation V = IR and wanted to know I.

 

We want to get rid of the R on the RHS so that I is left on its own.

 

So we divide both sides by R which gives us:

 

The R terms on the RHS cancel out because R/R = 1.  So we are left with:

 

 

Question 5

Rearrange these equations:

Equation

Subject

 V = IR

 R

p = mv

 v

 r = m/V

 m

 Q = CV

 C

Answer

 

Formulae with Four Terms

Triangle methods will not work with these.  Nor is there a magic square.  Instead use the same method as above.  Consider this formula:

 

  

 

Make r the subject.

 

Get rid of the l by dividing the whole equation by l:

The l terms cancel to give:

 

 

To get rid of the A downstairs we need to multiply both sides by A:

 

The A terms cancel to give us our final result:

 

 

 

Question 6

Rearrange these equations:

Equation

Subject

 

 

V

 

Dh

 

(Dh is a single term)

 

G

 

D

 

Answer

 

Equations with + or -

If there are terms which are added or subtracted, we need to progress like this:

 

 We want to find h.

 

To get rid of the F term we need to add it to both sides of the equation:

 Now we can get rid of the f on the RHS by dividing the whole equation by f:

 

Which gives us our final result of:

 

Question 7

Rearrange these equations:

Equation

Subject

v = u + at

 t

 E = V + Ir

 r

Answer

 

Dealing with Squares and Square Roots

If we have a square root, we get rid of it by squaring.  If there is a square, we get rid of it by taking the square root.

 

Consider this formula:

 

Suppose we want to find g.

 

Get rid of the square root by squaring the whole equation:

Now bring g upstairs by multiplying the equation on both sides and cancelling:

 

Then get rid of the T2 by dividing the whole equation by T2 and cancelling.

 

Question 8

Rearrange these equations:

Equation

Subject

 v

 

k

 

C

 

 

Answer

Question 9

(Challenge)  Make t the subject of this formula:

Answer