Measurement, errors and uncertainties

Calculation

Calculation in Physics

Performing addition and subtraction

Performing multiplication and division

Performing Calculation with lg and ln

Uncertainties and errors

Uncertainty and How to process it

How to determine uncertainty in a derived quantity when addition or subtraction is performed?

How to determine uncertainty in derived quantity when multiplication or division is performed.

How to determine uncertainty in a derived quantity when powers, root,etc are involved.

How to subtract one vector from another?

As we have seen in a previous post a vector is on object that has a magnitude and a direction. We have also seen how to add two vectors together. Today we are going to see how to subtract one vector from another  

Let us say that a vector A is as shown and has a magnitude of 4.

Then vector  -A will be a vector of the same magnitude and but in the opposite direction like in the diagram below

Now let us have a look at the two vectors A and B of magnitude 4 and 2 respectively.

If you want to perform the following subtraction ( A- B ) the first step is to change vector B to vector –B as shown in the diagram below.

Now that you have two vectors A and – B you can add them together as you did in the previous post on addition of vectors. You place the tail of vector –B on the tip of vector A  as shown below. 

The following mathematic is used. A – B = A + (-B)  As you can see you add the vector A to the vector –B  to obtain A - B.

The vector A – B is a vector that starts from the tail of vector A to the tip of vector –B  as shown in the diagram below.

If the magnitude of the two vectors are 4 and 2 respectively then the magnitude of the vector (A – B) = 4-2 = 2

If the two vectors were not to be horizontal or vertical then the same method should be used. Let say that we have two vectors as shown below and that you want to determine the vector (A –B)

Below are two vectors that are not horizontal or vertical. That is they are not parallel to each other.

How would you determine the vector A – B ?

First of all you will have to determine the vector –B. As you can see in the diagram below the vector –B has the same magnitude as the vector B but they have opposite direction.

Now if you want to determine (A – B) you will have to remember that

(A – B) = A +(-B)

Which means that if you want to perform the subtraction A – B you will have to add the vector A and –B.

Again you will have to add the two vectors by placing the the tail of vector –B on the tip of vector A as shown in the diagram below.

  As you have seen in the previous examples the vector (A – B) is the vector from the tail of A to the tip of –B. As shown in the diagram below.

Of course using the angles that –B makes to the horizontal,b, and the angle that A  makes to the horizontal ,a, (A –B) can be determined using cosine rule. Also the angle that the vector (A –B) makes with the horizontal can be determined.

Uncertainty and how to process it!

In physics all quantity that is measured come with its uncertainty. This is because the instrument, the experiment or the person performing the measurement is subjected to some limitations.

For example

1. Someone that is measuring a length with a metre rule cannot measure the length to a precision that is better than one mm. So a length that is measured with a metre rule will have an uncertainty of +-o.1 c m.
2. The limitation can either be due to the experiment itself, the apparatus,  or the experimenter.  Someone measuring the height of rebound will measure it with an uncertainty of +- 0.5 cm. Someone estimating the mass of an object can measure it with an uncertainty of +-0.5 kg. A voltmeter will give a reading with a percentage uncertainty of 1%.

[As you may have guess from 2 above not all measurements that are performed will have these uncertainty. These are for example only.]

As you can see all quantities that are measured will have uncertainties. And as a result any other quantities that are derived from the measured quantities will have their uncertainties.

We are going to see how to determine the uncertainty when the following operations are performed.

1. Addition and subtraction
2. multiplication and division
3. power, square root, square, etc
4. Other operations using the extreme-value method (under construction)

To know how to determine the uncertainty when addition and subtraction is performed see the next part of this series.

How to perform addition and subtraction in physics

In physics we often encounter calculations where you have to add two numbers together or we had to subtract one number from another. This is particularly important when data from practical sessions are processed even though additions and subtractions are often met in the syllabus.

However, as I have said in the introduction, the answer must be given to a certain number of decimal place. If you cannot say what the number of decimal place a number has, please refer to this post on decimal place.



Addition

Now let us see an example on how to add two numbers.

4.43

7.6    +

_____

12.03

____

12.0     Final answer.

So how do you give the answer to the appropriate number of decimal places? Add the two numbers and you get the answer 12.03. But this is not the final answer to the addition.

Before you write the final answer, look at the two numbers and determine the number of decimal places for each.

4.43    2 decimal places

7.6      1 decimal place

Now what is the smallest number of decimal places in the two numbers?

7.6 has 1 decimal place so the answer must have 1 decimal place. So the final answer is 12.0 with 1 decimal place.

Let us have a look at a second example.

67.54             2 decimal places

45.345            3 decimal places

______

112.885

112.89            Final answer2  decimal places

This is not the final answer. The two numbers used has 2 and 3 decimal places. So the final answer must be given to 2 decimal places. So the final answer is 112.89

Easy isn’t it.



Subtraction

Now we are going to see how subtraction is done. Here an example.

34.9

32.09  -

____

2.81

2.8      Final answer

So have you guessed how the answer is obtained?

The first number has 1 decimal place and the second number has 2 decimal places. As a result the answer must have 1 decimal place.

Let us have a look at a second example.

123.7             1 decimal place

3.433         3 decimal places

______

120.267

120.3             Final answer

1 decimal place

Have you got it? Let me summarise this with a rule.

When an addition or a subtraction is performed the answer is given to the same number of decimal places as the number used that has the smallest number of decimal places.

It is very important that you remember this law.

It is now time to do some exercises.

1. 12.45 + 34.555


2. 2.3 + 23.4445


3. 3.444 - 545


4. 34.3 – 34.111


5. 44.009 – 23.2

Do the following exercises and leave the answer in the comment section. I will give the correct answer later on when a few of you have submitted your answers.