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.

Scalar and vector quantities

A quantity is a characteristics of a body that can be measured with an instrument. Examples are length, area, temperature and so on.

All these quantities can be divided into two types: Scalar and Vector quantities. Normally it is quite easy to deduce whether a quantity is a scalar or a vector quantity but of you have any difficulties a list will be given later on.

Let us see how to identify a vector and a scalar quantities.

 

Scalar quantities

A scalar quantity is one that has a magnitude and a unit.

Remember the magnitude of a quantity is a number that represent its size.

Examples of scalar quantities are length, mass, temperature, etc

length = 19 m

mass = 4 kg

As you can see these quantities can only be represented by a magnitude and a unit.

 

Vector quantities

A vector quantity is one that has a magnitude, a unit and a unit.

Examples of vector quantity are acceleration, force, velocity, etc.

velocity = 100 km/h towards the north

Displacement = 100 m eastward

Force = 10 N 30⁰ to the horizontal

As you can see all quantities that can be represented using an arrow and in which a direction make sense is a vector quantity.

For example a mass will not make sense with a direction hence it is a scalar quantity. A force make sense with a direction hence it is a vector quantity.

How to know whether a quantity is a vector or a scalar quantity

• In case you do not know that a quantity is a scalar or a vector then you can consult this list[under construction]

 

• Two quantities can be added or subtracted from each other unless they are both scalar or vector quantities. And the answer is a scalar

Hence if  A = B –C

and C is a scalar quantity then B must be a scalar quantity and as a result A must be a scalar quantity. The same reasoning would have applied if addition was performed.

• If two quantities A and B are multiplied to get quantity C as in C = A*B then the nature of the quantity C would vary according to the table below.

 

A	B	C	Example
Scalar	Scalar	Scalar	Mass = density *volume
Scalar	Vector	Vector	Velocity = time *acceleration
vector	Vector

 

• If a quantity A is divided by another quantity B to obtain quantity C as in C = A/B then then the nature of quantity C would vary according to the table below.

A	B	C	Example
Scalar	Scalar	Scalar	Density = mass/volume
Vector	scalar	Vector	Acceleration = Velocity /time
Scalar	vector
Vector	Vector	scalar	time = velocity /acceleration

 

What is a vector and how to add vector?

A vector is an object that has a magnitude, size or length and direction. It is generally represented by an arrow that starts from a initial point (tail) to an end at a terminal point (tip).

 

Here the initial point is A and the terminal point is B. The magnitude is the length of the arrow. The longer the arrow the greater the magnitude of the vector. And the direction of the arrow is the direction of the vector.

 

However in physics the direction of the vector is often represented by an angle θ1 with respect to the horizontal or an angle θ2 with respect to the vertical.

A vector is named by using the initial and terminal point as shown in the diagram below. Hence the vector shown is .

The vector can also be name by using a single letter that may be bold or using the arrow on the letter as shown in the diagram below. Hence the vector can be named either using B or .

We are now going to see how vectors are subtracted, added and multiplied.

Addition

In order to show you how to add vectors we are going to use vectors that horizontal.

Let us have a look at two vectors A and B. Vector A has magnitude 4 and vector B has magnitude 2.

Here it is important to note that as vector A has a magnitude twice that of vector B then the length of vector A must be twice that of B.

A + B

To add vector A and B you place the tail of vector B on the tip of vector A. The result is as shown in the diagram. The two vectors drawn as such is A + B.

As you can see the length of the two vectors together is 6.

Hence the A + B = 6

What would happen if the two vectors are as shown below.

Then A + B is obtained by placing the tail of vector B on the tip of vector A as shown below.

The vector that starts from the tail of A to the tip of B is the vector (A + B). But as you can see here the magnitude of the vector A + B must be calculated using Pythagoras theorem.

If the two vectors A and B are not at right angle to each other, then the angles a and b  that the two vectors make to the vertical or the horizontal must be known as shown in the diagram below.

Hence to add these two vectors we just place the tail of vector B on the tip of vector A as shown below. Then the vector that starts from the tail of vector A to the tip of vector B is the vector (A + B)

In this example the two vectors A and B have angle a and b relative to the horizontal respectively. Hence using the two angles a and b, the angle between the two vectors can be found and as a result the vector A + B can be found using the cosine rule.

Speed and velocity

After distance and displacement that we have seen earlier, I am now going to talk about speed and velocity. They are quite different and to understand them it is important to understand clearly the difference distance and displacement before going forward.

 

Speed

Speed is a scalar quantity and the unit is the metre per second (m s^-1)

A boy moves from point A and walks a distance of 50 m to the point B in a time of 20 s.

Firstly what is the distance travelled in 1 second?

Yes the boy would walk a distance of 2.5 m in a time of 1 second.

The distance travelled in a time of 1 second is called the speed.

The equation to calculate the speed is shown below.

 

 

Can you determine the speed of the boy?

Hence whenever speed is calculated the distance should be used.

Velocity

Velocity is a vector quantity and the unit is also the metre per second(m s^-1)

Since it is a vector quantity then when it is determined direction is important and hence displacement should be used.

A boy moves from point A to point B performing a displacement of 50 m towards the right as shown in the diagram below.

The following equation is used to determine the velocity.

Hence the velocity of the boy can be calculated as shown below.

Please note the difference between the two. For the speed the distance travelled is used while for the velocity the displacement is used.

Distance and displacement

Distance and displacement are two quantities that are used to describe motion. We are going to have a look at both of them and see how to differentiate between the two.

Distance

Distance is a scalar quantity and the SI unit is the Metre (m)

Example 1

 

Now suppose a person moves from an initial position of A and travels to a final position of B. The line connecting the two points is the path that the person would take. Now the distance travelled is the length of the path taken by the person from the initial position to the final position.

Example 2

Now if the path is not a straight line like in the diagram shown above then the distance travelled is the length of the path  from A to B through all the turns  as measured with a measuring tape.

Example 3

The next example is shown above. The person moves from A to D. The path is a line moving from A to B to C and then finally to D. Then the distance travelled is the length of the path AB+ the length of the path BC + the length of the path CD

Displacement

Displacement is a vector quantity and the SI unit is the metre (m)

The displacement is a vector representing the shortest path from the initial position to the final position. It is thus a vector quantity. Let us have a look at the different motion above and see how the displacement is determined and how it is different from the distance.

Example 4

As we have seen in example 1 above the person moves from the initial position A to the final position B.

As you can see the person moves from the initial position A to the final position B and the vector representing this motion is as shown. And the displacement is represented by this vector. Hence the displacement will have both a magnitude which would be the length of the vector and a direction which would be the direction of the vector.

For example 4 above we can say that the displacement is 10 m to the east.

Displacement = 10 m to the east

or Displacement = 10 m to the right

As you can see the displacement would have both a magnitude and a direction since it is a vector quantity.

Example 5

This is the same motion that we used in example 2. The person moves from the initial position A and travels until he reaches the final position B. As you can see on the diagram above there is a vector from the initial position A to the final position B. This vector represent the displacement of the person where the length of the vector is its magnitude and the direction of the vector the direction of the displacement.

In this case the displacement of the person can be said to be

Displacement = 10 m 15^o clockwise from the horizontal,

Where the line AC represent the easterly direction,

Example 6

Now let us look at the same motion example that the person did in example 3. The person walks from A to B to C and finally arrives at D. You can see on the diagram below that I have drawn a vector AD that starts from the point A and ends at the point D. This vector is the displacement vector and as you can see it has a magnitude and a direction.

We can say for example that the displacement is as shown

Displacement = 15 m 15⁰ clockwise from the horizontal

As you have seen displacement and distance are two quantities easy to understand and to differentiate but as usual if you have any question you can leave them here and I would answer then as soon as possible.

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

As we have seen in the first two part of this series it is easier to determine the uncertainty when addition and subtraction is involved but also to determine uncertainty when multiplication and division is involved.

Today we are going to see how to determine the uncertainty when a power is involved.

Very often a derived quantity is determined basic or othere derived quantities using powers, roots, etc.

For example

 

As we are going to see the method to determine the uncertainty in these three cases are similar and can be adapted to other derived quantities.

Example

Let us have a look at how to determine the uncertainty in volume.

A cube has length of L = 20.4+_0.2cm

Determine the uncertainty in volume.

V = L³

V = L*L*L

=249 = 200 (1 sf because of 0.2 )

V = L³

=20.4³

=8489

=8500 (2 sf because of 200)

Hence V = 8500+_200 cm³

After you have studied this example are you able to know how to find the uncertainty in any other derived quantities?

If

Then the fractional uncertainty is

What if area

Then can you deduce the fractional uncertainty?

Yes it is

Can you deduce what would be the fractional uncertainty if?

Yes it is )

Lastly you deduce the fractional uncertainty if ?

Where K is a dimensionless constant with no uncertainty.

Yes it is also )

When there is a dimensionless constant with no uncertainty it does not enter in the equation to calculate the fractional uncertainty.

Question

The area of a circle is given by the equation

If the radius r =10.1+_0.5 cm

then calculate the fractional uncertainty and the uncertainty in A.

Please not that is a dimentionless constant with no uncertainty and hence will play no part in thecalculations.