what is a parallax error?

Parallax error is the error that is most committed when readings are taken in physics. You can thus understand why it is important to avoid it at all cost. One must be aware of its existence at all time so that it can be avoided and as a result the true value of the reading is obtained.

The concept of parallax error is related to the term parallax.

Imagine that we have  in a room a pelican and a flamingo as shown in fig 1 below.

Fig 1

Now Garfield is moving about in the room and each time is is somewhere in the room, he looks at the two birds. At the point A he sees the flamingo on the left of the pelican whereas when he is at position C he would see the flamingo on the right of the pelican. Only when he would be at position B would he sees the two birds one behind the other.

He would discover that each time he is at a different position, he would find that the position of the flamingo relative to the pelican has changed.

You can also have this effect when you are in front of a clock. If you move from side to side you would find that the time that you can read from the clock is different. 

So we can the understand that the parallax is the change in the apparent position of an object when the position of the observer changes.

Now let us look at the concept of parallax error. If you have placed a pencil on a metre rule and you are reading its length then just like in fig 1 above and in fig 2 below you can place you eye everywhere you want.

Fig 2

As you can see from fig 2 above I have chosen three position at which you can place your eye.

Clearly at these three position we can have the following reading

Reading at A  = 6.2 cm

Reading at B  = 5.8 cm

Reading at C = 5.5 cm

What you you think would be the correct reading?

The correct reading is would be obtained when the eye is placed at B.

 

 

Fig 3

Now there is a line from the tip of eye to the of the pencil  that continues up to the scale. this line is called the line of sight and the mark at which the line intersect the scale is the length of the pencil. This line of sight must be be at right angle to the scale. This is shown above in fig

If the line of sight and the scale are not ar right angle to each other then a parallax error is committed.

Similarly with a measuring cylinder the line of sight from the eye to the bottom of the meniscus must be at right angle to the scale as shown in fig 4 below. In this case the line of sight is horizontal and the scale vertical.

 

Fig 4

So as you can see above each time you are taking a reading you must ensure that the line of sight is perpendicular to the scale.

What is a zero error?

We have seen in a previous post what an error is. I am now going to talk about the zero error.

As the name suggest the error has a relation with the zero mark on a scale.

As you can see in fig 1 and fig 2 the scale on a measuring instrument can be either straight as on a meter rule or circular as on an ammeter.

Fig 1

Fig 2

Straight scale

Now when you measure using such instruments it is necessary for you to pay particular attention to the zero mark. If you are using a metre rule then one end of the object must be placed on the zero mark as shown in fig 3 below.

Fig 3

However some measuring instruments have the the zero mark that start slightly inside like in the second diagram in fig 3 above. Be careful.

Now if the object is not place on the zero mark as shown in fig 4 below then a zero error is committed. 

Fig 4

Can you read the scale and tell me what is the length of the object?

Now as you can see the  object is not placed starting on the zero mark. It is place on the o.2 cm mark. This 0.2 cm mark is the magnitude of the error. As you can also see the object’s other end is on the 2 cm mark. Since the other end of the object is not place on the zero mark, the length obtained will be greater that the true value.

Hence the length of the ruler = 2.0 – 0.2 = 1.8 cm

It is important to identify the magnitude of the error and then to  remove it from the reading with the error to get the true reading.

Circular scale

When there is a circular scale, there is always a pointer like you can see on an ammeter or voltmeter. Now if you want to have the correct reading it is important for the pointer to be on the zero mark before it is used. If the pointer is not initially on the zero mark then a zero error would be committed.

Fig 5                                                                      Fig 6                                                       

In fig 5 above you can clearly see that the pointer is on the zero mark before use, and as a result there is no zero error and the correct reading from fig 6 is 0.8.

Let us look at two examples where the pointer is not initially on the zero mark.

Fig 7                                                                            Fig 8                                             

 

In fig 7 the pointer is not on the zero mark before use. So in this case the solution is to adjust the pointer until the error is removed. If that is not possible then the magnitude of the error needs to be determined. In our example the the magnitude of the error is 0.3. Hence all measurement taken with this apparatus will be greater by 0.3. After the meter is used the reading is o.8. Then adjusted for the error

The true value = 0.8 – 0.3 =0.5

Fig 9                                                         Fig 10                                                    

In fig 9 the pointer is on the left side of the zero mark. The magnitude of the error is 0.2. But as you can guess when the meter is in used the reading obtained will be less than the true value. Hence the reading from the meter in fig 10 is smaller than the true value and as a result the magnitude of the error must be added to the reading. Hence

The true value = 0.8 + 0.2  = 1.0

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

This is the second part of a series of post on finding uncertainty in derived quantities. You can find the index here and the part on addition and subtraction here.

Now very often when you performed an experiment you meet quantities that need to be processed to obtain a derived quantity such as g, the acceleration due to gravity, or any other derived quantities.

You have already seen the first part on addition and subtraction and now you will see how to do it for multiplication and division.

If in an experiment is performed and the following quantities are measured with their uncertainties.

A = 10.2+-0.2 cm

B = 5.4 +-0.4 cm

Now if you need to process these quantities to find AB and A/B with their uncertainties how would you do it?

Uncertainty in multiplication

How would you determine the value of AB and its uncertainty?

You will have to calculate the value of AB first.

AB = 10.2 *5.4 = 55.o8 =55 (2 sf )

To determine the uncertainty in AB you will have to use the equation below.

Rearranging the equation will give you

Hence AB = 55+-5 cm²

Remember the uncertainty in A and B are to 1 sf hence the uncertainty in AB must be given to 1 sf.

Uncertainty in division

You are now going to determine the value of A/B and its uncertainty.

You will have to determine the value of A/B first.

A/B = 10.2 /5.4 =1.888 = 1.9

To determine the uncertainty of A/B  you will determine the equation below

Rearranging the equation will give you

Remember the uncertainty in A/B is given to 1 sf since the uncertainties in A and B are given to 1 sf.

Hence A/B = 1.9 +-0.2

With these two formula you can thus determine the uncertainty in any derived quantities that involves multiplication and division.

Now you can move to the next part where the method to determine uncertainty of derived quantities where powers,  square root, etc are involved.

Moment and couple

 

Moment

We have when earlier that a force acting on an object causes work done. Here we are going to see that if a force act on an object the a rotation can occur.

Let us look at some examples. You will see that if the force act on an object in a certain way the object would undergo a rotation about a centre of rotation.

Example 1

In this example a footballer would kick the ball and the ball would rotate about the centre of the ball the centre of rotation.

Example 2

In this example a person is opening a door by exerting a force on the door handle. The door will thus rotate about the hinge of the door.

So we can thus say that the moment is related to the turning effect of a force.

We are now going to have look at how the moment of a force is calculated.

In the diagram above we have the force F that is applied on the door handle. 

The force is applied on the handle and the point on the object where the force is applied is called point of application of the force. The distance between the point of application of the force and the pivot is the distance d.

Note that the force must be at right angle to the distance d as shown in the diagram below. If it is not at right angle as shown in the second diagram below then the component of the force that is at right angle to the distance must be determined. In the case of the second diagram below the component of the force that is at right angle to the distance is Fcosθ

 

We can now return to the diagram shown above, and we can say that the moment can be calculated as shown below.

Moment = Force * perpendicular distance from point of application of the force to the pivot

If the force is at right angle to the distance then

moment = F *d

If the force is not at right angle to the distance then

Moment = Fcosθ*d

          = Fd cosθ

 

Couple

A couple is when two forces of equal magnitude but acting in opposite direction on an object causing a rotation.

The best example is the car steering wheel or the handle of a motorcycle. As the diagram below shows two forces that are equal to each other in magnitude but acting in opposite direction act on an object. The point of applications of the two forces are separated by a distance d.

 

   Hence the couple can be calculated using the equations below.

Couple = Magnitude of one of the two forces * the distance between the point of application of the two forces

couple = F* d

Of course the same principle  will apply to couple if the forces are not at right angle to the distance. Then the component of the force at right angle to the distance must be used. The couple will then be calculated using the equation

couple = Fd Cosθ

Volume and volume of regular objects

The volume of an object is the space that the object occupies.

It is a scalar quantity and its SI unit is the cubic metre (m³)

The unit for the volume of object can also be derived from any other unit for length such as the cubic centimetre, cubic kilometre, etc

We are now going to see how the volume of an object is determined.  The method to determine the volume of the object depends on the nature of the object as shown below.

Volume of regular objects

Regular objects are those that have a plane of symmetry. It simply mean that if the object is cut along the plane of symmetry then the two parts will be similar to each other with the same volume.

The volume of regular objects is determined by measuring the dimensions of the object and finally using using an equation. The following are different regular objects and the equations used.

Cuboid

  

Cube

Volume = length * length * length

            = L³

Since all the sides of the cuboid are of equal length.   

Sphere

since r = d/2 then

As you can see all regular objects are determined by measuring the dimensions and then using equations to determine the volume. I would try with time to add as much of regular objects and their equations. If you require the equation for any regular shape then leave message in the comments below.

What is an error?

As we have seen in an earlier post, a physical quantity is a property of an object that can be measured with a measuring instrument.  Hence when you use a measuring instrument you would obtain a reading. The reading is simply a numerical value that you can read off a measuring instrument such as the volume off a measuring cylinder or the time off a stopwatch.

We must also introduce what is called the true value. The true value is the reading that you would obtain if the measurement is done in ideal conditions.

In order to obtain the the true value the following conditions must be present:

1. You must have the skills to use the instrument and know the steps that must be followed to obtain the reading.
2. You must be using instruments that are properly calibrated and are not damaged.
3. All the conditions that are required to obtain the true value must be present. For example 1/3 of the stem of the thermometer must be immersed in the liquid, the pressure must be one bar, the liquid used in the measuring cylinder must have a temperature of 20⁰C.
4. If a calculation is needed to determine the magnitude of a physical quantity, then the correct equation and the right constant must be used.

Now an error is made when the reading that you obtain is not equal to the true value.  And the magnitude of the error is the difference between the reading obtained and the true value.

Then we can thus introduce the term accuracy and precision.

A measurement is accurate if it is close to the true value. And if there are two values then the one that is closest to the true value is the most accurate.

The precision of an instrument is the smallest value that can be measured using the instrument. Hence if a length is measured using a metre rule then it is precise to 1 mm or it has a precision of 1 mm. However a length that is measure using a vernier caliper has a precision of 0.1 mm. Hence the length that is measure using a vernier caliper is most precise.

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.