A physics teacher: Physical quantities

Showing posts with label Physical quantities. Show all posts

Monday, December 14, 2009

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:

You must have the skills to use the instrument and know the steps that must be followed to obtain the reading.
You must be using instruments that are properly calibrated and are not damaged.
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.
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.

Tuesday, November 24, 2009

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.

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.

Saturday, November 21, 2009

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?

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.

Monday, October 5, 2009

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.

Saturday, September 5, 2009

Derived units

As we have seen in the last post there are 7 fundamental base units. From these 7 fundamental base units all the other units can be derived.

In fact if a quantity is not a basic quantity then it must certainly be a derived quantity and its unit a derived unit. The derived unit is based on one or more of the 8 fundamental base units. You must keep in mind that some quantities have units of their own based on famous scientists such as Joule (J) for energy or Newton (N) for force. For such quantities the unit can be used or the derived unit can be used.

We are now going to see several derived quantities and their derived units and how these derived units are obtained.

We are now going to see how these derived units are obtained. Once you know how to determine the derived units of common quantities like volume, force, etc then later on you will be able to determine the derived quantity for any other quantity.

So what is the method to obtain the derived unit of a quantity?

Example 1

We are going to start with a simple one: the derived unit for volume.

In order to deduce the derived unit of a quantity you need a formula to calculate the quantity. The formula must contain that you know the units in term of based unit.

Volume = length * width * height

Unit of [volume] = unit of [length * width * height]
= unit of [length] * unit of [width] * unit of [height]
= m * m *m
= m³

Example 2

What is the derived unit for acceleration?

In order to deduce the derived unit for acceleration, you must know a formula to calculate acceleration such as

Acceleration = velocity / time

The formula must contain quantities that you know the units in term of base units.

If in the new formula, there is quantity that you do not know the derived unit then you will also have to know the formula to calculate that quantity.

Velocity = displacement / time

You can now rewrite the formula for acceleration to

Acceleration = velocity / time
= (displacement / time) /time

So

the unit of acceleration = unit of [velocity / time]
= unit of [(displacement / time) /time]
= unit of [(displacement / time)] / unit of time
= (m/s) /s
= m *s^-1 * s^-1
= m s^-2

Was it easy?

Now let us have a look at another example to confirm your newly learned skills.

Example 3

What is the derived unit for pressure?

Have you worked it out?

Now you would remember that in order to find the derived unit you need to know a formula for the quantity in term of quantities that you already know the units.

Pressure = Force / surface are

Hence unit of pressure = unit of [force] / unit of [surface area]
= unit of [mass * acceleration] / unit of surface area]
= (kg * m/ s²)/m²
= (kg m * s^-2 * m ^-2
= Kg m^-1 s^-2

Do the following questions and post the answers in the comment section. I will give the correct answers later on after a few of you have submitted your answers.

Find the derived units for the following quantities.

1. Work done
2. Kinetic energy
3. Volume
4. Density

As you progress through the course you will meet more quantities that you would be able to derive the units.

Good luck and see you next time.

Kinetic energy

Kinetic energy is one if the eight forms of energy. It is simply the energy that a body possesses due to its motion.

The kinetic energy of a body is simply calculated using the equation

Kinetic energy Ek = ½ mv²

Where m is the mass of the mass of the object in kg
v is the velocity of the object in ms^-1

As you can see the kinetic energy of the object depends on both the velocity and the mass of the object. So it is possible for an elephant with a large mass to have the same kinetic energy as a small object moving at a high velocity.

Now let us see an example where the kinetic energy of an object is calculated.

Example 1

An elephant of mass _{3.00 x 10}³ kg is moving at a velocity of 3.00 ms^-1. What is the kinetic energy of the elephant?

So let us recall that the kinetic energy of the elephant is calculated using the equation

Ek = ½ mv²

Hence the kinetic energy Ek = ½ mv²
= ½(_3.00* ₁₀³)*(3.00)²
= 13500 J
= _{1.35 *10}⁴ J

Now you can be given the kinetic energy and be asked to calculate the mass or the velocity. Let us look at a second example.

Example 2

An object has a mass of 4.0 kg and a kinetic energy of 16 J. Determine the velocity of the object.

Ek = ½ mv²
16 = ½ (4.0)*v²

Making v the subject of formula

v² = (16 *2)/4.0
v = (16*2/4.0)^½
v = 2.8 ms^-1

Now to test your newly acquired you can do the following questions

1. Calculate the kinetic energy possessed by an aeroplane of mass _{3.4 * 10}⁵ kg flying at 110 ms.
2. If a body has _{3.4 x 10}⁵ J of kinetic energy and mass _{3.2* 10}² kg, what is its velocity?

Base units

Units are the third aspect that is important when measuring in Physics. It can be the mistake that will ruin the whole measurement process.

Let us recap.

You need to identify the quantity that is to be measured eg length of the book.
you need to use the correct instrument to measure the quantity. In this case the 30 cm ruler.
Lastly you need to measure the length and write it with appropriate unit.

Imagine that you write the length of a 20 cm long book as 20 inches. Then it ruins the whole measurement process.

SI units

To avoid such mistakes scientist around the world have decided to use what is called the SI system of units. It is a standard that is used by all scientists such that when a particular quantity is being measures everyone uses the same units although the Americans and the British continue to use the old systems. This has resulted in costly mistakes and loss of human lives in the space industry.

Base units

The SI system is based on fundamental base units on which all other units can be derived.

The 7 fundamental base units are given in the table below.

Basic Quantity	Base unit	Symbol of base unit
Length	Metre	m
Time	Second	s
Mass	Kilogram	kg
Thermodynamic temperature	Kelvin	K
Luminous intensity	Candela	Cd
Electric current	Ampere	A
Amount of matter	Mole	Mol

These units are the 7 fundamental base units that are used in the SI system of units. Hence even though the quantities in the above table can be measured using other units, it is recommended that only these units are used.

The next part this series on derived units can be found here.

See you later my students.

Sunday, August 16, 2009

What is a physical quantity?

Physical quantities

A physical quantity is a property of an object that can be measured with a measuring instrument.For example length, width, time, weight, etc.

It is important for you to be able to say which instrument is used to measure which physical quantity.The table below shows a few examples of physical quantities and the instruments used to measure them.

Physical quantity Instrument

Length Metre rule

Temperature Thermometer

Time Stopwatch

Volume Measuring cylinder

Of course as we go through the different chapters we will be able to introduce more physical quantities and their measuring instruments.

Basic and derived quantities

Quantities can be divided into two types :

1. Base quantities

2. Derived quantities

Basic quantities

Basic quantities are the fundamental quantities that are not related to each other and that are use to derive all other quantities.

There are seven basic quantities. They are

1. length

2. time

3. mass

4. Thermodynamic temperature

5. electric current

6. amount of substance

7. luminous intensity

Derived quantities

Derived quantities are just quantities that are derived from one or more basic quantities.

For example area is a derived quantity because it is derived from the basic quantity length.

Area = length * length

Volume is a derived quantities because it is derived from the basic quantity length.

volume = length * length * length

density is a derived quantity because it is derived from length and mass, two basic quantities.

density = mass/(length * length * length)

As you can see all other quantities apart from the 7 basic quantities are derived from the seven basic quantities and it can easily be shown as in the example above.

Can you show what are the basic quantities that are used to derive the following derived quantities?

a) force

b) kinetic energy

c) power

d) pressure

I would give the answer after a few of you would have give your answers.

Good luck and see you next time.