FIELD FORCE

CONTENT

1. The Concept of Fields
2. Types of Fields
3. Properties of a Force Field
4. Acceleration Due to Gravity
5. Determination of Acceleration Due to Gravity
6. The Shape and Dimension of the Earth

The Concept of Fields

A field is a region under the influence of some physical agencies such as gravitation, magnetism and electricity.

Types of Fields

There are two types of field:

(i) Vector field

(ii) Scalar field.

Vector Fields

A vector field is that field which is usually represented by lines of force; while a scalar field is that field that is not represented by lines of force.

Examples of vector fields include gravitational field, magnetic field and electric field.

Examples of scalar fields include regions with distribution of temperature, density, etc.

(i) Gravitational Field

Gravitational field is a region of space or a force field surrounding a body that has the property of mass. In this region, any object that has mass will experience a force of attraction, called gravitational force.

Gravitational force is responsible for the fact that any object thrown up must definitely fall back. This force of gravity pulls every object towards the centre of the earth. That is to say, gravitational force causes a body which is not in contact with the earth to fall to the ground. This therefore means that the earth exerts an attractive force on every object either on it or near it.

Similarly, two objects of different masses exert equal and opposite forces of attraction on each other.

The radial field near a planet (e.g, earth) is shown below:

(ii) Magnetic Field

Magnetic field is a region around a magnet where it exerts force on other magnets. It is also a region where magnetic force is felt.

The patterns of the magnetic lines of force are shown below:

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Magnetic Field Patterns

1. Field of a bar magnet:

2. Attraction between unlike poles:

1. Repulsion between like poles:

NP means Neutral Point. In this point, no magnetic influence is felt

(iii) Electric Field

An electric field is a region around an electric charge where it exerts force on other charges. It is a field where an electric influence is felt.

The patterns of the electric lines of force are shown below:

1. Isolated positive and negative charge field lines:

2. Attraction between unlike charges:

1. Repulsion between like charges:

NP means Neutral Point. In this point, no electrical influence is felt.

EVALUATION

1. What is a field?
2. State the two types of field.
3. List the examples of vector field.
4. What is neutral point?

Properties of a Force Field

(i) Properties of Gravitational Field

(a) The lines of force are directed towards the centre of the planet; hence, it is a radial field.

(b) The gravitational force field (field strength) ‘g’ at a point is the force per unit mass placed at that point. i.e, g=Fm in N/kg but the S.I unit is m/s²

(c) Any force acting on a body falling towards the centre of the earth is given by F = mg

(d) Gravitational field is a vector quantity.

(ii) Properties of Magnetic Field

(a) Direction: When a magnet is freely suspended, it comes to rest in the South-North direction of the earth.

(b) Attraction: A magnet has the ability to attract magnetic materials e.g, steel, iron, etc.

(c) Force: A magnet exerts force on other magnets in such a manner that like poles repel and unlike poles attract.

(d) The inseparable nature of poles on the magnetic dipoles: If a magnet is broken into small pieces, however small it may be, it will still have a North and South Poles. The smallest bit of a magnet is a dipole.

(e) Magnetic lines of force originate from the North pole and terminate at the South pole.

(iii) Properties of Electric Field

(a) Electric lines of force originate from a positive charge and terminate in a negative charge.

(b) Electric lines of force never cross each other.

(c) They repel each other side ways.

(c) They are in a state of tension which tends to shorten them.

(d) The electric field at a point is defined as the force per unit charge placed at that point. i.e, ε=Fq measured in Newton per Coulomb N/C

EVALUATION

1. State two properties each of the three vector fields discussed.
2. What is the direction of the magnetic lines of force?
3. What is the unit of electric field strength?

GENERAL EVALUATION

1. Discuss the properties of the magnetic flux.
2. Define the electric field strength.
3. Itemise the three vector fields.
4. Why is electric When a field is represented by lines of force, it is then calledlines of force a vector quantity?

Acceleration Due to Gravity

When an object is dropped from the top of a hill or even a tree, the body moves and increases in velocity until it touches the ground with a velocity of finite value. Such movement is influenced by the earth’s gravitational field. The increase in velocity is therefore due to acceleration due to gravity which is usually represented by ‘g’. The motion of such body under gravity is always described as motion under free fall.

However, when two bodies of different masses are released from a height above the ground level, they do hit the ground at the same time. This is because acceleration due to gravity at a location is the same for all bodies irrespective of their masses and thus reach the ground at the same time.

This constant acceleration is called acceleration due to gravity and has a value of  or .

When a body is released from a height so that it falls towards the centre of the earth, ‘g’ is positive; but when a body is thrown upward, it goes against ‘g’ thereby decreasing in velocity until it momentarily comes to rest at the maximum height. For upward movement, ‘g’ is negative.

The equations connecting acceleration due to gravity, ‘g’ are as follows:

For downward movement, v2=u2+2gs and  s=ut+12gt2

For upward movement, v2=u2–2gs and s=ut−12gt2

When a body is released from rest at a certain height so that it falls towards the centre of the earth,

For upward movement, s=ut+12gt2

Since,  u = 0, 2s = gt²

∴ t2=2sg

Hence, t=2sg−−√

This equation shows that the time to reach the ground does not depend on the mass of the object.

Determination of Acceleration Due to Gravity

The value of ‘g’ could be determined using:

1. Formula method: A body is released from a height ‘s’  and the time t is taken; then use s=12gt2 to get the value of ‘g’.
2. Simple Pendulum Experiment method: The value of ‘g’ could also be determined using this experiment.

The period T for the oscillation is given by: T=2πlg−−√

By linearizing this formular, we have T2=4π2(lg)

When T² is plotted against l, the equation is T2=(4π2g)l

A. Hence, the slope for such graph is 4π2g

When l is plotted against T², the equation is l=(g4π2)T2

And the slope for such graph is g4π2

In any case, from the slope, you get the value of ‘g’.

(NB: Educator should carry out the two experiments with the students.)

EVALUATION

1. What is the value of acceleration due to gravity?
2. What is the mathematical relationship between the period of oscillation T and the length of the string used l in a simple pendulum experiment?

The Shape and Dimension of the Earth

The earth is one of the nine planets in the solar system. It is spherical is shape. It is also divided into two hemispheres – the Northern and Southern hemispheres. There are two major types of lines that run through the earth. They include:

• The latitude lines and
• The longitude lines

The latitude lines are imaginary lines running from the east to the west, north or south of the equator. This means that they increase towards the North or South. Examples are:

• Tropic of cancer
• Tropic of capricon
• Artic circle
• Antartic circle.
• The equator line on zero degree.[mediator_tech]

The longitude lines are imaginary lines running from the North pole to the South pole, east or west of the Greenwich meridian. They increase towards the east or west. E.g, the Greenwich meridian on zero degree running through Ghana and London.

However, the earth has a radius of approximately 6400km.

EVALUATION

1. State two differences between latitude lines and longitude lines.
2. Mention two examples of the lines of latitude.
3. What is the approximate radius of the earth?

THE PARTICLE NATURE OF MATTER

CONTENT

1. The Definition of Matter
2. Structure of Matter
3. Evidence of the Particle Nature of Matter
4. Experimental Evidence of the Particle Nature of Matter
5. Explanation of Brownian Motion
6. Simple Atomic Structure
7. The Nature and Size of Molecules

The Definition of Matter

Matter is anything that has weight and occupies space. Every object or substance is made up of matter. Many of the properties and behavior of substances can best be explained by assuming that all substances are composed of small particles called molecules. The assumption that matter is made up of tiny particles (molecules) which are in constant motion is known as the “molecular theory of matter.”

Structure of Matter

Evidence of the Particle Nature of Matter

(i) Many substances in solid form can easily be crushed to powder form e.g. piece of chalk, lump of clay, charcoal and piece of stone.

(ii) A dry stick or dry wood is easily broken into smaller bits

(iii) Solubility – if you drop a cube of sugar into a cup of water and turn the water, the sugar “disappears”. That is, it dissolves in water.

(iv) If you scrape the surface of a piece of chalk, you will see thousands of very tiny particles flake off and float through the air

(v) If a beam of light (e.g. sunlight) is entering a dusty room through a window, you will observe a chaotic motion of the dust particles in the air.

Experimental Evidence of the Particle Nature of Matter

Experimental evidence of the atomic or molecular nature of matter is the Brownian motion named after the botanist, Robert Brown, who discovered the phenomenon in 1827. While he was observing tiny pollen grains suspended in water under a microscope, Brown noticed that the tiny pollen grains moved about in zigzag paths even though the water appeared to be perfectly still. The here and there by the molecules of water pollen grains were being knocked about which were vigorously moving about.

Brownian motion can be demonstrated by the smoke-cell experiment. The apparatus is shown in the figure below.

Molecular motion in gas: Brownian (random) motion

Collect some smoke from a smoldering piece of cloth or wood by means of a syringe and introduce it into the cell. Replace the cover quickly and adjust the focus of the microscope until the fine particles come into view clearly. You will observe the smoke particles as black dots which move about irregularly like a (drunkard) drunken man staggering about. The particles dart from one place to another very suddenly, some going out of focus, others coming into focus, but always in motion.

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Explanation of Brownian Motion

The irregular movement of the smoke particles is due to the motion of the invisible air molecules which bombard each particle from all sides. The particle is very small and the number of molecules of air hitting one side is not balanced by the number of molecules hitting the opposite side at the same instant. Therefore, the particle moves in the direction of the resultant force and when it moves to another place, the same thing happens. Why can’t the motion of the particles be due to convection? In that case the particles would move upwards continually and not zigzag from side to side.

Brownian motion in liquids can be demonstrated similarly as follows:

Place on a clean microscope slide a few drops of diluted aquadag (finegraphite particles suspended in water) or photopake (a similar suspension used for blacking negatives) and cover the liquid with a cover-slip. Project an image of the slide on a screen using a micro projector so that the particles can be seen. The graphite particles are then seen to be moving about in an irregular manner, thus showing Brownian motion in a liquid. In this case, the irregular motion of the graphite particles is due to their bombardment by the surrounding water molecules which are constantly moving about in different directions. Brownian motion is important for two reasons.

(i) It provides evidence for the existence of the tiny particles of matter called molecules

(ii) It gives evidence that molecules are in a constant state of random motion.

Simple Atomic Structure

Definition of an Atom

An atom is the smallest indivisible particle of an element which can take part in a chemical change.

The figure shows the simple structure of an atom. It consists of two parts: The nucleus and the electrons. The nucleus is the heavy portion of the atom and is made up of two types of particles called protons and neutrons. The protons carry a positive charge while neutrons carry no charge. The electrons carry a negative charge and circle in orbits around the heavy nucleus. The numbers of orbits depend on the substance, for example hydrogen has only one orbit while oxygen has two. An electron is very light (about 1 /1840 of the mass of the proton). The negative charge of an electron is equal to the positive charge of a proton and the number of electrons in an atom is equal to the number of protons. The atom is therefore electrically neutral.

The below table summarizes the properties of the elementary particles in the atom of an element.

The important points to keep in mind are as follows:

(i) Protons and neutrons have almost the same mass, while the electron is approximately 2000 times lighter.

(ii) Protons and electrons carry charges of equal magnitude, but opposite charge. Neutrons carry no charge (they are neutral).

EVALUATION

1. What are the elementary particles contained in the nucleus of an atom?
2. Write down their masses and charges.
3. Which particle is located in the outermost shell?

The Nature and Size of Molecules

All matter is made up of tiny particles called molecules. These molecules themselves are made up of tinier particles called atoms. Both molecules and atom are too tiny to be seen with the naked eyes.

One mole of every substance is believed to contain about 6.02 × 10²³ molecules. One molecule of a substance could be found from the combination of two or more elements of that substance.

Definition of Molecule

Hence, we define a molecule as “the smallest unit of matter that is capable of independent existence”. This means that a molecule of a substance could exist alone.

Definition of Atom

An atom is defined as the smallest unit of matter that can take part in a chemical reaction and is not capable of independent existence.

Definition of Element

An element is any substance in which everything could be built up. It is a substance which consists of only one kind of matter and cannot be broken down into anything simpler by any chemical means.

EVALUATION

1. (a) Define the following terms: (i) Molecule (ii) An atom (iii) An element.
2. Which of the above is capable of dependent existence?
3. Which of them is capable of independent existence?

GENERAL EVALUATION

1. Differentiate an atom from a molecule.
2. Highlight the states of matter.
3. Enumerate the factors capable of influencing diffusion rate.
4. What were the evidences to prove the particulate nature of matter?

THE PARTICLE NATURE OF MATTER

CONTENT

1. Molecules
2. Definition of Molecules
3. Structure, Nature and Size of Molecules
4. Some Ideas about Molecular Size
5. Estimating the Size of a Molecule
6. States of Matter
7. The Kinetic Molecular Theory of Matter
8. Fundamental Assumptions of the Kinetic Molecular Theory
9. Basic Assumptions of the Kinetic Theory of Gases
10. Characteristics of the Three States of Matter
11. Crystalline and Amorphous Substances
12. Crystals
13. Non-Crystalline and Amorphous Solids
14. Differences between Amorphous and Crystalline Substances

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Molecules

Definition of Molecules

A molecule is the smallest particle of a substance which can have a separate existence and still retain the properties of that substance.

Structure, Nature and Size of Molecules

(i) Most substances cannot exist by themselves as individual atoms, rather they combine their atoms with themselves or with other atoms to form molecules. Thus a molecule may be made up of similar atoms of the same element or different atoms of two or more elements. For example a molecule of hydrogen is made up of two atoms of hydrogen but a molecule of water consists of two atoms of hydrogen and one atom of oxygen.

(ii) The molecules of any pure substance are identical they have the same structure, the same mass and the same mechanical properties.

(iii) Molecules are formed by atoms combining in simple proportions

(iv) The simplest model of a molecule is that of a rigid sphere (like a small billiard ball) which can move and collide with other molecules or with a wall and exert attractive or repulsive forces on neighboring molecules. The molecular forces decrease as the distance separating the molecules increase.

(v) Molecules are in constant motion. The motion is random (haphazard or zigzag) in liquids and gases but oscillatory or vibrational in solids.

(vi) The size of a molecule is extremely small – of the order of 10^-9– 10^-10 m. as a result of the small size, molecules cannot be see with the naked eyes or even with the aid of a microscope: Again because of the small size, one gram of an element contains several millions of molecules. For example one gram of hydrogen contains about 1023 molecules.

Some Ideas about Molecular Size

It is difficult to imagine how tiny the diameter of a molecule actually is. Even the numerical value estimated to be of the order of 10-10 m (10-9 cm) is also not easily and accurately conjectured. The following may enable you to have a better idea of the size of a molecule:

(i) If a fine hair is magnified until its thickness is that of a wide street, a molecule in the hair would then look like a speck of dust in the street

(ii) The tip of a pin contains millions of molecules.

(iii) Two grains of hydrogen contain 6 x 1023 molecules.

If the whole population of the world were to count such a huge number of molecules individually at the rate of five molecules per second, it would take 100 years to count all of them.

Estimating the Size of a Molecule

You can estimate the size or diameter of a molecule by performing the oil film experiment. The principle of the oil film experiment was discovered by Lord Raleigh in 1890. It was known that certain oils when dropped on the surface of water, would spread to form a circular film with the molecules standing up – right. Lord Raleigh argued that if a drop of oil is placed on top of a water surface, the oil will spread out on top of the water surface until the thickness of the oil film is one molecule thick. He, therefore, used this reasoning to obtain the first estimate of the diameter of a molecule. You can repeat Lord Raleigh’s experiment as follows:

(i) Fill a shallow tray with water and allow it to stand until the water is at rest.

(ii) Sprinkle some lycophodium powder lightly on the surface of the water

(iii) Using a graduated pipette with a fine bore, take up a small volume of olive oil and note the reading on the pipette scale

(iv) Drop a very small quantity of the oil on the water surface. Note again the reading on the pipette and obtain the volume of the oil dropped by subtracting the second from the first pipette reading.

(v) Allow the oil to spread, pushing the lycopodium powder outwards and forming a clear thin circular film of oil on thewater surface.

(vi) Measure the diameter of the oil film to the nearest centimeter using a half millimeter scale. Calculate the thickness of the oil film as follows:

Let diameter of oil film = dcm

Let volume of oil drop = Vcm3

Area of oil film = pd 2  = 2 cm2

.’. Thickness of oil film =volumeareaV2=pd2=4Vcm

If you have performed the experiment accurately you should get a value of about 2 × 10^-7cm as the thickness of the oil film. Hence the size of an oil molecule is taken as about 2 × 10^-7cm

States of Matter

Matter exists in three main states, namely solid, liquid and gas.

Solids have fixed shape and volume. They cannot be poured.

Liquids have fixed or constant volumes but they assume the shape of the container. They can be poured.

Gases have no fixed shape, but always occupy the shape of the container. They can be poured.

The Kinetic Molecular Theory of Matter

The theory states that matter is made up of tiny particles called molecules which are in constant motion.

Fundamental Assumptions of the Kinetic Molecular Theory

(i) Matter exists either in solid, liquid or gaseous state.

(ii) All substances consist of molecules, the smallest particle which can exist independently.

(iii) In solids the molecules vibrate about a mean or fixed position. The forces between the molecules are strong and may be attractive or repulsive. All true solids have a crystalline structure in which the atoms are arranged in regular patterns called lattices.

(iv) In liquids the molecules move freely in all directions. In addition to vibrational energy, they have translational energy. The Kinetic energy of the liquid molecules is greater than in solids.

(v) In gases the molecules are in constant motion and are further apart than in solids and liquids.  They move at high speeds and have translational, vibrational and in addition rotational energy if the molecules are made of two or more atoms. The attractive or cohesive force is negligible, so, gases are perfectly free to expend and completely fill the vessels containing them. Gas molecules have the greatest Kinetic energy. Because the intermolecular forces are small, the motion of molecules in the gaseous state is linear until collision takes place either with other molecules or with the walls of the container.

Basic Assumptions of the Kinetic Theory of Gases

The Kinetic theory of matter has been more completely developed for gases than for solids and liquids. This is because the problems involved are much simpler in the case of gases. The simplest substance to which the theory has been applied is the ideal gas. The fundamental assumptions of the theory are as follows:

(i) Gases consist of many very small particles called molecules, which are like perfectly elastic spheres and are usually in constant random motion.

(ii) Molecules exert no forces on one another except when they collide. Therefore, between collisions with other molecules or with the walls of the container, they move in straight lines.

(iii) Collisions of molecules with one another or with the walls of the container are perfectly elastic. This means that the total Kinetic energy of two molecules before collision is the same as that after collision, and that when a molecule collides with the wall its Kinetic energy is unchanged.

(iv) The duration of a collision is negligible compared with the time between collisions

(v) Molecules are separated by distances which are very large compared with the size of the molecules (or the volume of the molecules is negligible when compared with the volume of the container); they are, however, distributed uniformly throughout  the container.

(vi) Any finite volume of the gas contains a very large number of molecules. This assumption is supported by experimental evidence because under standard temperature and pressure (s.t.p), there are about 3 x 1019 molecules per cm3 of any gas.

Characteristics of the Three States of Matter

Crystalline and Amorphous Substances

Solids are usually classified into two groups:

(i) Crystals or crystalline solids;

(ii) Noncrystal or non-crystalline solids;

The difference between crystals and non-crystals is the arrangement of atoms or molecules in the solid.

Crystals

Definition of Crystals

A crystal is a piece of solid matter in which the atoms, molecules or ions are arranged in a highly regular repeating pattern called lattice.

Crystal Lattice

The particles in a crystal are arranged in regular 3-dimensional framework or pattern called crystal lattice which repeats over and over again in all directions. The high degree of regularity and order in the arrangement of the molecules is the principal feature distinguishing solids from liquids. Particles in a liquid are jumbled and highly disorganized as they move about. They are even more disorganized in a gas. Examples of common crystals are: sodium chloride, zinc sulphide, chromium, iron and platinum salts.

Structure of Simple Crystals

A simple crystal is made up of a huge number of simple basic units or building blocks called unit cells. If you stack these units up and, down, side by side and in all directions, you can build the whole lattice. Unit cells are of three types, giving rise to 3 types of lattice and hence 3 types of crystals, simple cubic lattice, face-centered cubic lattice, and body-centered cubic lattice.

(a) Simple Cubic Crystal

Here the atoms or molecules or ions are placed at the corners of imaginary cubes stacked side by side, up and down like building blocks. An example is the Sodium chloride (NaCl) crystal. In the lattice, the atoms of Na and CI alternate positions in the cube in each of the 3 directions. Each atom within solid, thus has six immediate neighbors.

(b) Face-Centered Cubic Crystal

The unit cell has identical particles at each of the corners plus another particle in the center of each face as shown in the figure above. A typical face-centered cubic.

Crystal is Zinc sulphide (ZnS). Other crystals in this group include crystals of common metals like copper, silver, aluminum, lead, etc. Body-centered cubic crystal

Zinc sulphide ZnS

(c) Body-centered Cubic Crystal

The unit cell has identical particles at each corner of the cube plus one in the center of the cell as illustrated in the figure include; chromium, iron and platinum salts.

Non-Crystalline and Amorphous Solids

The atoms of non-crystals are not regularly arranged as in the case of crystals. They are said to be “amorphous” that is having no definite shape or form; not organized. In a number of ways amorphous substances resemble liquids more than solids. Examples of amorphous solids are glass and plastics. Amorphous solids never form crystals.

They are usually made up of long, chain-like molecules that are intertwined in the liquid state just like strands of earthworms

Chain-like molecules of amorphous substances

Note: Crystalline substances have high melting points because much heat is required to break the strong intermolecular forces binding the molecules together.

Differences between Amorphous and Crystalline Substances

EVALUATION

1. Define molecule.
2. State the kinetic molecular theory of matter.
3. What are crystals?

THE PARTICLE NATURE OF MATTER

CONTENT

1. Molecules
2. Definition of Molecules
3. Structure, Nature and Size of Molecules
4. Some Ideas about Molecular Size
5. Estimating the Size of a Molecule
6. States of Matter
7. The Kinetic Molecular Theory of Matter
8. Fundamental Assumptions of the Kinetic Molecular Theory
9. Basic Assumptions of the Kinetic Theory of Gases
10. Characteristics of the Three States of Matter
11. Crystalline and Amorphous Substances
12. Crystals
13. Non-Crystalline and Amorphous Solids
14. Differences between Amorphous and Crystalline Substances

Molecules

Definition of Molecules

A molecule is the smallest particle of a substance which can have a separate existence and still retain the properties of that substance.

Structure, Nature and Size of Molecules

(i) Most substances cannot exist by themselves as individual atoms, rather they combine their atoms with themselves or with other atoms to form molecules. Thus a molecule may be made up of similar atoms of the same element or different atoms of two or more elements. For example a molecule of hydrogen is made up of two atoms of hydrogen but a molecule of water consists of two atoms of hydrogen and one atom of oxygen.

(ii) The molecules of any pure substance are identical they have the same structure, the same mass and the same mechanical properties.

(iii) Molecules are formed by atoms combining in simple proportions

(iv) The simplest model of a molecule is that of a rigid sphere (like a small billiard ball) which can move and collide with other molecules or with a wall and exert attractive or repulsive forces on neighboring molecules. The molecular forces decrease as the distance separating the molecules increase.

(v) Molecules are in constant motion. The motion is random (haphazard or zigzag) in liquids and gases but oscillatory or vibrational in solids.

(vi) The size of a molecule is extremely small – of the order of 10^-9– 10^-10 m. as a result of the small size, molecules cannot be see with the naked eyes or even with the aid of a microscope: Again because of the small size, one gram of an element contains several millions of molecules. For example one gram of hydrogen contains about 1023 molecules.

Some Ideas about Molecular Size

It is difficult to imagine how tiny the diameter of a molecule actually is. Even the numerical value estimated to be of the order of 10-10 m (10-9 cm) is also not easily and accurately conjectured. The following may enable you to have a better idea of the size of a molecule:

(i) If a fine hair is magnified until its thickness is that of a wide street, a molecule in the hair would then look like a speck of dust in the street

(ii) The tip of a pin contains millions of molecules.

(iii) Two grains of hydrogen contain 6 x 1023 molecules.

If the whole population of the world were to count such a huge number of molecules individually at the rate of five molecules per second, it would take 100 years to count all of them.

Estimating the Size of a Molecule

You can estimate the size or diameter of a molecule by performing the oil film experiment. The principle of the oil film experiment was discovered by Lord Raleigh in 1890. It was known that certain oils when dropped on the surface of water, would spread to form a circular film with the molecules standing up – right. Lord Raleigh argued that if a drop of oil is placed on top of a water surface, the oil will spread out on top of the water surface until the thickness of the oil film is one molecule thick. He, therefore, used this reasoning to obtain the first estimate of the diameter of a molecule. You can repeat Lord Raleigh’s experiment as follows:

(i) Fill a shallow tray with water and allow it to stand until the water is at rest.

(ii) Sprinkle some lycophodium powder lightly on the surface of the water

(iii) Using a graduated pipette with a fine bore, take up a small volume of olive oil and note the reading on the pipette scale

(iv) Drop a very small quantity of the oil on the water surface. Note again the reading on the pipette and obtain the volume of the oil dropped by subtracting the second from the first pipette reading.

(v) Allow the oil to spread, pushing the lycopodium powder outwards and forming a clear thin circular film of oil on thewater surface.

(vi) Measure the diameter of the oil film to the nearest centimeter using a half millimeter scale. Calculate the thickness of the oil film as follows:

Let diameter of oil film = dcm

Let volume of oil drop = Vcm3

Area of oil film = pd 2  = 2 cm2

.’. Thickness of oil film =volumeareaV2=pd2=4Vcm

If you have performed the experiment accurately you should get a value of about 2 × 10^-7cm as the thickness of the oil film. Hence the size of an oil molecule is taken as about 2 × 10^-7cm

States of Matter

Matter exists in three main states, namely solid, liquid and gas.

Solids have fixed shape and volume. They cannot be poured.

Liquids have fixed or constant volumes but they assume the shape of the container. They can be poured.

Gases have no fixed shape, but always occupy the shape of the container. They can be poured.

The Kinetic Molecular Theory of Matter

The theory states that matter is made up of tiny particles called molecules which are in constant motion.

Fundamental Assumptions of the Kinetic Molecular Theory

(i) Matter exists either in solid, liquid or gaseous state.

(ii) All substances consist of molecules, the smallest particle which can exist independently.

(iii) In solids the molecules vibrate about a mean or fixed position. The forces between the molecules are strong and may be attractive or repulsive. All true solids have a crystalline structure in which the atoms are arranged in regular patterns called lattices.

(iv) In liquids the molecules move freely in all directions. In addition to vibrational energy, they have translational energy. The Kinetic energy of the liquid molecules is greater than in solids.

(v) In gases the molecules are in constant motion and are further apart than in solids and liquids.  They move at high speeds and have translational, vibrational and in addition rotational energy if the molecules are made of two or more atoms. The attractive or cohesive force is negligible, so, gases are perfectly free to expend and completely fill the vessels containing them. Gas molecules have the greatest Kinetic energy. Because the intermolecular forces are small, the motion of molecules in the gaseous state is linear until collision takes place either with other molecules or with the walls of the container.

Basic Assumptions of the Kinetic Theory of Gases

The Kinetic theory of matter has been more completely developed for gases than for solids and liquids. This is because the problems involved are much simpler in the case of gases. The simplest substance to which the theory has been applied is the ideal gas. The fundamental assumptions of the theory are as follows:

(i) Gases consist of many very small particles called molecules, which are like perfectly elastic spheres and are usually in constant random motion.

(ii) Molecules exert no forces on one another except when they collide. Therefore, between collisions with other molecules or with the walls of the container, they move in straight lines.

(iii) Collisions of molecules with one another or with the walls of the container are perfectly elastic. This means that the total Kinetic energy of two molecules before collision is the same as that after collision, and that when a molecule collides with the wall its Kinetic energy is unchanged.

(iv) The duration of a collision is negligible compared with the time between collisions

(v) Molecules are separated by distances which are very large compared with the size of the molecules (or the volume of the molecules is negligible when compared with the volume of the container); they are, however, distributed uniformly throughout  the container.

(vi) Any finite volume of the gas contains a very large number of molecules. This assumption is supported by experimental evidence because under standard temperature and pressure (s.t.p), there are about 3 x 1019 molecules per cm3 of any gas.

Characteristics of the Three States of Matter

Crystalline and Amorphous Substances

Solids are usually classified into two groups:

(i) Crystals or crystalline solids;

(ii) Noncrystal or non-crystalline solids;

The difference between crystals and non-crystals is the arrangement of atoms or molecules in the solid.

Crystals

Definition of Crystals

A crystal is a piece of solid matter in which the atoms, molecules or ions are arranged in a highly regular repeating pattern called lattice.

Crystal Lattice

The particles in a crystal are arranged in regular 3-dimensional framework or pattern called crystal lattice which repeats over and over again in all directions. The high degree of regularity and order in the arrangement of the molecules is the principal feature distinguishing solids from liquids. Particles in a liquid are jumbled and highly disorganized as they move about. They are even more disorganized in a gas. Examples of common crystals are: sodium chloride, zinc sulphide, chromium, iron and platinum salts.

Structure of Simple Crystals

A simple crystal is made up of a huge number of simple basic units or building blocks called unit cells. If you stack these units up and, down, side by side and in all directions, you can build the whole lattice. Unit cells are of three types, giving rise to 3 types of lattice and hence 3 types of crystals, simple cubic lattice, face-centered cubic lattice, and body-centered cubic lattice.

(a) Simple Cubic Crystal

Here the atoms or molecules or ions are placed at the corners of imaginary cubes stacked side by side, up and down like building blocks. An example is the Sodium chloride (NaCl) crystal. In the lattice, the atoms of Na and CI alternate positions in the cube in each of the 3 directions. Each atom within solid, thus has six immediate neighbors.

(b) Face-Centered Cubic Crystal

The unit cell has identical particles at each of the corners plus another particle in the center of each face as shown in the figure above. A typical face-centered cubic.

Crystal is Zinc sulphide (ZnS). Other crystals in this group include crystals of common metals like copper, silver, aluminum, lead, etc. Body-centered cubic crystal

Zinc sulphide ZnS

(c) Body-centered Cubic Crystal

The unit cell has identical particles at each corner of the cube plus one in the center of the cell as illustrated in the figure include; chromium, iron and platinum salts.

Non-Crystalline and Amorphous Solids

The atoms of non-crystals are not regularly arranged as in the case of crystals. They are said to be “amorphous” that is having no definite shape or form; not organized. In a number of ways amorphous substances resemble liquids more than solids. Examples of amorphous solids are glass and plastics. Amorphous solids never form crystals.

They are usually made up of long, chain-like molecules that are intertwined in the liquid state just like strands of earthworm

FLUIDS AT REST AND IN MOTION

CONTENT

1. Properties of Fluids at Rest
2. Definition and Effects of Surface Tension
3. Experiment to Demonstrate Surface Tension
4. Methods of Reducing Surface Tension
5. Effects of Surface Tension
6. Capillarity
7. Definition of Capillarity
8. Illustration of Capillarity
9. Demonstration of Capillary Action
10. Explanation of Capillarity
11. Cohesive and Adhesive Forces
12. Diffusion of Gases
13. Osmosis
14. Viscosity
15. Terminal Velocity
16. Applications of Surface Tension and Viscosity
17. Basic Assumptions of the Kinetic Theory of Matter (Gases)

Properties of Fluids at Rest

Definition and Effects of Surface Tension

Surface tension can be defined as the force per unit length normal or perpendicular to a line on the surface of a liquid.

Surface tension exists because of the molecular attraction between the liquid molecules. Consider a vessel of water with molecules P and Q as shown in the figure below

Molecular Forces in a Liquid

Molecule Q is attracted by equal number of molecules all around and so it is in a state of equilibrium. Molecule P is nearer the surface of the liquid than Q. Therefore part of its sphere of molecular attraction is in the air and part is in the liquid. Since the liquid has much more molecules than the air, Q will be attracted towards the liquid by many more molecules than towards the air. The resultant force on Q will be towards the liquid; hence tension exists on the surface of the liquid.

Experiment to Demonstrate Surface Tension

Apparatus:

Beaker, water, steel needle, filter paper and grease.

Procedure:

1. Apply the grease to the steel needle so that water will not wet it and place it on the filter paper
2. Carefully place it on the water surface

Observation:

The paper will absorb water and eventually sinks to the bottom of the beaker leaving behind the needle floating on the water provided the water is not disturbed. The water surface will also be depressed under the needle. The needle floats on the water surface due to surface tension.

Methods of Reducing Surface Tension

1. Increasing the temperature of the liquid
2. Adding soap or detergent to the liquid
3. Adding alcohol
4. Adding camphor

Effects of Surface Tension

1. Ants and pond skaters are able to move on water surface because of surface tension
2. Small objects like razor blade and needle can be made to float on water when carefully placed as a result of surface tension
3. Mercury forms spherical droplets when spilled on glass because of surface tension
4. The hairs of a paint brush spread out and come together when dipped in clean water and removed respectively as a result of surface tension
5. Water drops slowly from a loosely closed tap and forms a bag-like structure as a result of surface tension
6. The rise and depression of liquid in a narrow tube is as a result of surface tension
7. Tarpaulin and umbrella are able to keep off rainwater as a result of surface tension.

EVALUATION

1. Define surface tension
2. Mention some effects of surface tension

Capillarity

Definition of Capillarity

Capillarity (or capillary action) is the tendency of a liquid to rise or fall in a narrow tube

Illustration of Capillarity

Demonstration of Capillary Action

(i) Dip three capillary tubes with fine bores but with different diameters into clean water as in the figure above. You will observe that the water rises in the tubes but the narrower the bore the greater the height to which the water rises.

(ii) Repeat the experiment with soap solution. You will observe a similar situation except that the heights of soap solution are lower than in the case of water.

(iii) Repeat the experiment again with mercury. You will observe that the level of mercury falls in the three tubes. The mercury level is depressed below the level of mercury outside the tubes (i.e. in thecontainer). Again the narrower the tube the lower the level of mercury .

Explanation of Capillarity

(i) The meniscus of water or soap solution is curved upwards (concave) because the adhesion of water and soap solution to glass is greater than the cohesion of water or soap. Therefore water or soap solution wets the glass tube and so spreads a thin film of water/soap solution on the inner surface of the tube. The adhesive forces thus force the water (soap solution) to creep up the inside of the tube. The water/soap solution is held up as it creeps by surface tension forces acting around the circumference of the meniscus. The water/soap solution thus keeps rising in the tube until the weight of the column of water/soap solution balances the surface tension acting at the top of the column.

(ii) In the case of mercury the cohesion of mercury molecules which is greater than the adhesion of mercury to glass causes the mercury level to be depressed in the tube. The surface tension forces acting around the circumference of the tube holds down the mercury column as it is depressed by cohesive forces of the mercury molecules. The depression continues until the weight of the mercury column in the tube is equal to the surface tension.

Cohesive and Adhesive Forces

Force of cohesion is the force of attraction existing among molecules of the same substance while force of adhesion is the force of attraction that exists between molecules of different substances. These forces can be used to explain why water wet glass and mercury does not.

The force of adhesion of water molecules to glass molecules is stronger than the force of cohesion of water molecules. This makes water to wet the glass when it is spilled on it. The force of cohesion of mercury molecules is greater than the force of adhesion between mercury and glass molecules. Hence, when mercury is spilled on glass, it does not wet the glass but forms spherical droplets. For the same reason, the water surface in a glass vessel is convex i.e curves downward while the mercury in a glass vessel is concave.(it curves upward) as shown in the diagrams below.

EVALUATION

1. State four assumptions of the kinetic theory of gases
2. Explain why water poured in a glass tube curve downward and when the liquid in the glass is mercury, the mercury curve upward

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Diffusion of Gases

The molecular kinetic theory can be used to explain a number of phenomena. Some of these phenomena are:

1. Brownian Motion
2. Osmosis
3. Diffusion
4. Viscosity
5. Surface tension

Let us consider Diffusion

Definition of Diffusion

Diffusion is the process whereby gas molecules mix up intimately with one another as a result of the kinetic energy of the molecules. According to Graham’s law of diffusion, the time rate at which gases diffuse is inversely proportional to the square root of its density provided that temperature is kept constant. This is written mathematically as:

tα1d√t=kd√

t = rate of diffusion

d = molecular density of the gas

This means that, the heavier the gas, the slower the movement of the gas.

Pressure of a Gas

If the volume of a fixed mass of a gas is reduced at constant temperature, the molecules will be compressed and become more closely packed together as shown in the figure below. Under this condition, the time rate of collision of the molecules with themselves and the walls of the container increases thereby increasing the pressure of the gas. If on the other hand, the volume is increased by raising the piston up at constant temperature, the time rate of collision of the molecules will decrease. They will take longer time to collide with the walls of the container. This will lead to a decrease in the pressure. This agrees with Boyle’s law which states that, ‘the volume of a fixed mass of a gas at constant temperature is inversely proportional to its pressure.’

Osmosis

Definition of Osmosis

Osmosis is the tendency of a solvent to pass from a dilute solution, through a semi permeable membrane into a concentrated solution. Semi-permeable membrane is a substance such as cellophane, parchment or vegetable material which would allow some molecules of liquid to diffuse through them but not others. Such a membrane may allow the molecules of a solvent to pass through it but not those of a solute.

Demonstration of Osmosis

(i) Tie a semi-permeable membrane across the mouth of a thistle funnel and pour some concentrated sugar solution into the funnel.

(ii) Immerse the funnel in a beaker of water as shown in the figure and note the level of sugar solution in the funnel.

(iii) Allow the set up to stand for a day, and check the level of sugar solution in the funnel again. Test the water in the beaker for sugar. You will observe that the level of the sugar solution in the inverted thistle funnel has risen and that the water in the beaker is free from sugar.

Explanation

The molecules of sugar (the solute) cannot pass through the semipermeable membrane, but molecules of water (the solvent) can do so. Sugar and water molecules are bombarding the membrane on one side and only water molecules on the other side. Thus, more water molecules move up into the funnel than down out of it and so the level of liquid in the thistle funnel rises.

Viscosity

Viscosity by definition is internal friction between layers of fluids in motion. Liquids that are dense pour more slowly than those that are less dense. E.g honey will pour more slowly than water because it is denser than kerosene. This means that honey is a more viscous liquid than water. Viscosity can be demonstrated if we consider a ball bearing falling through some liquids.

Experiment to Demonstrate Viscosity

Apparatus:

Beaker, two different liquids say, engine oil and kerosene

Procedure:

Pour the engine oil into the beaker and drop the ball bearing into it. Observe the ball bearing as it moves to the bottom of the beaker. Do the same thing using the other liquid (kerosene). You will observe that the ball bearing gets to the bottom of the beaker much earlier than it does in engine oil. Therefore viscosity in engine oil is higher than that in kerosene.

Terminal Velocity

A ball that is made to fall through a liquid is under the influence of three forces namely

1. its weight (W = mg) that acts vertically downwards,
2. the upthrust (U) of the liquid on the ball acting upwards the viscous force (V) opposing it motion.

The resultant force acting on the ball can be written as:

ma=W–V–U

Where ‘a’ is the acceleration of the ball through the liquid and m is the mass of the ball. At a certain time in the motion of the ball, its velocity becomes uniform or constant. At this stage acceleration ‘a’ is 0 so that the above equation becomes

W–V–U=0 or Mg–V–U=0V=Mg–U

The velocity-time graph of the motion of the ball is given as shown below

Applications of Surface Tension and Viscosity

1. The knowledge of surface tension is applied in industries in making some materials such as, umbrellas, canvass, rain coats and waterproof tents
2. It is difficult to wash dirty clothes or oily clothes with water only. That is why we use soap and detergents to wash. Soap and detergents weakens the surface tension of water and enables it to float away dirt or oil from the material
3. Viscous liquids are used as lubricants. Examples are grease and engine oil.
4. Viscosity is applied in the design of boats, ships and aircraft.

EVALUATION

1. Differentiate between viscosity and surface tension
2. Give two applications each of viscosity and surface tension

Basic Assumptions of the Kinetic Theory of Matter (Gases)

1. The attractions between molecules are negligible
2. The molecules of a gas are in a state of constant random motion, colliding with one another and the walls of their container
3. The collisions of the molecules are perfectly elastic
4. The duration of a collision is negligible compared with the time between collision
5. The molecules possess kinetic energy by virtue of their motion which is directly proportional to their temperature.

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EVALUATION

1. Explain why the pressure of a gas at constant temperature increase with a decrease in the volume of the gas
2. What do you understand by diffusion?
3. Can diffusion take place in solids, liquids and gases? Explain
4. Explain these phenomena using the kinetic theory of matter: (a) Brownian motion (b) surface tension  (c) osmosis
5. Consider at least four different liquids. E.g palm oil, ground nut oil, engine oil, e.t.c and test their degree of viscosity as was done earlier using a suitable ball bearing. Now answer these questions
6. In which liquid did the ball travel fastest?
7. Which liquid did the ball travel slowest?
8. Hence, list the liquids in their order of viscosity starting from the lowest to the highest.
9. Mention two other low viscous liquids and two other high viscous liquids.

RELATED TO: Fluids at Rest and in Motion

Chain-like molecules of amorphous substances

Note: Crystalline substances have high melting points because much heat is required to break the strong intermolecular forces binding the molecules together.

Differences between Amorphous and Crystalline Substances

EVALUATION

1. Define molecule.
2. State the kinetic molecular theory of matter.
3. What are crystals?

ELECTRIC FIELD

CONTENT

1. Electric Current
2. Electric Circuit
3. Potential Difference
4. Electromotive Force
5. Resistance
6. Types of Resistors
7. Sources of Electric Current
8. Arrangement of Resistors
9. Ohm’s Law
10. Calculations

Electric Current

Definition of Electric Current

Electric current (l) is defined as the time rate of flow of electric charge along a conductor.

I=Qt

Q is the quantity of charge measured in Coulomb,  ‘t’ is the time in second . I is the current in Ampere(A).There are submultiples of Ampere

1mA = 10^-3A

1μA = 10^-6A

Ammeter is an instrument used for measuring current. The electric symbol for ammeter is

Milliameter measures smaller current.

Galvanometer are used to detect very small current.

Electric Circuit

Definition of Electric Circuit

An electric circuit is the path provided for the flow of electric current.

An electric circuit is a system that consists of the source of electricity, the key or switch and the connecting wires, ammeter to measure the current, voltmeter to measure the potential difference, Resistor or load and a rheostat to adjust the flow of current

Circuit Diagram

Closed Circuit

It is a circuit in which there is no gap (key closed) along the conducting path.

Open Circuit

It is a circuit in which there is a gap (key open) along the conducting path.

Short Circuit

A short circuit is a closed circuit without a load. The terminals of the cell are connected together.

Potential Difference (V)

The potential difference between any two points in an electric field is defined as the work done in moving a positive charge of 1 coulomb from one point in the electric field to another. Potential difference is measured in volts.

Voltmeter is used to measure potential difference.

V=WorkCharge

Electromotive Force

Electromotive force is defined as the total work done in driving one coulomb of electricity round a closed circuit or the total energy per coulomb obtained from a cell or battery.

E.m.f=WorkCharge

Electromotive force can also be defined as the potential difference across the terminal of a cell when it not delivering current to an external circuit or the potential difference across the terminal of a cell when it is in an open circuit.

Resistance

This can be defined as the opposition to the flow of charges (electrons) or current. Its S.I unit is Ohm. It is measured using Ohmmeter.

Types of Resistors

1. Fixed/standard resistor: They have fixed resistance. The electrical symbol is

2. Variable resistor: They are those resistors whose resistance can be varied such as Resistance box and Rheostat. The electrical symbols are

EVALUATION

1. Define the following terms (i) Resistor (ii) Electromotive force (iii) Current (iv) Lines of force (v) Potential difference

Sources of Electric Current

Electric current can be generated from the following sources:

1. Chemical energy: Electrical cells store chemical energy. There are two types of electrical cell. The primary cell and the secondary cell. The primary cell cannot be recharged while the secondary cell can be recharged.
2. Heat Energy: Electricity can be generated by thermoelectric effect using a thermocouple, which consists of two different metallic wires joined and dipped in hot water while the other end is connected to a sensitive galvanometer.
3. Mechanical Energy: Current can be obtained from the generator. The generator converts mechanical  energy to electrical energy by the principle of electromagnetic induction.
4. Solar Energy: Electricity can be generated from solar energy using the solar cell. In the solar cell solar energy is converted to electrical energy.

GENERAL EVALUATION

1. Explain at least three sources of generating electricity.

(a) Define electric field.

(b)  Draw the electric field pattern around two unlike charges.

1. (a) Define the electromotive force and terminal potential difference of a battery.

(b) Explain why the electromotive force of a cell is not always the same as the potential difference between its terminals.

Arrangement of Resistors

Resistors can be arranged in series and in parallel.

When resistors are arranged in series in a circuit, the same current flows but, they have different potential differences.

When they are connected in parallel, they have the same potential difference but different current.

Ohm’s Law

The electric current passing through a metallic conductor is directly proportional to the potential difference applied between its end provided temperature and other physical property of the conductor remains constant.

Current =Potential differenceResistanceV=IR

R is a constant of proportionality and depends on the nature of the material. The unit of resistance is ohm.

Calculations

1. A potential difference of 240V is applied to a lamp of 60 ohms resistance. What amount of current will flow in the circuit?

Solution

Current =Potential differenceResistance=24060=4A

1. (a) Calculate the effective resistance in the diagram shown below.

(b) Calculate the current flowing in the circuit above

Solution

(a) Solve the parallel first,

1RP=1R1+1R21RP=12+121RP=22=1RP=1ΩR=RP+4=1+4=5Ω

(b) Total resistance = 5Ω.

From Ohm’s law,

Current =Potential difference Resistance=205=4A

1. A current of 3A flows in a circuit when a p.d of 24V is applied to it. The resistance across the circuit is

Solution

I=3AV=24VR=?V=IR24=3×RR=243=8Ω

THE SOLAR COLLECTOR

CONTENT

1. Role of the Sun in Energy Production
2. The Importance of Solar Energy
3. Solar Cell
4. Solar Panel

Role of the Sun in Energy Production

The sun is a spherical central body of the solar system that radiates light and heat. It has a diameter of 1,384,000km and lies at an average distance of 148,800,000 km from the earth. It is liquid internally and gaseous outwards. The sun produces an enormous output of energy through nuclear fusion. The temperature of the sun is estimated to be about 6000k.

The Importance of Solar Energy

1. Plants need energy from the sun for photosynthesis’
2. Direct heating warms our body, early morning sun is a source of vitamin D.
3. It can be used to dry our cloth, food items, preservation of food.
4. It can be used to generate electricity as in solar panel.
5. It can be used to heat water as in solar water heater.
6. it can be used for cooking as in solar furnace

Solar Cell

This is a device used for the purpose of producing electric power. It consists of semi-conductor like silicon, copper, copper (1) oxide. It converts solar energy to electricity by the use of photo voltaic effect.

Solar Panel

The essence of solar panel is to trap sunlight to generate electricity. It consists of millions of solar cells joined together to generate electricity of high voltage.

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EVALUATION

1. State at least five uses of solar energy?
2. Mention materials used in the construction of a solar panel and state the reasons why those materials are being used.

EQUATIONS OF UNIFORMLY ACCELERATED MOTION

CONTENT

1. Velocity – Time v-t Graph
2. Relative Motion
3. Derivations of Equation of Uniformly Accelerated Motion
4. Derivation of Equations of Uniform Motion
5. Application of the Equations of Uniform Accelerating Bodies
6. Motion of Bodies Under Gravity

Velocity – Time v-t Graph

1. Gradient of a v-t graph = acceleration

acceleration=gradient=change in velocitychange in timeacceleration(a)=ΔvΔt

1. Area under a v-t graph = distance.

1. Total distance covered during the motion = area of trapezium 0edc
2. Distance covered during acceleration = area of triangle 0ea
3. Distance covered during constant velocity = area of rectangle aedb
4. Distance covered during deceleration = area of triangle bdc
5. Acceleration = slope of line 0e, a=ae0a
6. Deceleration =  slope of dc, −a=bdbc

Example 1:

A car starts from rest and accelerates uniformly to 15ms^-1 in 5 s. It then continues at this velocity for the next 10s before decelerating back to rest in another 8 s.

Use the information to answer the following questions

1. Sketch the velocity time graph of the motion of the car
2. Calculate the acceleration of the car
3. Calculate the deceleration of the car
4. What is the total distance travelled by the car
5. Estimate the average speed of the car.

Solution:

1.

1. Acceleration a=ae0aa=155=3ms−2
2. Deceleration −a=bdbc−a=1523−15=158−a=−1.875ms−2
3. Total distance = area under the graph = area of trapezium

s=12(a+b)hs=12(15−5)+(23−0)15s=12(10+23)15s=33×152=4952s=247.5m

1. Average speed v=total distancetotal time

v=247.523=10ms−1

Example 2:

A body at rest is given an initial uniform acceleration of 8.0ms² for 30s after which the acceleration is reduced to 5.0ms² for 30s. The body maintains the speed attained for 60s after which it is brought to rest in 20s.

(a)  Draw the velocity-time graph of the motion using the information given above.

Using the graph, calculate (b) maximum speed attained during the motion.  (c) average retardation as the body is brought to rest.  (d) total distance travelled during the first 60s  (e) average speed during the same intervals as in (c)

Solution

(a)

(b) There are two stages of acceleration

Stage 1: Acelecation = gradient

a=8ms−2a=V1−030−08=V130

Cross multiplying,

V1=8×30=240ms−1

Stage 2: a=5ms−2a=V2−V160−305=V2−V130

Cross multiplying,

V2−V1=150

But V1=240V2−240=150V2=150+240=390ms−1

The maximum velocity is 390ms−1

(c) Average retardation  is equal to  gradient

−a=V2−0140−120

But V2=390ms−1−a=390−020−a=19.5ms−2

Average retardation =−19.5ms−2

(d) Distance is in the first 60sec = area of triangle + area of the next trapezium

s=12(time)V1+12(V1+V2)times=12(30)(240)+12(240+390)30s=3600+9450=13050m

(d) Average speed V=total distancetotal time

V=1305060=217.5ms−1

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Relative Motion

This is the motion of a body with respect to another. All motion is relative. The motion of a car on the road is with respect to the earth or any other frame of reference in which the motion of the car is being observed.

Resultant Velocity of Relative Motions

• Consider two cars X and Y travelling in the same direction and at the same speed, a commuter in X will observe that Y is stationary (not moving)

Vx=Vy

Relative velocity Vx−Vy=0

• If car X is to be travelling at a speed V_x which is greater than the speed of V_y, a commuter in car Y will observe the speed of car X to be

Vx−Vy= relative velocity of car X with respect to Y

A commuter in X will observed the relative velocity of Y to be

Vy−Vx

This value will be negative. This means that to an observer in X, the car Y will appear to be going backward (going the opposite direction with a speed of /Vy−Vx/

• But if car X and Y were to be travelling in opposite direction, the relative velocity of X with respect to Y will be

Vx−Vy= relative velocity of X with respect to Y

Vy−Vx= relative velocity of Y with respect to X

N.B. note that the relative velocity of X with respect to Y, V_xy is equal in magnitude but opposite in direction to the relative velocity of Y with respect to X, V_yx.

Vxy=−Vyx

Examples

1. Two racing cars A and B travelling in the same direction at 300m/s and 340mls respectively. What is the relative velocity of A with respect to B?

Solution:

Va=300km/hVb=340km/h

Relative velocity of a with respect to B, Vab=Va−Vb=300−340=−40km/h

(Note that this is negative. A appears to be travelling in the opposite direction to B)

1. A boat whose speed is 8 km/h sets course on a bearing 060⁰. If the tide is running at a speed of 3 km/h from a bearing of 330⁰, find;
2. The actual speed of the boat(i.e, relative speed of the boat)

The direction of travel

To obtain the relative velocity (actual velocity), draw the component velocity such that the head of one point to the end of the other. Draw the relative velocity to beginning from end of the first to the head of the last.

Using Pythagoras theorem

V2rel=82+32V2rel=64+9=73V2rel=73−−√=8.54kmh−1

Let ϴ be the angle between the relative velocity and the direction of the boat.

tanθ=VtVb=38θ=tan−1[0.375]=20.6o

EVALUATION

1. A train runs at a constant speed of 20m/s for 300s. and then accelerate uniformly to a speed of 30m/s over a period of 20s. this speed is maintained for 300s before the train is brought to rest with uniform deceleration is 30s. draw the velocity – time graph to represent the journey describe above. From the graph find,

• The acceleration while the speed changes from 20m/s to 30m/s.
• The total distance travelled in the time described
• The average speed over the time described. (J.M.B)

1. A car travels at a uniform velocity of 20m/s for 4s. if the brakes were applied to bring the car to rest in the next 8 s. draw the velocity time graph for the motion. How far does the car travel after the brakes were applied?

Derivations of Equation of Uniformly Accelerated Motion

Analysis of Rectilinear Motion

Supposing a body moving at an initial velocity  later attains a final velocity  in time.
Its acceleration is given as:

a=change in velocitytime taken

Change in velocity =v−u

∴ a=v−ut– – – – – -(1)

By making v subject of the formula,

We have v=u+at– – – – – – (2)

Recall that speed=distancetime

∴v=st

∴s=vt

Since the body above experienced two velocities, u & v, thus, the average velocity is v+u2

Hence, s=(v+u)t2– – – – – – (3)

Putting (2) into (3), we have

s=(u+at+u)t2=2ut+at22=2ut2+at22

Hence, s=ut+12(at2)– – – – – – (4)

From (1), t=v−ua– – – – – – (5)

Putting (5) into (3), we have

s=v+u2×v−ua=(v+u)(v−u)2a

But (v+u)(v−u) is a difference of two squares, implying that (v+u)(v−u)=v2–u2

Hence, s=v2−u22a

Making  subject of the formula, we have

v2=u2+2as– – – – – – (6)

Thus, in summary, the equations of motion include:

v=u+ats=(v+u2)ts=ut+12at2s=vt−12at2v2=u2+2as

Under gravity, for a body descending, gis +ve . Therefore, the above equations become:

v=u+gt

s=ut+12gt2

v2=u2+2gs

Under gravity, for a body ascending, gis −ve. Therefore, the above equations become:

v=u−gt

s=ut−12gt2

v2=u2−2gs

EVALUATION

1. Define the following terms (i) average speed (ii) average velocity (iii) uniform acceleration (iv) constant velocity
2. State the value of the acceleration of a body moving with uniform velocity.

Derivation of Equations of Uniform Motion

Recall that,

Equation (i) ——-V=U+at

Equation (ii) ——s=(V+U)t2

Substituting equation (i) into equation (ii)

s=[(U+at)+U]t2s

s=12[U+at+U]t

s=12[2U+at]t

s=12[2Ut+at2]

s=Ut+12at2 ——- (iii)

Again from equation (i),

V=U+at

V−U=at

Dividing both sides by a,

V−Ua=t ——-(3)

Substituting equation (3) into equation (ii)

(ii) ——- s=(V+U)2t

becomes

s=(V+U)2×(V−U)a

s=(V+U)(V−U)2a

Expanding the bracket in the numerator,

s=V2−U22a

Cross multiplying,

V2−U2=2as

V2=U2+2as ——-(iv)

Summarily, the equations of uniformly accelerating bodies are:

N.B.: Note that these equations can only be applied to solve problems on bodies moving with constant/uniform acceleration. Problems on bodies moving with non-uniform acceleration can be solved using differential calculus.[mediator_tech]

EVALUATION

1. State the equations of uniformly accelerating bodies.
2. Derive the (iv) of uniformly accelerating motion.

Application of the Equations of Uniform Accelerating Bodies

1. A train starts from rest and accelerate until it attains a velocity of 8m/s is 10 s. calculate the acceleration of the train.

Solution:

For a body at rest velocity is zero.

Initial velocity U= 0

Final velocity V= 8m/s

Time t = 10 s

Acceleration a = ?

{You use any of the four equations that has U,V, t, a, as identified from the question}

V=U+at8=0+a×108=10a

Dividing both side by 10

a=0.8m/s2

1. A horse rider moving with constant acceleration covers the distance between two point 70.0m apart in 7.0 s. if his speed as he passes the second point is 15.0 m/s. what is its speed at the first point?

Distance S = 70.0m

Time t = 7.0s

Initial speed U = ?

Final speed V = 15.0m/s

{The equation containing S, t, U, and V is S=(V+U)t2}

S=(V+U)t270=(15+U)72

Cross multiplying

(15+U)7=140

Dividing both sides by 7

15+U=20U=20−15U=5m/s

1. A body starts with an initial velocity of 26m/s and moves down it with uniform acceleration of 7m/s² for 25 s. find the total distance moved in metres

Solution: Initial velocity U = 26m/s

Acceleration a = 7m/s²

Time t = 25 s

Distance S = ?

{The equation containing U, a, t and s is S=Ut+12at2}

S=Ut+12at2S=26×25+12×7×252S=650+3.5×625S=650+2187.5S=2837.5m

Motion of Bodies Under Gravity

Neglecting air resistance, motion of bodies moving under gravity (either vertical upward or downward) is an example of uniformly accelerating motion.

1. A body thrown vertically upward in the earth gravitational field

When a body is thrown vertically upward from the earth surface, it retards uniformly (with acceleration of a = -g) until it attain it maximum height where its final velocity is zero. (V = 0)

If U is the initial velocity with which the body was projected vertically upward and H=S is the maximum height where it the velocity is zero (i.e, temporarily at rest before coming down)

g – acceleration due to gravity

V2=U2+2as

V=0, s = H, a = -g  is negative (retardation) where g is the acceleration due to gravity.

0=U2+2(−g)H0=U2–2gH2gH=U2H=U22g
H is the maximum height

Again, using V=U+atV=0a=−g0=U+(−g)t0=U−gtgt=Ut=Ug

T is the time to reach the maximum height.

If the body is thrown vertically upward and allowed to return to the point of projection, the total time of flight is given as

t=2Ug

1. Motion of a bodies falling freely under gravity

The body was initially at rest, hence the initial velocity is zero. As it falls, its velocity increases i.e it accelerates,   a = g

Using V2=U2+2asV2=0+2gHV=2gH−−−−√

This is the velocity of the body just as it it about to reach the ground

Again, using S=Ut+12at2H=0×t+12gt2H=12gt2t=2Hg−−−√

This is the time to reach the ground.

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EVALUATION

1. A stone was thrown vertically upward with an initial speed U. If g is the acceleration of free fall, show that the time taken for the ball to return to its point of projection is
2. A ball is thrown vertically upward with a velocity of 19.6m/s. What distance does it travels before coming to rest momentary at the maximum height?
3. With what velocity must a ball be projected vertically upward for it return to it point of projection in 5s?
4. A vehicle which starts from rest is accelerated uniformly at the rate of 5m/s² for 5 s. It attains a speed which is maintained for 60 s. the vehicle is then brought to rest by a uniform retardation after another 3 s. Determine the total distance covered.