• Introduction
• Length
  • Measurement of length
  • Measurement of curved length
  • Estimation of Length
• Area
  • Measurement of area (Accurate Measurement)
  • Approximation of area of irregular bodies
• Volume
  • Converting Volume Units
  • Measurement of volume
  • Measurement of volume of liquids
  • Measurement of volume of irregular objects
• Mass
  • Measurement of mass
• Density
  • Measurement of Density
  • Density of common substances
  • Density bottle
  • Density of Mixtures
• Time
  • Measurement of time
• Accuracy and errors

Introduction

Scientists from various parts of the world were giving measurements in different units and languages. This made it impossible for them to compare discoveries.

In 1960 they met in France and agreed on one Universal System of units called system international d’ unites (International system of units) SI units which assigned seven basic quantities as shown below.

Basic Physical Quantity -This are quantities that cannot be obtained by any other quantity e.g. mass, time, length.

Derived Quantity -This are quantities obtained by multiplication or division of basic physical quantities e.g. Area, Volume, Density.

Length

This is the distance between two fixed points.

The S.I Unit is metre (M)

Other units of length include;

Measurement of length

1. Approximation – This includes estimating the length.
2. Using a standard measure(instruments)
   • Metre rules and half metre rules are used.
   • They are graduated in centimetres and millimetre.
   • They are made of wood, plastic or steel.

Use of a Ruler

1. Put the zero (0) mark to coincide with the start of the object to be measured.
2. Look perpendicular to the edge end of the measurement taken
3. For accurate reading, always place your eyes vertically above the mark to avoid parallax.
   
   

When using a ruler the following should not be done;

• Never drop a metre rule
• Never use it as a walking stick
• Never use it as a cane
• Put it at the centre of the table.
• Keep it in a dry place away from corrosive substances

Tape measure

It is graduated in millimetre (mm) or centimetre (cm)

They are three types;

1. Tailor’s tape measure
2. Carpenter’s tape measure
3. Surveyor’s tape measure

NOTE: The choice of a tape measure depends on accuracy required and the size of object to measure.

A tape measure can be made up of cloth, steel or flexible plastic. Always ensure that the tape measure is taut when measuring.

Measurement of curved length

Curved length can be measured using a thread. The thread is placed along the required length and the length is found by placing the thread on a scale.

EXPERIMENT: Measuring the circumference of a cylinder using a thread.

APPARATUS: A cylinder, a thread and a metre rule

PROCEDURE

• Wrap a thin thread say 10 times around the cylinder
• Mark with ink the beginning and end of turns as shown
  
  
• The circumference of the cylinder will be given by;
  
  Circumference=^{[length of thread]}/₁₀
  
  Circumference=πD or 2πr (where r is the radius of the cylinder)

Estimation of Length

EXPERIMENT: To estimate the height of a tree

APPARATUS: A metre rule, tape measure

PROCEDURE

• Measure the length of the metre rule when upright using a tape measure followed by measuring its shadow.
• Measure the shadow of the tree in the school compound.

RESULTS

Height of metre rule = ............Cm
Height of shadow of metre rule=............Cm
Height of shadow of the tree =...............Cm

Estimation of the height of the tree is given by;

^{Height of the tree}/_{Height of the metre rule} x ^{Length of the shadow of the tree} / _{Length of the shadow of the metre rule}

Area

Area refers to a measure of surface covered by a body.

It is measured in metre-square (m²), that means its a derived quantity.

Other multiples and sub-multiples are; cm², mm², km²

CONVERTING

1. mm² to m²
   1m²=1000*1000
   = 1000000mm² (Divide by 1 million)
   1mm² = {¹/_1000000} m² 
2. m² to mm² 
   1m² = 1000000mm²{multiply by 1 million}
3. cm² to m² 
   1cm=0.01m
   1cm²=0.0001m²{multiply by 0.0001}
4. m² to cm²
   1m=100cm
   1m²=100cm*100cm
   =10000cm²{multiply by 10000}

EXERCISE

1. Convert 7.5m² to cm²
2. Convert 940mm² to cm² 
3. Convert 12000mm² to m²

Measurement of area (Accurate Measurement)

Area of a regularly shaped object can be found by applying an appropriate formula e.g.

Approximation of area of irregular bodies

We trace their outline on the square paper of 1cm² e.g.

Full squares=............cm²

¹/₂ full squares=...........cm²

AREA=full square+ ¹/₂ full squares

Volume

Volume is the amount of space occupied by space

The SI unit of volume is cubic metres [M³ ].It is a derived quantity of length

Multiples and submultiples are; mm³, cm³ and Km³

Converting Volume Units

1. From m³ to mm³
   1m =1000mm
   1m³=1000mm*1000mm*1000mm
   =1000000000mm³
   To change m³ to mm³ you multiply by 1 billion
   
   
2. From mm³ to m³
   To change m³ to mm³ you divide by 1 billion i.e. ¹/_10000000000

Example

• Express 9cm³ in m³
• Express 9000000000mm³ in m³
• Express 0.0546m³ to cm³

Measurement of volume

The volume of regularly shaped solids can be obtained by applying the appropriate formula i.e.

1. Cuboids L X W X H
2. Cylinder πr2h
3. Triangular prism ¹/₂bhl
4. Sphere ⁴/₃ πr3

Measurement of volume of liquids

Liquids have no definite shape but they assume the shapes of the container in which they are put.

One of the methods which can be used to measure the volume of liquids is to pour the liquids into a container with a uniform cross-section as shown,

Volume=Area of cross-section x height

=Ah where A = l X b

=lbh

Instruments can also be used to measure the volume of liquids. They include; Burette, Pipette, Measuring cylinder, graduated beaker and Volumetric flask:

NOTE: The scale of the Burette begins from zero at the top and increases downwards to the maximum value e.g. a reading of 31.0ml on the burette means that volume of the liquid is [50-31] ml

Measurement of volume of irregular objects

1. Using a measuring cylinder
   
   
   PROCEDURE
   • Fill the measuring cylinder with water.
   • Record the volume of water as V₁-
   • Submerge gently a stone [irregular object] tied around a thread.
   • Record the volume of water and the stone as V₂.
     
     
     
     
   • Volume of the stone =V₂-V₁
2. Using a Eureka can
   
   A Eureka or displacement can is a container with a spout from the side.
   
   Apparatus; Eureka can, measuring cylinder, irregular object e.g. a stone, water
   
   Procedure
   • Fill the Eureka can with water until it flows out of the spout.
   • Place a measuring cylinder under the spout of the can.
   • Tie the solid [irregular object] with a thread and submerge it gently inside the can.
   • The result [water] collected to the measuring cylinder is the volume of the irregular object.
     

Mass

Mass is a quantity of matter in a body.

Its S.I unit is kilogrammes (Kg)

It is measured using a beam balance or top pan balance.

The multiples and submultiples include;

The mass of an object is the same everywhere because the number of particles in an object remains constant.

Measurement of mass

There are two common types of balances for measuring mass; Electrical and mechanical types.

Electrical types are very accurate and the mass of the object is read on display (Top Pan Balance).

A Mechanical type(Beam Balance), the object whose mass to be measured is balanced against a known standard mass on an equal level.

The three balances used in measuring are;

1. Top Pan Balance
   
2. Beam balance
   
   
3. Lever balance
   
   
   In a lever balance combination of levers moves the pointer along a scale when the mass is placed on it.

Density

The density of a substance is defined as its mass per unit volume. Its symbol is rho(ρ).

The SI unit is kilogram per cubic metre (Kg/m³)

Conversion from kg/m³ to g/cm³

1g/cm³ = 1000kg/m³

EXAMPLE

A Block of glass of mass 187.5g is 5cm long, 2.0cm and 7.5cm high. Calculate the density of the glass block.

Solution

Density = ^m/_v = ^187.5g/_{5 x 2 x 7.5}

=2.5g/cm3 or 2500kg/m3

Measurement of Density

The density of an object is calculated from the formula;

Density =^mass/_volume

Density of common substances

Density bottle

A Density bottle is a small glass bottle fitted with a glass stopper which has a hole through which excess liquid flows out.

Normally, the density bottle has its capacity indicated on the side.

To find the density of the liquid using a density bottle, measure the mass m₁ of a dry clean density bottle with its stopper.

Fill the bottle with liquid and replace the stopper. Dry the bottle on outside (excess liquid overflows through the hole in the stopper).

Measure the mass m₂ of the bottle plus the liquid.

If the volume of the liquid is V then;

Density =^(m₂-m₁)/_V

PRECAUTIONS

• The bottle is held by the neck when wiping it dry. This is because when held in hands, it may expand due to warmth from the hand.
• The outside of the bottle must be wiped carefully.
• Ensure that there are no air bubbles when the bottle is filled with liquid

To measure the density of a solid using a density bottle

This method is used for solids in form of grains, beads or turnings

Apparatus: density bottle, lead shots and beam balance.

PROCEDURE

• Measure the mass m₀ of a clean dry empty density bottle.
• Fill the bottle partly with the solid (lead shots) and measure mass m₁.
• Fill up the bottle with water up to the neck and measure its mass as m₂.
• Empty the bottle and rinse it
• Fill it with water and replace it with the stopper, wipe outside dry and measure the mass m₃ of the bottle filled with water.

RESULTS

Mass of water = (m₃-m₀)g

Volume of water= (m₃-m₀) cm³ (since density of water is 1g/cm³)

Mass of lead shots (solid)=(m₁-m₀)g

Mass of water present when the bottle is filled with lead and water = (m₂-m₁)g

Volume of water = (m₂-m₁) cm3

Volume of lead shots= (m₃-m₀)-(m₂-m₁) cm³ (since density of water is 1g/cm³)

Therefore density of lead shot= (m₁-m₀)-{(m₃-m₀)-(m₂-m₁)}

NOTE: This method is unsuitable for solids which are either soluble or react with water.

EXAMPLE

1. The mass of a density bottle is 20g when empty and 45g when full of water. When full of mercury, its mass is 360g. Calculate the density of mercury.
   SOLUTION
   Mass of water = 45-20 =25g
   Volume of water=25g/1g/cm³
   =25cm³
   Volume of bottle = 25cm³
   Mass of mercury = 360-20 =340g
   Volume of mercury= 25cm³
   Density of mercury=340/25
   =13.6g/cm³ or
   =13600kg/m³

Density of Mixtures

A Mixture is obtained by putting together two or more substances such that they do not react with one another.

The density of the mixture lies between the densities of its constituent substances and depends on their proportions.

Density of the mixture=^{mass of the mixture}/_{volume of the mixture}

EXAMPLE

100cm³ of fresh water of density 1000kg/m³ is mixed with 100cm³ of sea water of density 1030kg/m³. Calculate the density of the mixture.

Solution

Mass of fresh water= density x volume
=1g/cm³ x100cm³
=100g

Mass of sea water = 1.03 x 100
=103g

Mass of the mixture= 100+103
=203g

Volume of the mixture=100+100
=200cm³

Density of the mixture =203/200
=1.015g/cm³

Time

It is a measure of duration of an event. Some ancient measuring instruments were the sundial and the hour glass

The SI unit of time is seconds (s)

Multiples and submultiples of time

Measurement of time

Time is measured using either a stopwatch (digital) or stop clock. They are used depending on the accuracy required.

Accuracy and errors

Accuracy is the closeness of a measurement to the correct value of the quantity being measured. It is expressed as an error. An error is therefore the deviation of measurement to the correct value being measured. The smaller the error the accurate the measurement.
% error = (^sensitivity / _{size measured}) × 100.