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Measurement is the process of assigning numbers to observations or events
Physical Quantities
Physical quantity is a property of a material that can be quantified by measurement.
There are two types of physical quantities namely.
1. Fundamental quantities
2. Delivered quantities
Fundamental Quantities
Are the basic physical quantities which cannot be obtained from other physical quantities.
SI UNITS
This is the System International of Units. They are specifically chosen units which have been agreed upon internationally to be used for measurements.
BASIC QUANTITIES
These are quantities that cannot be expressed in terms of other physical quantities. These quantities are sometimes referred to as Fundamental quantities and examples of the basic quantities with their SI units are given in the table below;
Fundamental Quantities and their SI Unit
| Quantities | SI unit | Unit symbol |
| Length | Metre | m |
| Mass | Kilogram | kg |
| Time | Second | S |
| Electric current | Ampere | A |
| Temperature | Kelvin | K |
| Amount of substance | Mole | mol |
| Luminous intensity | Candela | Cd |
Length
Is the distance between two points.
The SI unit of length is metre (m).
It is measured by metre rule, tape measure, Vernier calliper and micrometer screw gauge
Metre Rule
Metre rule is a mainly wooden graduated in 100 centimeters or 1metre.
The reading should be perpendicular to the mark otherwise the parallax error occurs
Parallax Error
Is the apparent motion of one object related to another when the position of the eye is varied
Vernier Caliper
Is an instrument used to measure length to the nearest accuracy of 0.01cm
It is used to measure lengths to the range of 1.0 cm to about 12.0 cm
Using vernier calipers:
Place the object between the jaws of the calipers and close the jaws until they just grip it.
Read the main scale value just before the zero mark on the vernier scale.
Read the divison on the vernier scale that coincides with the mark on the main scale. The number on this mark on the vernier scale is reading for the hundredth of centimeters. (Always divide the number read on vernier scale by 100)
Add the reading of the main scale to that of the vernier scale to obtain the vernier caliper reading.
Object to be measured
Therefore, the reading of the above vernier caliper is 1.30 + 0.02 = 1.32cm
Examples
1.What is the reading of the vernier caliper below?
Main scale = 3.70cm
Vernier scale = 0.07cm
Vernier reading = 3.77cm
2. What is the reading of the vernier caliper below?
Main scale = 0.70cm
Vernier scale = 0.08cm Vernier caliper reading = 0.78cm
3. What is the reading of the vernier caliper below?
Main scale = 11.20cm
Vernier scale = 0.08cm Vernier caliper reading = 11.28cm
Scale of Vernier Calliper has two scales
- Main (Fixed) scale
- Vernier scale
NB:
Fixed scale gives reading in centimeter (cm) or millimeter (mm).
Vernier scale gives reading in hundredth of a centimeter (0.01cm) or thousands of millimeter (0.001mm)
The reading should be taken in the parallel mark between fixed scale and Vernier scale then convert it to cm or mm
Total reading is obtained by Summing up the main scale (M.S) and Vernier scale (V.S)
Before using a vernier caliper, close its jaws to determine if it contains zero error
Zero error is the error arises when scale is not starting from zero mark
Example
1. From the fig below, determine the diameter of the object.
Solution:
Give: Main scale, m.s = 9.9cm , Vernier scale, v.s = 2 x 0.01 = 0.02cm
Micrometer Screw Gauge
Is an instrument used to measure the length to the nearest accuracy of 0.001cm or 0.01mm
It is used to measure the diameters of wires and ball bearings
It can measure small lengths up to about 2.5 cm
Scale of Micrometer Screw Gauge:-
a. Main scale (mm)
a. Thimble scale
NB:
Fixed scale gives reading in centimeter (cm) or millimeter (mm).
Thimble scale gives reading in thousandth of a centimeter (0.001cm).
Before to use micrometer screw gauge close its jaws to determine if it contains zero error
Example
1. From the fig below, determine the diameter of an object.
Solution:
Given: Main scale, m.s = 9.5mm = 0.95cm , Thimble scale, v.s = 31 x 0.001 = 0.031cm
Mass
Is the quantity of matter in a substance.
It is measured by beam balance.
Other units of mass are milligram, gram, tones etc
Their equivalence: 1t = 1000kg 1kg = 1000g 1g = 1000mg
Types of Beam Balance
1. Lever arm balance (uses the principle of moments to measure the mass)
2. Triple beam balance (uses the principle of moments to measure the mass)
3. Digital balance (measures the mass to an accuracy of the thousandth (0.001g) of a gram
Difference between Mass and Weight
| Mass | weight |
| Is the quantity of matter in an object | Is a force of gravity on an object |
| It is constant | It varies with environment |
| It is a fundamental quantity | It is a derived quantity |
| Its SI unit is kilogram (kg) | Its SI unit is Newton (N) |
| It is measured by beam balance | It is measured by spring balance |
| Is a scalar quantity | Is a vector quantity |
Time
Time is the rate at which an event happens.
It is measured by using clock or wristwatch or stopwatch
Stopwatch
Is a device that is held in the hand to show the time elapsed
Types of Stopwatches
1. Mechanical stopwatch
2. Digital stopwatch
N.B: Digital stopwatch is more accurate than mechanical stopwatch. They include date and time
Ways of reducing errors during measurement
1. Taking several readings and then find the average
2. Avoiding parallax error by positioning te instrument properly on the table with eyes perpendicular to the scale
3. Some instruments cab be adjusted to eliminate zero error
Delivered Quantities
Are the physical quantities which are expressed in terms of the fundamental quantities
Examples are area, volume, weight, pressure etc
Volume
Is the quantity of space that an object occupies.
Its SI unit is cubic meter (m3)
N.B
1L = 1000 cm3 1L = 1000 ml 1L = 1dm3
Volume of a solid regular object
Regular object is the object with known shape.
For example, cylinder, rectangular prism, cube etc.
The Volume of an object is given by: -
Whereby:
A = area of a regular object h = height of a regular object
Since w = h = b
Volume of Rectangular prism
Volume of Cylinder
Volume of Sphere (h = r)
Example
1. Calculate the volume of rectangular block of sides 15cm, 8cm and 7cm Solution:
V = 15 cm x 8 cm x 7 cm = 840 cm3
Question
Calculate the volume of cylinder whose radius and height are 5 cm and 14 cm respectively. Given that π = 3.14. (ANS: V = 1099 cm3)
Volume of Liquid
Littre is the standard unit used for measuring the volume of liquids.
Burette, Pipette, measuring cylinder are examples of the instruments or apparatus used to measure the volume of liquids
During measurement the eye should be in the same line with the meniscus of the liquid
Volume of Gas
The volume of gas is obtained by measuring the volume of the container into which it is put
And the volume of the container can be determined from its dimensions or by filling it with water and then pouring the water into a graduated cylinder
Thus VGAS = V (CONTAINER + GAS) – V(CONTAINER)
Volume of an irregular object
Irregular object is the object with unknown shape.
For example, stone, human body etc.
The volume of irregular object is obtained by displacement method or immersion method
Displacement Method
Volume of irregular object is based on the principle that when an object is completely submerged in water it displaces a volume of water equal to its own volume.
The volume of irregular object can be measured by using:
(a). A Graduated cylinder
(b). A Eureka can or overflow can
Graduated Cylinder
Suppose you want to measure the volume of a small stone. The following steps are necessary:-
Fill a graduated cylinder to known mark (let it be 300ml)
Carefully measure the initial volume of water (V1)
Gently lower the stone into the water
Measure the final volume of water (V2)
Lastly find the difference between the final and initial volume of water.
This gives the volume of a stone.
That is VSTONE = V2 – V1
Example
1. When an irregular solid was immersed in 65cm3 of water the water level rises to 81cm3. What was the volume of the solid?
Solution:
Volume of the solid, V = V2 – V1 = 81 – 65 = 16 cm3
Using Eureka Can (Over flow can)
Consider the following steps: -
Fill the overflow can with water up to the level of the spout
Tie the irregular solid (stone) with a string
Gently drop the irregular solid into water using a string
The irregular solid (stone) will displace some water which will be collected in the beaker
Transfer the displaced water into a graduated cylinder
Measure the volume of water, which is the volume of irregular solid
Density
Density is the mass per unit volume of a substance.
Uses of density:
The following are uses of density;
- Indentify materials
- Determine the purity of a material
- Choose light gases for filling balloons
- Finding the volume of a substance
Measuring density
Using the formula
Measure the mass and volume then calculate the density using the formula
The table gives densities of some common substances;
| Substance | Density | |
| gcm-3 | kgm-3 | |
| Water | 1.0 | 1000 |
| Mercury | 13.6 | 13600 |
| Kerosene | 0.8 | 800 |
| Hydrogen | 0.000089 | 0.089 |
| Glass | 2.5 | 2500 |
| Lead | 11.3 | 11300 |
| 1ron | 7.86 | 7860 |
| Ice | 0.92 | 920 |
| Silver | 10.5 | 1050 |
Measuring density of a liquid
b) Using a beaker:
Measure the mass m1 of a clean dry beaker using a balance.
Measure a known volume, V of the liquid using a measuring cylinder and transfer it to the beaker.
Measure the mass m2 of the beaker with the liquid.
Calculate the mass of the liquid m = m2 – m1.
Density of the liquid is then calculated as follows
b) Using a density bottle:
Measure the mass, m1 of a clean density bottle with its stopper.
Fill the density bottle with water and replace the stopper.
Measure the mass, m2 of the density bottle and water.
Calculate the mass of water = m2 – m1.
Determine the volume of water which is the volume of density bottle = m2 – m1 since the density of water = 1gcm-3.
Fill the density bottle with a liquid and replace the stopper.
Measure the mass, m3 of the density bottle and the liquid.
Calculate the mass of a liquid = m3 – m1 and density of a liquid,
The SI unit of density is kg/m3 or g/cm3
Density of Regular Solid Object
The density of regular object can be found easily.
For example, to measure the density of rectangular block
Procedure:
- Measure the mass, m of the solid
- Measure the volume, v = l x h x b
- Calculate density, ρ
Density of irregular solid Object
The density of irregular object can be obtained by:-
Measuring its mass using a triple beam balance or digital balance
Determining the volume through the displacement (immersion) method
Example
1. A stone has a mass of 50 g. When it is totally immersed in water of volume 60 cm3, the final volume is read 70 cm3. Calculate the density of the stone.
Solution:
Example
1. A stone has a mass of 50 g. When it is totally immersed in water of volume 60 cm3, the final volume is read 70 cm3. Calculate the density of the stone.
The Table showing Densities of Different Substance
| Substance | Density (g/cm3) | Substance | Density (g/cm3) |
| Aluminium | 2.7 | Silver | 10.5 |
| Copper | 8.3 | Steel | 7.9 |
| Gold | 19.3 | Cork | 0.2 |
| Iron | 7.8 | Ice | 0.92 |
| Lead | 11.3 | Alcohol | 0.8 |
| Glass | 2.5 | Milk | 1.03 |
| Brass | 8.5 | Kerosene | 1.0 |
| Mercury | 13.6 | Fresh water | 1.0 |
| Sea water | 1.03 | Sand | 2.5 |
Density of Liquids
It can be determined by using burette or density bottle by the following steps
Measure the mass of an empty burette or density bottle, m1
Fill the liquid in the burette or density bottle and measure its mass, m2
Calculate the mass of liquid by, m = m2 –m1
Either by graduated cylinder or overflow can Measure volume of liquid, V
Calculate the density of liquid, ρ
Measuring volume of liquids:
Liquids have no definite shape but assume the shapes of the containers in which they are put. The following instruments are used in measuring the volume of liquids and these are measuring cylinder, pipette and burette, conical flasks, beakers, round bottom flasks e.tc.
Measuring cylinders are made of glass or transparent plastic and graduated in cm3 or ml and measuring flasks, pipettes, burettes and beakers are used to transfer known volumes of liquids.
Measuring the volume of an irregular object:
Volumes of irregular solids are measured using the displacement method.
The method works with solids that are not soluble in water i.e. do not absorb water or react with water.
1. Using a measuring cylinder
Partly fill a measuring cylinder with water and note the volume V1 of water.
Tie a stone whose mass m is known with a thread and lower it gently into the cylinder until it is wholly submerged.
Note and record the new volume V2.
Calculate the volume of the stone, V = V2 – V1.
Density can be calculated from,
2. Using density bottle:
Measure the mass, m1 of a clean dry empty density bottle.
Fill the density bottle partly with lead shot and measure the mass, m2.
Fill up the density bottle with water and measure its mass, m3.
Empty the bottle and fill it with water and measure its mass, m4.
Mass of water = (m4 – m1)g.
Mass of lead shots = (m2 – m1)g.
Mass of water added to lead shots = (m3 – m2)g.
Volume bottle = volume of water = (m4 – m1) [since density of water = 1gcm-3].
Volume of water added to lead shots = (m3 – m2).
Volume of lead shot = (m4 – m1) – (m3 – m2)
Density of Granules
It is difficult to determine the density of very small and fine particles such as sand or lead shots. Density bottle is used to determine the density of granules.
Procedures:
a. Find the mass of an empty bottle by a beam balance (m0)
b. Put some sand in the bottle (see diagram (b))
c. Record the mass of the bottle when partly filled with sand (m1)
d. Pour water into the bottle until it is full
e. Record the new mass m2 of the bottle with its contents
f. Record the mass m3 when the density bottle is filled with water only
Calculate the density of granules
1. Given that
Mass of empty density bottle = 4.0 g Mass of density bottle with sand = 94g
Mass of density bottle with sand and water = 110g Mass of density bottle full of water = 70g
Find the density of sand from above readings
Relative Density of a Substance
Relative density is the ratio of density of substance to the density of water.
It has no SI unit.
This shows that how many times a substance is denser than water
Example,
An empty density bottle weighs 20g. When full of water it weighs 70g and when full of liquid it weighs 60g. Calculate
The relative density of the liquid (b) Its density Solution
M0 = 20 g
M1 = 70 g M2 = 60 g
Importance of Measurement
1. It is used in architecture and engineering for designing of bridges, flyovers and other structures
2. It is used in school to determine the number of students
3. Measurement for length are used for fitting clothes in the fashion industry
4. In trade – exact quantities for export or import are to be known
5. Helps to identify the space occupied by substance
6. Helps us to know the rate of working
7. Helps us to identify the size of substance
8. It helps in decision making
Assignment
Where necessary use g = 10 m/s2
1. A silver cylindrical rod has a length of 0.5 m and radius of 0.4 m, find the density of the rod if its mass is 2640 kg. (ANS: Density = 10509 kg/m3)
2. The relative density of some type of wood is 0.8. Find the density of the wood in kg/m3 (ANS: Density = 800 kg/m3)
3. A stone has a mass 112.5 g. When the stone totally immersed in water contained in measuring cylinder displaced water from 50 cm3 to 95 cm3 . Find the density of the stone. (ANS: Density = 2.5 g/cm3)
4. A piece of anthracite has a volume of 15 cm3 and a mass of 27 g. What is its density (a) in g/cm3 (b) in kg/m3 (ANS: a. 1.8 g/cm3 b. 1800 kg/m3)
5. A 30 ml density bottle was filled with kerosene and found to weigh 86 g. If the mass of empty dry bottle was 62 g, find the density of kerosene (D = 0.8 g/cm3)
6. A solid ball has a mass of 50 g and a volume of 20 cm3. What is its density? (ANS: Density = 2.5 g/cm3)