Population Forecasting

Population forecasting is a method to predict or forecast the future population of an area. Design of water supply and sanitation scheme is based on the projected population of a particular city, estimated for the design period. Any underestimated value will make system inadequate for the purpose intended; similarly overestimated value will make it costly. Change in the population of the city over the years occurs and the system should be designed taking into account of the population at the end of the design period. Factors affecting changes in population are given below.

• Increase due to births
• Decrease due to deaths
• Increase/ decrease due to migration
• Increase due to annexation

The present and past population record for the city can be obtained from the census population records. After collecting these population figures, the population at the end of design period is predicted using various methods as suitable for that city considering the growth pattern followed by the city. The various population forecasting methods are mentioned below. 

1) Arithmetical Increase Method 

The arithmetical Increase Method is mainly adopted for old and developed towns, where the rate of population growth is nearly constant. Therefore, it is assumed that the rate of growth of the population is constant. It is similar to simple interest calculations. The population predicted by this method is the lowest of all. If it is used for small, average or comparatively new cities, it will give low result than actual value. In this method the average increase in population per decade is calculated from the past census reports. This increase is added to the present population to find out the population of the next decade. 

Hence, 

i.e. rate of change of population with respect to time is constant. 

Therefore, Population after n^th decade will be

Pn = Po + n x̄ 

where, 

    Po - last known population 

    Pn - population (predicted) after 'n' number of decades 

    n - number of decades between Po and Pn 

    x̄ - the rate of population growth 

Example Question

Predict the population for the year 2021, 2031 and 2041 from the following population data. 

Year	1961	1971	1981	1991	2001	2011
Population	8,58,545	10,15,672	12,01,553	16,91,538,	20,77,820,	25,85,862

Solution

Year	Population	Increment
1961	858545	-
1971	1015672	157127
1981	1201553	185881
1991	1691538	489985
2001	2077820	386282
2011	2585862	508042
	Average increment, x̄	345463

     Population after n^th decade is Pn = Po + n x̄ 

     Population in year 2021 is, P₂₀₂₁ = 2585862 + 345463 x 1 

                                                                       = 2931325 

Similarly,                                 P₂₀₃₁ = 2585862 + 345463 x 2 

                                                                = 3276788 

                                                   P₂₀₄₁ = 2585862 + 345463 x 3 

                                                               = 3622251 

2) Geometrical Increase Method or Geometrical Progression Method 

This method is adopted for young and developing towns, where the rate of growth of population is proportional to the population at present (i.e., dP/dt ∝ P). Therefore, it is assumed that the percentage increase in population is constant. It is similar to compound interest calculations. The population predicted by this method is the highest of all. Geometric mean increase is used to find out the future increment in population. Since this method gives higher values and hence should be applied for a new industrial town at the beginning of development for only few decades. 

The population at the end of n^th decade ‘Pn’ can be estimated as 

where, 

     Po - last known population 

     Pn - population (predicted) after 'n' number of decades 

     n - number of decades between Po and Pn 

     r - growth rate in percentage 

r could be found as 

a) Arithmetic Mean Method 

b) Geometric Mean Method 

Note: According to Indian standards 'r' should be calculates using geometric mean method.

Example Question

Considering data given in above example predict the population for the year 2021, 2031 and 2041 using geometrical progression method. Solution

Solution

Year	Population	Increment	Geometrical increase Rate of growth
1961	858545	-
1971	1015672	157127	(157127/858545) = 0.18
1981	1201553	185881	(185881/1015672) = 0.18
1991	1691538	489985	(489985/1201553) = 0.40
2001	2077820	386282	(386282/1691538) = 0.23
2011	2585862	508042	(508042/2077820) = 0.24

By Geometric Mean Method 

 

                                                                 = 0.235 i.e., 23.5% 

Population in year 2021 is, P₂₀₂₁ = 2585862 x (1+ 0.235)¹

                                                                 = 3193540 

Similarly, for year 2031 and 2041 can be calculated  by,

                                            P₂₀₃₁ = 2585862 x (1+ 0.235)²  

                                                      = 3944021 

                                           P₂₀₄₁ = 2585862 x (1+ 0.235)³  

                                                      = 4870866 

3) Incremental Increase Method 

This method is adopted for average sized towns under normal conditions, where the rate of population growth is not constant i.e., either increasing or decreasing. It is a combination of the arithmetic increase method and geometrical increase method. Population predicted by this method lies between the arithmetical increase method and the geometrical increase method. While adopting this method the increase in increment is considered for calculating future population. The incremental increase is determined for each decade from the past population and the average value is added to the present population along with the average rate of increase.

Hence, population after n^th decade is 

where, 

      Po - last known population 

     Pn - population (predicted) after 'n' number of decades 

     n - number of decades between Po and Pn 

     x̄ - mean or average of increase in population 

     ȳ - algebraic mean of incremental increase (an increase of increase) of population

Example Question

Considering data given in the above example, predict the population for the year 2021, 2031 2041 using incremental increase method.

Year	Population	Increase (X)	Incremental Increase (Y)
1961	858545	-	-
1971	1015672	157127	-
1981	1201553	185881	+28754
1991	1691538	489985	+304104
2001	2077820	386282	-103703
2011	2585862	508042	+121760
	Total	1727317	350915
	Average	345463	87729

Population in year 2021 is P₂₀₂₁ = 2585862 + (345463 x 1) + {(1 (1+1))/2} x 87729

                                                                = 3019054

            For year 2031      P2031 = 2585862 + (345463 x 2) + {((2 (2+1)/2)}x 87729

                                                           = 3539975

                                         P2041 = 2585862 + (345463 x 3) + {((3 (3+1)/2)}x 87729

                                                      = 4148625 

4) Decreasing Rate of Growth Method 

Since the rate of increase in population goes on reducing as the cities reach towards saturation, this method is suitable. In this method, the average decrease in the percentage increase is worked out and is then subtracted from the latest percentage increase for each successive decade. This method is applicable only when the rate of growth shows a downward trend. 

Decreasing Rate of Growth Method Formula 

where, 

     Pn - population at required decade 

     P(n-1) - population at previous decade (predicted or available) 

     r (n-1) - growth rate at previous decade 

     S - average decrease in growth rate 

Due to the very nature of the formula, which requires population data at the previous decade i.e., P(n-1), this method requires the calculation of population at each successive decade (from the last known decade) instead of directly calculating population at the required decade.

Example Question

Considering data given in the above example, predict the population for the year 2021, 2031 2041 using incremental increase method. 

Year	Population	Increase in population	Growth rate (r) (%)	Decrease in Growth rate (%)
1961	858545	-	-	-
1971	1015672	157127	18	-
1981	1201553	185881	18	0
1991	1691538	489985	40	-22
2001	2077820	386282	23	17
2011	2585862	508042	24	-1

Average of decrease in growth rate 

                       S = (0-22+17-1)/4 

                          = -1.5 S 

                          = 0.015% 

By using the equation, 

                                                                      = 3206081 

(Here r₍₂₀₃₁₎ is directly found as (24 - 0.015) i.e., r₍₂₀₂₁₎ - S, which equals to 23.985.

                                                                    = 3974579 

                         r₍₂₀₄₁₎ = 23.985 - 0.015

                                    = 23.97

                                                                      = 4926689

5) Graphical Method 

In this method, the population of last few decades are correctly plotted to a suitable scale on a graph. The graph is plotted from the available data between time and population, the curve is then smoothly extended up to the desired year. It is to be done by an experienced person and is almost always prone to error. As per the graph shown in Fig.1, the population up to the year 2001 is known and the population of the year 2021 can be found by smoothly extending the graph.

Fig.1 Graphical Method of Population Forecasting

6) Comparative Graphical Method 

Cities of similar conditions and characteristics are selected which have grown in similar fashion in the past and their graph is plotted and then mean graph is also plotted. This method gives quite satisfactory results. In this method, the population of a town is predicted by comparing it with a similar town. The advantage of this method is that the future population can be predicted from the present population even in the absence of some of the past census report.

Example Question

Let the population of a new city ‘X’ be given for decades 1970, 1980, 1990 and 2000 were 32000, 38000, 43000 and 50000 respectively. The cities A, B, C and D were developed in similar conditions as that of city X. It is required to estimate the population of the city X in the years 2010 and 2020. The population of cities A, B, C and D of different decades were given below.

1. City A was 50000, 62000, 72000 and 87000 in 1960, 1972, 1980 and 1990 respectively.
2. City B was 50000, 58000, 69000 and 76000 in 1962, 1970, 1981 and 1988 respectively.
3. City C was 50000, 56500, 64000 and 70000 in 1964, 1970, 1980 and 1988 respectively.
4. City D was 50000, 40000, 58000 and 62000 in 1961, 1973, 1982 and 1989 respectively. 

Population curves for the cities A, B, C, D and X were plotted. Then an average mean curve is also plotted by dotted line as shown in the Fig.2. The population curve X is extended beyond 50000 matching with the dotted mean curve. From the curve the populations obtained for city X are 58000 and 68000 in year 2010 and 2020.

Fig.2 Comparative Graphical Method

7) Master Plan Method 

The big and metropolitan cities are generally not developed in haphazard manner, but are planned and regulated by local bodies according to master plan. The master plan is prepared for next 25 to 30 years for the city. According to the master plan, the city is divided into various zones such as residence, commerce and industry. The population densities are fixed for various zones in the master plan. From this population density total water demand and wastewater generation for that zone can be worked out. So by this method it is very easy to access precisely the design population.

8) The Ratio Method or Apportionment Method

In this method, the city’s census population record is expressed as the percentage of the population of the whole country, in order to do so, the local population and the country’s population for last 4 - 5 decades is obtained from the census records. The ratios of local population to national population is worked out a graph is then plotted between those ratios and time and extended up to the design period and then ratio is multiplied by expected national population at the end of design period. This method does not take into consideration abnormalities in local areas.

9) The Logistic Curve Method

This method is given by P.F. Verhulst. This method is mathematical solution for logistic curve. This method is used when the growth rate of population due to births, deaths and migrations takes place under normal situation and it is not subjected to any extraordinary changes like epidemic, war, earth quake or any natural disaster etc. the population follow the growth curve characteristics of living things within limited space and economic opportunity. If the population of a city is plotted with respect to time, the curve so obtained under normal conditions is look liked ‘S’ shape curve and is known as logistic curve.

Fig. 3 Logistic Curve

In Fig. 3, the curve shows an early growth AB at an increasing rate i.e. geometric growth or log growth, dP/dt ∝𝑃, the transitional middle curve BD follows arithmetic increase i.e. dP/dt = constant and later growth DE the rate of change of population is proportional to difference between saturation population and existing population, i.e. dP/dt ∝ (Ps-P). Verhaulst has put forward a mathematical solution for this logistic curve AE.

The population at any time t from the start is given by

where,

      P_S = Saturation population

      P = Population at any time ‘t’ from start point

     P_O =Population at the start point of the curve

P_O, P₁, P₂ are population at times t₀, t₁, t₂ and t₂ = 2 t₁.

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Percapita Demand (q) and Coincident Draft

It is the annual average amount of daily water required by one person and it includes the domestic use, industrial and commercial use, public use, wastes, thefts etc.

Mathematically;

A city’s total annual water demand can be calculated using the above formula if the per capita demand is known or assumed. Per capita demand depends on a variety of factors and varies with consumer’s living conditions and the size and type of industries that have developed or may be developed in the region. For an average Indian city, as recommended by I.S. 1172-1983, per capita demand (q) can be assumed as in the following table.

Table 1 Rate of Demand for Various Uses

Sl. No.	Type of Use	Rate of Demand in lpcd (Litre per Capita per Day)
1	Domestic Use	200
2	Industrial Use	50
3	Commercial Use	20
4	Civil or Public Use	10
5	Waste and Theft, etc.	55
	Total	335

(As per IS 1172-1983, the domestic consumption in India accounts for 135 lpcd (liters/capita/day) without full flushing system. The value is 200 lpcd with full flushing system as indicated in the Table 1.) 

Multiplying this value of 335 liters/person/day by the projected population at the end of the planning period gives the city’s annual average daily water demand. Multiply this by 365 to get your annual water requirement in liters.

Factors Affecting Water Demand or Percapita Demand 

The average annual water demand (per capita demand) varies greatly from city to city. It is generally 100-360 liters/person/day by Indian standards. These variations in total water use in different cities and communities depend on a variety of factors that need to be thoroughly investigated and analyzed before determining the per capita requirements for planning and design purposes. Total water demand is affected by following factors.

1) Size of the City 

Demand increases with size of city. Larger cities generally have a higher per capita demand than smaller cities. Large cities require large amounts of water to maintain a clean and healthy environment. Similarly, large cities generally require more commercial and industrial activity and require more water. The wealthy living in air-cooled homes can also increase city water use.

The population indirectly affects the size of a city. Because even small cities can have high water consumption if they are fully industrialized or have industries that require huge amounts of water or are inhabited by wealthy people. On average, the per capita demand in Indian cities varies by population, as shown in the table below.

Table 2 Variation in Per Capita Demand (q) with population in India

Sl. No.	Population	Per Capita Demand in lpcd (Litre per Capita per Day)
1	Less than 20,000	110
2	20,000 - 50,000	110 - 150
3	50,000 - 200,000	150 - 240
4	200,000 - 500,000	240 - 275
5	500,000 - 1,000,000	275 - 335
6	Over 1,000,000	335 - 350

2) Climatic Conditions

At hotter and dry places, the consumption of water is generally more, because more of bathing, clearing, air-coolers, air-conditioning, lawns, gardens, roofs etc. are involved. Similarly, in extremely cold countries, more water may be consumed, because the people may keep their taps open to avoid freezing of pipes and there may be more leakage from pipe joints since metals contract with cold.

3) Types of Gentry and Habits of People

Rich and upper class communities generally consume more water due to their affluent living standards. Middle-class communities consume average amounts of water, while poor slum-dwellers consume very little. Thus, water consumption is directly dependent on the consumer’s economic status.

4) Industrial and Commercial Activities

The pressure of industrial and commercial activities at a particular place increase the water consumption by large amount. Many industries require very large amounts of water (much more than households need), which greatly increases the demand for water. As mentioned earlier, the demand for industrial water is not directly related to population or city size, but generally, there is more industries in big cities, increasing the per capita demand in big cities. However, for well-planned and zoned cities, estimating industrial and commercial needs separately can help to predict water needs more accurately.

5) Quality of Water Supplies

If the quality and taste of the supplied water is good, it will be consumed more, because in that case, people will not use other sources such as private wells, hand pumps etc. Similarly, certain industries such as boiler feeds etc., which require standard quality waters will not develop their own supplies and will use public supplies, provided the supplied water is up to their required standards.

6) Pressure in the Distribution Systems

If the pressure in the distribution pipes is high and sufficient to make the water reach at 3^rd or even 4^th storage, water consumption shall be definitely more. This water consumption increases because of two reasons.

• People living in upper storage will use water freely as compared to the case when water is available scarcely to them.
• The losses and waste due to leakage are considerably increased if their pressure is high. For example, if the pressure increase from 20 m head of water (i.e. 200 kN/m²) to 30 m head of water (i.e. 300 kN/m²), the losses may go up by 20 to 30 percent.

7) Development of Sewerage Facilities

The water consumption will be more, if the city is provided with ‘flush system’ and shall be less if the old ‘conservation system’ of latrines is adopted.

8) System of Supply

Water may be supplied either continuously for all 24 hours of the day or may be supplied only for peak period during morning and evening. The second system, i.e. intermittent supplies, may lead to some saving in water consumption due to losses occurring for lesser time and a more vigilant use of water by the consumers. 

Water may be supplied continuously for 24 hours a day or only during morning and evening peak hours. Supplying the water only during the peak hour (morning and evening) can lead to saving in water consumption due to losses occurring in a shorter time and consumers paying more attention to their water consumption. However, in many locations, intermittent delivery fails to provide greater savings than continuous delivery for the following reasons. 

• In intermittent supply systems, water is generally stored by the consumer in tanks, barrels, utensils, etc. for the time it is not being supplied. This water, even if not used, is discarded when fresh supplies are restored. This greatly increases rejections and losses. 
• People usually tend to leave the faucet open during off-hours so that they know when the supply is restored which leads to waste.

9) Cost of Water

If the water rates are high, lesser quantity may be consumed by the people. This may not lead to large savings as the affluent and rich people are little affected by such policies.

10) Policy of Metering and Method of Charging

When the supplies are metered, people use only that much of water as much is required by them. Although metered supplies are preferred because of lesser wastage, they generally lead to lesser water consumption by poor and low income group, leading to unhygienic conditions. Water tax is generally charged in two different ways. 

• On the basis of meter reading (meters fitted at the head of the individual house connections and recording the volume of water consumed). 
• On the basis of a certain fixed monthly flat rate. 

In the second case, i.e. when the delivery is not counted and the fee is fixed, people think that they only need to pay a fixed amount regardless of how much water they use, so generally doesn’t save water. Therefore, they generally consume water and on multiple occasions, their taps are left unclosed. All this leads to a lot of waste and a lot of water consumption. Moreover, meters put unnecessary hindrances to the flow, resulting in loss of pressure and increased cost of pumping. Meters are also liable to be stolen and the cost of installing, repairing and reading the meters is generally high. 

Factors Affecting Losses and Waste 

The various factors on which losses depend and the measure to control them are given below. 

• Water tight joints 
• System of supply 
• Unauthorized connections

Fluctuations in Rate of Demand

Average Daily Per Capita Demand (q)

If this average demand is supplied at all the times, it will not be sufficient to meet the fluctuations. The variations in water demand is listed below.

1) Seasonal Variation

The demand peaks during summer. Fire breakouts are generally more in summer, increasing demand. So, there is seasonal variation.

2) Daily Variation

It depends on the activity. People draw out more water on Sundays and Festival days, thus increasing demand on these days.

3) Hourly variations

These are very important as they have a wide range. During active household working hours i.e. from six to ten in the morning and four to eight in the evening, the bulk of the daily requirement is taken. During other hours the requirement is negligible. Moreover, if a fire breaks out, a huge quantity of water is required to be supplied during short duration, necessitating the need for a maximum rate of hourly supply.

Fig. 1 Variation of Water Demand with respect to Time

So, an adequate quantity of water must be available to meet the peak demand. To meet all the fluctuations, the supply pipes, service reservoirs and distribution pipes must be properly proportioned. The water is supplied by pumping directly and the pumps and distribution system must be designed to meet the peak demand. The effect of monthly variation influences the design of storage reservoirs and the hourly variations influences the design of pumps and service reservoirs. As the population decreases, the fluctuation rate increases.

Variation in Demand 

Smaller towns have more variation in the demand. The shorter the period of draft, the greater is the departure from the mean. 

(A) Maximum Daily Consumption 

                                  Maximum daily consumption = 1.8 x Avg. daily consumption 

                                                                                                 =1.8 q 

(B) Maximum hourly Consumption  

This is taken as 150% of its average. 

Maximum hourly consumption of maximum daily Peak Demand 

                                                                                  = 1.5 x Maximum daily consumption

Coincident Draft 

For general community purposes, the total draft is not taken as the sum of maximum hourly demand and fire-demand, but is taken as sum of maximum daily demand and fire demand or the maximum hourly demand, whichever is more. The maximum daily demand when added to the fire demand is known as the ‘Coincident Draft’.

Example Question

A water supply scheme has to be designed for a city having a population of 1,00,000. Estimate the important kinds of drafts which may be required to be recorded for an average water consumption of 250 lpcd. Also record the required capacities of the major components of the proposed water works system for the city using a river as the source of supply. Assume suitable data.

Solution

(i) Average daily draft

(per capita average consumption in litre/person/day) x population

                                 Average daily draft = 250 x 1,00,000 litres/day

                                                                         = 250 x 10⁵ litres/day

                                                                         = 25 MLD

(ii) Maximum daily draft 

It maybe assumed as 180% of annual average daily draft

                            

                                                                                           = 45 MLD

(iii) Maximum hourly draft of the maximum day

It may be assumed as 270 percent of annual average

                                                                                                                             = 67.5 MLD

(iv) Fire flow may be worked out from

                                                                              = 41733 litre/min

where P = population in thousands

                                                                                     = 61 MLD

              Coincident draft = maximum daily draft +fire draft

                                                 = 45 +61

                                                 =106 MLD

which is greater than the maximum hourly draft of 67.5 MLD

Hence ok.

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Composition and Resolution of Vectors

The process of finding a single vector which will have the same effect as a set of vectors acting on a body is known as composition of vectors. The resolution of vectors is exactly the opposite process of composition i.e., it is the process of finding two or more vectors which will have the same effect as that of a single vector acting on the body.

Parallelogram Law of Vectors

The parallelogram law of vectors enables us to determine the single vector called resultant vector which can replace the two vectors acting at a point with the same effect as that of the two vectors. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc.

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig.1 Representation of Parallelogram Law of Vectors

In the Fig. 1, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

Triangle Law of Vectors

Referring to Fig. 1. (b), it can be observed that the resultant AD may be obtained by constructing the triangle ABD. Line AB is drawn to represent F1 and BD to represent F2. Then AD should represent the resultant of F1 and F2.

Thus we have derived the triangle law of forces from the fundamental law of parallelogram. The Triangle Law of Forces (vectors) may be stated as ‘if two forces (vectors) acting on a body are represented one after another by the sides of a triangle, their resultant is represented by the closing side of the triangle taken from the first point to the last point.’

Polygon Law of Forces (Vectors)

If more than two forces (vectors) are acting on a body, two forces (vectors) at a line can be combined by the triangle law and finally resultant of all forces (vectors) acting on the body may be obtained.

Fig. 2 Representation of Polygon Law of Vectors

A system of four concurrent forces acting on a body are shown in Fig. 2. AB represents F1 and BC represent F2. Hence according to triangle law of forces AC represents the resultant of F1and F2, say R1. If CD is drawn to represent F3, then from the triangle law of forces AD represents the resultant of R1 and F3. In other words, AD represents the resultant of F1, F2 and F3. Let it be called as R2.

Similarly, the logic can be extended to conclude that AE represents the resultant of F1, F2, F3 and F4. The resultant R is represented by the closing line of the polygon ABCDE in the direction from A to E. Thus we have derived the polygon law of the forces (vectors) and it may be stated as ‘if a number of concurrent forces (vectors) acting simultaneously on a body are represented in magnitude and direction by the sides of a polygon, taken in an order, then the resultant is represented in magnitude and direction by the closing side of the polygon, taken from the first point to the last point’.

Graphical (Vector) Method for the Resultant Force

It is another name for finding out the magnitude and direction of the resultant force by the polygon law of forces. It is done as discussed below.

1) Construction of space diagram (position diagram)

It means the construction of a diagram showing the various forces (or loads) along with their magnitude and lines of action.

2) Use of Bow’s notations

All the forces in the space diagram are named by using the Bow’s notations. It is a convenient method in which every force (or load) is named by two capital letters, placed on its either side in the space diagram.

Fig. 3 Denoting a Force by Bow’s Notation

3) Construction of vector diagram (force diagram)

It means the construction of a diagram starting from a convenient point and then go on adding all the forces by vector addition one by one (keeping in view the directions of the forces) to some suitable scale. Now the closing side of the polygon, taken in opposite order, will give the magnitude of the resultant force (to the scale) and its direction.

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Basic Laws of Engineering Mechanics

The structure of engineering mechanics rests on relatively few basic laws. They are given below.

1) Newton’s Laws of Motion

2) Newton’s Law of Gravitation

3) Principle of Transmissibility of Forces

4) Parallelogram Law of Forces

5) Principles of Physical Independence of Forces

6) Principles of Superposition

1) Newton’s Laws of Motion

i) Newton’s First Law of Motion

Newton’s first law states that ‘everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external agency acting on it’. This leads to the definition of force as ‘force is an external agency which changes or tends to change the state of rest or uniform linear motion of the body’.

ii) Newton’s Second Law of Motion

Magnitude of force is defined by Newton’s second law. It states that ‘the rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of the force acting on it’. As the rate of change of velocity is acceleration and the product of mass and velocity is momentum we can derive expression for the force as given below.

From Newton’s second law of motion,

                                            Force ∝ rate of change of momentum

                                            Force ∝ rate of change of (mass × velocity)

Since mass do not change,

                                           Force ∝ mass × rate of change of velocity

                                                      ∝ mass × acceleration

                                                   F ∝ m × a

                                                   F = k × m × a

where 'F' is the force, 'm' is the mass, 'a' is the acceleration and 'k' is the constant of proportionality.

In all the systems, unit of force is so selected that the constant of the proportionality becomes unity. For example, in S.I. system, unit of force is Newton, which is defined as the force that is required to move one kilogram (kg) mass at an acceleration of 1 m/sec².

                                 ∴ One newton = 1 kg mass × 1 m/sec²

   Thus k = 1

                                                     F = m × a

ii) Newton’s Third Law of Motion

Newton’s first law gave definition of the force and second law gave basis for quantifying the force. Newton’s third law states that ‘for every action there is an equal and opposite reaction’.

Consider the two bodies in contact with each other. Let one body apply a force F on another. According to this law the second body develops a reactive force R which is equal in magnitude to force F and acts in the line same as F but in the opposite direction. Fig.1 shows the action of a ball on the floor and the reaction of floor to this action. In Fig. 2 the action of a ladder on the wall and the floor and the reactions from the wall and the floor are shown.

Fig. 1 Action of a Ball on the Floor and the Reaction of Floor

Fig. 2 Action of a Ladder on the Wall and the Floor and the Reactions from the Wall and the Floor

2) Newton’s Law of Gravitation

It states that everybody attracts the other body. ‘The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them’. Thus the force of attraction between the bodies of mass m1 and mass m2 at distance ‘d’ between them as shown in Fig. 3 is

where G is the constant of proportionality and is known as constant of gravitation.

Fig. 3 Force of Attraction between Two Bodies

From above equation,

It has been proved by experiments that the value of G = 6.673 × 10^–11 Nm²/kg². Thus if two bodies one of mass 10 kg and the other of 5 kg are at a distance of 1 m, they exert a force 

on each other.

Similarly, 1 kg-mass on earth surface experiences a force of

Since, mass of earth = 5.96504 × 10²⁴ kg and radius of earth = 6371 × 10³ m. This force of attraction is always directed towards the centre of earth. In common usage the force exerted by earth on a body is known as weight of the body. Thus weight of 1 kg-mass on/near earth surface is 9.80665 N, which is approximated as 9.81 N for all practical problems. Compared to this force the force exerted by two bodies on each other is negligible. Thus in statics

• Weight of a body W = mg
• Its direction is towards the centre of the earth, in other words, vertically downward. 
• The force of attraction between the other two objects on the earth is negligible.

3) Principle of Transmissibility of Forces

According to this law ‘the state of rest or motion of the rigid body is unaltered, if a force acting on the body is replaced by another force of the same magnitude and direction but acting anywhere on the body along the line of action of the replaced force’.

Let F be the force acting on a rigid body at point A as shown in Fig. 4. According to this law, this force has the same effect on the state of body as the force F applied at point B, where AB is in the line of force F.

Fig. 4 Representation of Principle of Transmissibility of Forces

In using law of transmissibility it should be carefully noted that it is applicable only if the body can be treated as rigid. Hence if we are interested in the study of internal forces developed in a body, the deformation of body is to be considered and hence this law cannot be applied in such studies.

3) Parallelogram Law of Forces

The parallelogram law of forces enables us to determine the single force called resultant force which can replace the two forces acting at a point with the same effect as that of the two forces. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc. 

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig. 5 Representation of Parallelogram Law of Forces

In the Fig. 5, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

4) Principles of Physical Independence of Forces

It states that the action of a force on a body is not affected by the action of any other force on the body.

5) Principles of Superposition of Forces

It states that ‘the net effect of a system of forces on a body is same as the combined of individual forces acting on the body’. Since a system of forces in equilibrium do not have any effect on a rigid body this principle is stated in the following form also: ‘The effect of a given system of forces on a rigid body is not changed by adding or subtracting another system of forces in equilibrium.’

Fig. 6 Representation of Principle of Superposition of Forces

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Scalar and Vector Quantities

Scalar Quantities

The scalar quantities (or sometimes known as scalars) are those quantities which have magnitude only such as length, mass, time, distance, volume, density, temperature, speed, energy work etc.

Vector Quantities

The vector quantities (or sometimes known as vectors) are those quantities which have both magnitude and direction such as force, displacement, velocity, acceleration, momentum etc. Following are the important features of vector quantities.

Representation of a Vector

A vector is represented by a directed line as shown in Fig. 1. It may be noted that the length OA represents the magnitude of the vector OA. The direction of the vector OA is from O (i.e., starting point) to A (i.e., end point). It is also known as vector P.

Fig. 1 Vector OA

Unit Vector

A vector, whose magnitude is unity, is known as unit vector.

Equal Vectors

The vectors, which are parallel to each other and have same direction (i.e., same sense) and equal magnitude are known as equal vectors.

Like Vectors

The vectors, which are parallel to each other and have same sense but unequal magnitude, are known as like vectors.

Addition of Vectors

Consider two vectors PQ and RS, which are required to be added as shown in Fig. 2 (a). Take a point A and draw line AB parallel and equal in magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to vector RS to the same scale. Join AC which will give the required sum of vectors PQ and RS as shown in Fig. 2 (b).

Fig. 2 Method of Addition of Vectors

This method of adding the two vectors is called the Triangle Law of Addition of Vectors. Similarly, if more than two vectors are to be added, the same may be done first by adding the two vectors, and then by adding the third vector to the resultant of the first two and so on. This method of adding more than two vectors is called Polygon Law of Addition of Vectors.

Subtraction of Vectors

Consider two vectors PQ and RS in which the vector RS is required to be subtracted as shown in Fig. 3 (a). Take a point A, and draw line AB parallel and equal in magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to the vector RS, but in opposite direction, to that of the vector RS to the same scale. Join AC, which will give the resultant when the vector PQ is subtracted from vector RS as shown in Fig. 3 (b).

Fig. 3 Method of Subtraction of Vectors

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