LECTURE -4 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

PLASTICS:

A Plastic is a synthetic material which can be moulded into desired shape when soft and then hardened to produce a durable article. Plastics are also polymers. Plastics are made from petroleum products called petrochemicals. 

Some of the examples of plastics are: Polythene, polyvinyl chloride (PVC), bakelite, melamine and Teflon. Nylon is also a plastic.

Polythene: 

It is a plastic obtained by the polymerization of a chemical compound known as ethene. Polythene is tough and durable. Polythene is used in making Polythene bags, waterproof plastic sheets, bottles, buckets and dustbins. Polythene is also used for packaging.

Polyvinyl chloride (PVC): 

It is a strong and hard plastic. It is not as flexible as polythene. PVC is used for making insulation of electric wires, pipes, garden hoses, rain coats, seat covers etc.

Bakelite: 

It is a very hard and tough plastic. Bakelite is a poor conductor of heat and electricity. Bakelite is used for making the handles of various cooking utensils. 

Bakelite is used for making handles cooking utensils because

1. It is a poor conductor of heat and

2. It does not become soft on getting heated as it is a thermosetting plastic.

Bakelite is also used for making electrical fittings such as electric switches, plugs and sockets etc. because

1. It does not conduct electricity and

2. It does not become soft on getting heated.

Melamine: 

It is a plastic which can tolerate heat better than other plastics and resists fire. 

Melamine is used for making floor tiles, unbreakable kitchenwares, ashtrays and fire resistant fabrics. 

Melamine is a fire resistant plastic, hence, the uniforms of fire man heavy coating of melamine plastic to make them fire resistant. 

Special plastic cookwares made of melamine is used in microwave oven for cooking food. In microwave oven the heat cooks the food but does not affect the plastic vessel.

Teflon:

Teflon is a special plastic on which oil and water do not stick. Oil and water do not stick on Teflon plastic because it has a slippery surface. Teflon also withstands high temperature. Teflon is used for giving non stick coating on cookwares like non stick frying pans. Teflon is also used for making soles of electric irons.

TYPES OF PLASTICS:

There are two types of plastics

1. Thermoplastics

2. Thermosetting plastics

a. Thermoplastics:

A Plastic which can be softened repeatedly by heating and can be moulded in two different shapes again and again, is called a thermoplastic. 

Properties of thermoplastics:

Thermoplastics are flexible so they can be bent easily without breaking. Thermoplastics are also known as thermosoftening plastics. Some of the examples of thermoplastics are: polythene and polyvinyl chloride (PVC).

Uses of thermoplastics:

Thermoplastics are used for making those articles which do not get too hot and are flexible. Thermoplastics are used for making insulation of electric wires and cables, various types of plastic containers like plastic bottles, plastic jars, combs, toys, plastic bags, raincoats, seat covers, bristles of brushes, packaging materials and chairs. 

Thermoplastics are used for making the insulation of electric wires because

1. They do not conduct electricity, and

2. They are flexible. 

b. Thermosetting plastics:

A plastic which once set, does not become soft on heating and cannot be moulded a second time, is called a thermosetting plastic. Once Seth in a given shape and solidified, a thermosetting plastic cannot be re-softened or remoulded. Thus, and article or object made of thermosetting plastic will retain its original shape permanently, even on heating. Thermosetting plastics are also known as thermosets.

Some of the examples of thermosetting plastics are: bakelite and melamine. 

Properties of thermosetting plastics:

Most of the thermosetting plastics hard and rigid. Thermosetting plastics are not flexible. Therefore, thermosetting plastics cannot bend. When an article made of thermosetting plastic is forced to bend, it breaks. Thermosetting plastics do not become soft on heating. Thermosetting plastics are used for making those articles which may get too hot during use and are hard and rigid so that they do not bend at all. 

Uses of thermosetting plastics:

Thermosetting plastics are used for making handles of cooking utensils, plates, cups, floor tiles, electrical fittings like electrical switches, plugs and sockets, ballpoint pens and telephone instruments. 

Reasons of using thermosetting plastics for making the handles of cooking utensils:

Thermosetting plastics are used for making the handles of cooking utensils because

a. They do not softened on getting heated, and

b. They are poor conductors of heat.

Reasons of using thermosetting plastics for making electrical fittings:

Thermosetting plastics are used for making electrical fittings such as electric switches, plugs and sockets because

a. They do not become soft on getting heated, and

b. They do not conduct electricity.

Plastics use for making toothbrush:

The handle and bristles of a toothbrush cannot be made of the same plastics, because the handle of a toothbrush has to be hard and rigid whereas the bristles of a toothbrush have to be soft and flexible. This means that the handle of a toothbrush should be made of thermosetting plastic whereas its bristles should be made of thermoplastic.

Structures of thermoplastics and thermosetting plastics:

In thermoplastics long polymer chains are not cross linked with one another but in thermosetting plastics the long polymer chains are cross linked with one another.

As in thermoplastic long polymer chains are not cross-linked, hence, the individual polymer chains can slide over one another and thermoplastic material become soft and ultimately melts.

As in thermosetting plastics, long polymer chains are cross linked, these cross-links prevent the displacement of individual polymer chains on being heated. Due to this, thermosetting plastics do not become soft on heating once they have been set into to a particular shape.

Useful properties of plastics:

Some of the most useful properties of plastics are as follows. 

a. Plastics are chemically unreactive. 

Plastics do not react with air and water. Therefore, plastics are resistant to corrosion. Due to this property, plastic containers are used to store various kinds of materials, including many chemicals.

b. Plastics are bad conductors of heat and electricity:

Plastics neither conducts heat nor electricity. They are used as insulators. This properties make it the material for making of handles of the cooking utensils and electrictical fittings like switches, plugs and sockets as well as it is used as the covering of the electrical wires. 

c. Plastics can be moulded into different shapes.

Since plastics can be very easily moulded, they are used to make a large variety of articles like buckets, mugs, furniture, bags, sheets, slippers, electrical fittings, toys, combs, toothbrushes etc.

d. Plastics are quite cheap and easily made:

Plastics a generally cheaper than metals and wood. Plastics production is also easy comparing to the extraction and purification of metals. These property made plastics as the first choice for making many of the household as well as industrial articles. It replaces metals in most of the cases. 

e. Plastics are light, strong and durable:

As the plastics has low density, they are lighter than metals. Plastics has good strength and they are durable too. Due to their low cost, high strength, easy availability, lightweight, long durability and corrosion resistant properties, plastics are now-a-days widely used for the making of most of the articles we use. 

BIODEGRADABLE & NON-BIODEGRADABLE MATERIALS:

A material which gets decomposed through natural processes such as action of bacteria and other other natural factors are called biodegradable material. They are environment friendly. Plant and animal wastes, paper, cotton, cloth, wool, jute, wood are some examples of biodegradable materials. 

A material which is not easily decomposed by natural processes are called non-biodegradable material. They are not environment friendly. They directly or indirectly pollute nature. Plastics, glass, metal foils, aluminium cans are some examples of non-biodegradable materials. 

PLASTICS POLLUTE ENVIRONMENT:

Plastic materials have a bad effect on the environment. The excessive use of plastic materials affect our environment in the follwing ways. 

a. Plastics are non-biodegradable, so, the waste plastic articles keep on accumulating in the surroundings and pollute the environment badly.

b. The waste plastic articles thrown in the drain choked them and causes them to overflow and create an unhygienic condition. 

c. Plastics may be eaten by the animals, which causes permanent damage to the animals. 

d. If plastic materials are burnt they produce toxic gases which pollutes air and ultimately become the causes of many diseases.

The ways to prevent environment pollution from excessive plastic wastes:

Plastics are very useful materials but use of plastic articles is not good for the environment. Therefore, some steps should be taken to save the environment from plastic waste. They are as follows:

1. We should try to reduce or minimise the use of plastic by using other alternative materials. Like instead of using Polythene bags we can use jute or paper bags.

2. We should not throw Polythene bags, wrappers of chips, biscuits or other eatables in water bodies, on the roads, in parks or picnic places. We should use dustbin for solid waste.

3. We should reuse the plastic containers which come with jams, pickles, oils and other packed food. 

4. Plastic wastes should be recycled. All the plastic waste in in the different places should be collected and sent for recycling to plastic making factories. In plastic factories, the waste plastic articles are melted and used to make new plastic articles. All the recycled plastic product are given certain colours, so that buyers can understand they are recycled plastic product and they don't use them for stories of food.

5. We should follow 3R principle. It means Reduce, Reuse and Recycle for plastic products.

LECTURE: 3 : CLASS XI: PHYSICS : UNITS & MEASUREMENTS

CLASS XI   |    PHYSICS    |    CHAPTER 2

      notes prepared by subhankar Karmakar

                                                                                  

Accuracy and precision:

Accuracy: it refers to the closeness of a measurement to the true value of the the physical quantity. It indicates the relative freedom from errors. As we reduce the errors, the measurement becomes more accurate.

Precision: it refers to the resolution or the limit to which the quantity is measured. Precision is determined by the least count of the measuring instrument. The smaller the list count, greater is the precision. 

Errors in measurement: 

The error in a measurement is equal to the difference between the true value and the measured value of the quantity.

      Error = true value - measured value

Different types of errors:

1. Constant errors

2. Systematic errors

     a. Instrumental errors

     b. Imperfection in experimental technique

     c. Personal errors

     d. Errors due to external causes

3. Random errors

4. Least count error

5. Gross errors or mistakes

1. Constant errors: the errors which affect each observation by the same amount are called constant errors. 

2. Systematic errors: the errors which tend to occur in one direction, either positive or negative, are called systematic errors. Systematic errors are classified as follows:

     a. Instrumental errors: These errors occur due to the inbuilt defect of the measuring instrument. 

     b. Imperfections in experimental technique: These errors are due to the limitations of the experimental arrangement. 

     c. Personal errors: These errors arise due to to individual's bias, lack of proper setting of apparatus or individual's carelessness in taking observations without observing proper precautions.

    d. Errors due to external causes: These errors arise due to the the change in external conditions.

3. Random errors: The errors which occur irregularly and at random, in magnitude and direction, are called random errors.

4. Least count error: This error is due to the limitation imposed by the the least count of the measuring instrument.

5. Gross errors or mistakes: These errors are due to either carelessness of the person or due to improper adjustment of the apparatus.

Different types of error measurement: 

a. True value of a physical quantity: arithmetic mean of all the measurements can be taken as the true value of the measured quantity. 

If a₁, a₂, a₃, a₄, a₅ ...... aₙ be the n measured values of a physical quantity, then is true value 

aₘₑₐₙ or ā = (a₁+ a₂ + a₃ + a₄ + a₅ +......+ aₙ )/n

b. Absolute Error: The magnitude of the difference between the true value of the quantity measured and the individual measured value is called absolute error. 

|∆a₁| = |ā - a₁|

|∆a₂| = |ā - a₂|

|∆a₃| = |ā - a₃|

............................

|∆aₙ| = |ā - aₙ|

c. Mean or final absolute error:

The arithmetic mean of the positive magnitudes of all the absolute errors is called mean absolute error. It is given by

∆ā = (|∆a₁|+ |∆a₂| + |∆a₃|  +......+ |∆aₙ| )/n

The final result of the measure of a physical quantity can be expressed as

    a = ā ± ∆ā

d. Relative error:

The ratio of the mean absolute error to the true value of the measured quantity is called relative error. 

Relative error, δa = ∆ā /ā

e. Percentage error:

The relative error expressed in percent is called percentage error. 

Percentage Error = (∆ā/ā) x 100%

COMBINATION OF ERRORS:

a. Error in the sum of two quantities:

Let ∆A and ∆B be the absolute errors in the two quantities A and B respectively. Then, 

Measured value of A = A ± ∆A

Measured value of B = B ± ∆B

Consider the sum, Z = A + B

The error ∆Z in Z is then given by 

Z ± ∆Z = (A ± ∆A) + (B ± ∆B)

            = (A + B) ± (∆A + ∆B)

            = Z ± (∆A + ∆B)

∴ ∆Z = (∆A + ∆B)

b. Error in the difference of two quantities

Consider the difference, Z = A - B

The error ∆Z in Z is then given by 

Z ± ∆Z = (A ± ∆A) - (B ± ∆B)

            = (A - B) ± ∆A ∓ ∆B

            = Z ± ∆A ∓ ∆B

For error ∆Z to be maximum, ∆A and ∆B must have the same sign, therefore

∴ ∆Z = (∆A + ∆B)

c. Error in the product of two quantities:

Consider the product , Z = AB

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)(B ± ∆B)

            = AB ± A∆B ± B∆A ± ∆A. ∆B

Dividing LHS by Z and RHS by AB [∵ Z = AB]

1 ± ∆Z/Z = 1 ± ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)

As the last term is very small, it can be neglected. 

 ± ∆Z/Z =  ± (∆B/B + ∆A/A)

∴ ∆Z/Z =  (∆B/B + ∆A/A)

d. Error in the division or quotient

Consider the product , Z = A/B

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)/(B ± ∆B)

            = A(1 ± ∆A/A)/{B(1 ± ∆B/B)}

            = (A/B)(1 ± ∆A/A)(1 ± ∆B/B)⁻¹

            = Z(1 ± ∆A/A)(1 ∓ ∆B/B)

 [∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]

Dividing both sides by Z 

1 ± ∆Z/Z = 1 ∓ ∆B/B ± ∆A/A ± (∆A/A)(∆B/B)

As the last term is very small, it can be neglected. 

∴ ∆Z/Z =  (∆B/B + ∆A/A)

e. 1. Error in the power of a quantity:

Consider. Z = Aⁿ

The error ∆Z in Z is given by

Z ± ∆Z = (A ± ∆A)ⁿ = Aⁿ (1 ± ∆A/A)ⁿ

                                  = Z (1 ± n∆A/A)

[∵ (1 + x)⁻¹ ≃ 1 + nx when x <<1]

Dividing both sides by Z, we get

1 ± ∆Z/Z = 1 ± n(∆A/A)

or  ∆Z/Z = n(∆A/A)

   2. General rule:

   Consider. Z = Pᵃ Qᵇ / Rᶜ

Then ∆Z/Z = a(∆P/P) + b(∆Q/Q) + c(∆R/R)

Numericals :

Q1. The length of a rod as measured in an experiment was found to be 2.48 m, 2.46 m, 2.49 m, 2.50 m, 2.48 m. Find the (a) average length, (b) the absolute error in each observation and (c) the percentage error.

Soln. (a) Average length 

= (2.48 + 2.46 + 2.49 + 2.50 + 2.48)/5

= 12.41/5 = 2.482 = 2.48

∴ true length, ā = 2.48 m

(b) The absolute errors in different measurements are:

|∆a₁| = |ā - a₁| = |2.48 - 2.48| = 0.00 m

|∆a₂| = |ā - a₂| = |2.48 - 2.46| = 0.02 m

|∆a₃| = |ā - a₃| = |2.48 - 2.49| = 0.01 m

|∆a₄| = |ā - a₄| = |2.48 - 2.50| = 0.02 m

|∆a₅| = |ā - a₅| = |2.48 - 2.48| = 0.00 m

(c) the absolute error, |∆ā| 

= (0.00 + 0.02 + 0.01+ 0.02 + 0.00)/5

= 0.01 m

∴ correct length, ā ± |∆ā| = 2.48 ± 0.01 m

∴ percentage error = (0.01/2.48)x 100%

                                 = 0.40%

Q2. In successive measurements, the readings of the period of oscillation of a simple pendulum were found to be 2.63 s, 2.56 s, 2.42 s, 2.71 s and 2.80 s in an experiment. Calculate (a) mean value of the period of oscillation

(b) absolute error in each measurement

(c) mean absolute error

(d) relative error

(e) percentage error and

(f) Express the result in proper form.

Soln. (a) mean period of oscillation 

= (2.63 + 2.56 + 2.42 + 2.71 + 2.80)/5

= 13.12/5 = 2.624 s ≃ 2.62 s

(b) absolute errors in different measurement,

|∆a₁| = |ā - a₁| = |2.62 - 2.63| = 0.01 s

|∆a₂| = |ā - a₂| = |2.62 - 2.56| = 0.06 s

|∆a₃| = |ā - a₃| = |2.62 - 2.42| = 0.20 s

|∆a₄| = |ā - a₄| = |2.62 - 2.71| = 0.09 s

|∆a₅| = |ā - a₅| = |2.62 - 2.80| = 0.18 s

(c) mean absolute error, |∆ā|

= (0.01 + 0.06 + 0.20 + 0.09 + 0.18)/5

= 0.11 s

(d) relative error δā = |∆ā|/ā

= 0.11/2.62 = 0.04

(e) percentage error = 0.04 x 100% = 4%

(f) in terms of absolute error, 

(2.62 ± 0.11) s

In terms of percentage error, 

(2.62 ± 4%) s.

Homework:

Q3. In an experiment, refractive index of glass was observed to be  1.45, 1.56, 1.54, 1.44, 1.54 and 1.53. Calculate (a) mean value of refractive index, (b) mean absolute error, (c) fractional error aur relative error, (d) percentage error, 

(e) express the result in terms of absolute error and percentage error.

Q4. In an experiment to measure focal length of a concave mirror, the value of focal length in successive observations turns out to be 17.3 cm, 17.8 cm, 18.3 cm, 18.2 cm, 17.9 cm and 18.0 cm. Calculate the mean absolute error and percentage error. Also, express the result in a proper way. 

Numericals on combination of errors:

Q5. Two resistances R₁ = 100 ± 3 Ω and R₂ = 200 ± 4 Ω are connected in series. What is their equivalent resistance?

Q6. Two different masses are determined as (23.7 ± 0.5) g and (17.6 ± 0.3) g. What is the sum of their masses?

Q7. The initial and final temperatures of a water bath are (18 ± 0.5)°C and (40 ± 0.3)°C. What is the rise in temperature of the bath?

Q8. The resistance R =V/I, where V = 100 ± 5 V and I = 10 ± 0.2 A. Find the percentage error in R.

Q9. The percentage errors in the measurement of mass and speed are 2% and 3% respectively. How much will be the maximum error in the estimate of kinetic energy obtained by measuring mass and speed?

Q10. The length, breadth and height of a rectangular block of wood were measured to be :

l = 12.13 ± 0.02 cm;

b = 8.16 ± 0.01 cm and

h = 3.46 ± 0.01 cm

Determine the percentage error in the volume of the block.

Q11. The period of oscillation of a simple pendulum is T = 2π √(L/g). Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?

Q12. A physical quantity X is given by 

X = (a²b³)/(c√d). If the percentage errors of measurement in a, b, c and d are 4%, 2%, 3% and 1% respectively, then calculate the percentage error in X.

    

LECTURE -3 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

ADVANTAGES OF SYNTHETIC FIBRES

Important properties and advantages of synthetic fibres as follows:

1. Synthetic fibres are very strong whereas natural fibres like cotton, wool and silk have low strength. 

2. Synthetic fibres are more durable. Synthetic fibres have high resistance to abrasion and hence, the clothes made of synthetic fibres are very durable whereas natural fibres like cotton, wool and silk have low abrasion resistance due to which the clothes made of natural fibres are not much durable. They do not last long.

3. Synthetic fibres absorb very little water and hence, the clothes made of synthetic fibres dry up quickly. On the other hand, natural fibres like cotton, wool and silk absorb a lot of water. So, the clothes made a natural fibres do not dry up quickly.

4. Synthetic fibres are wrinkle resistant and hence, class made of synthetic fibres do not get crumpled easily during washing or wear. They keep permanent creases. But, natural fibres like cotton, wool and silk are not wrinkle resistant. So, the clothes made of natural fibres get crumpled easily during washing and wear.

5. Synthetic fibres are quiet lightweight. Whereas natural fibres are comparatively heavy.

6. Synthetic fibres are extremely fine. So the fabrics made from synthetic fibres have a very smooth texture. But, natural fibres are not so fine. Therefore, the fabrics made from natural fibres do not have a very smooth texture. 

7. Synthetic fibres are not attacked by moths. But natural fibres are damaged by moths.

8. Synthetic fibres do not shrink. So, the clothes made of synthetic fibre retain their original size even after washing. On the other hand, natural fibres shrink after washing.

9. Synthetic fibres are less expensive and readily available as compared to natural fibres.

10. Clothes made from synthetic fibres are easier to maintain as compared to those made from natural fibres.

11. The manufacturing of fully synthetic fibres is helping in the conservation of forests. As the synthetic fibres are made from petrochemical products, so no trees have to be cut down for making them. On the other hand, semi synthetic fibres like Rayon are made from wood pulp request cutting down of forest trees.

DISADVANTAGES OF SYNTHETIC FIBRES:

There are several disadvantages of synthetic fibres. They are as follows:

1. Synthetic fibres always melt on heating. Therefore, if a person is wearing clothes made of synthetic fibres and his clothes catch fire accidentally, then the synthetic fibres of clothes melt and stick to the body of the person causing severe burns. So, it is quite safe to wear clothes made of natural fibres while working in the kitchen or in a science laboratory.

2. The clothes made of synthetic fibres are not suitable for wearing during hot summer weather. As the synthetic fibres are extremely fine so the clothes made of synthetic fibres do not have sufficient pores for the sweat to come out, evaporate and coo our body. Due to this, clothes made of synthetic fibres make us feel hot and uncomfortable during summer. 

LECTURE: 2 : CLASS XI: PHYSICS : UNITS & MEASUREMENTS

CLASS XI   |    PHYSICS    |    CHAPTER 2

      notes prepared by Subhankar Karmakar

Conversion table from degree to radian:

a.   1° = 1.745 x 10⁻² rad

b.   1' = 2.91 x 10⁻⁴ rad

c.   1" = 4.85 x 10⁻⁶ rad

 

Q1. The moon is observed from two diametrically opposite points A and B on the earth. The angle θ subtended at the moon by the two directions of observation is 1°54'. Given the diameter of the earth to be 1.276 x 10⁷ m, compute the distance of the Moon from the Earth. 

 

Soln. Here the parallactic angle 

θ = 1°54' = 1.745 x 10⁻² + 54 x 2.91 x 10⁻⁴ rad

                = 3.32 x 10⁻² rad.

Here, b = AB = 1.276 x 10⁷ m

The distance of the Moon from the Earth,

S = b/θ = 1.276 x 10⁷/3.32 x 10⁻²

             = 3.84 x 10⁸ m

 

Q2. The angular diameter of the sun is 1920". If the distance of the sun from the earth is 1.5 x 10¹¹ m, what is the linear diameter of the sun?

 

Soln. Distance of the sun from the earth

          S = 1.5 x 10¹¹ m 

          Angular diameter of the sun

          θ = 1920" = 1920 x 4.85 x 10⁻⁶ rad

                           = 9312 x 10⁻⁶ rad

Linear diameter of the sun

          D = Sθ = 1.5 x 10¹¹ x 9312 x 10⁻⁶ m

                           = 13968 x 10⁵ m

                           = 1.4 x 10⁶ km

 

 

• DIMENSION OF A PHYSICAL QUANTITY:

 

All the derived physical quantities can be expressed in terms of some combination of the seven fundamental or base quantities. We call these fundamental quantities as the seven dimensions of the world, which are denoted with square brackets [ ]. 

 

• Dimension of length = [L]

• Dimension of mass = [M]

• Dimension of time = [T]

• Dimension of electric current = [A]

• Dimension of thermodynamic temperature = [K]

• Dimension of luminous intensity = [cd]

• Dimension of amount of substance = [mol]

 

The dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent that quantity completely. 

For example, 

Density = Mass/Volume = Mass/ (Length x breadth x height) 

Dimensions of density = [M]/([L] x [L] x [L])

= [M¹L⁻³T⁰]

 

·        Area = [M⁰L²T⁰] = m²

·        Volume = [M⁰L³T⁰] = m³

·        Density = [M¹L⁻³T⁰] = kg m⁻³

·        Speed or Velocity = [M⁰L¹T⁻¹] = m/s

·        Acceleration = [M⁰L¹T⁻²] = m/s²

 

DIFFERENT TYPES OF VARIABLES AND CONSTANTS: 

 

There are two types of variables

1. Dimensional variables: 

 

The physical quantities which possess dimensions and have variable values are cal dimensional variables. For example, area, volume, velocity, force, power, energy etc.

 

2. Dimensionless variables: 

 

The physical quantities which have no dimensions but have variable values are called dimensionless variables. For example, angle, specific gravity, strain etc.

 

There are two types of constants:

 

1. Dimensional constants: 

 

The physical quantities which possess dimensions and have constant values are called dimensional constants. For examples, gravitational constant, Planck's constant, electrostatic constant etc.

 

2. Dimensionless constants: 

 

The constant quantities having no dimensions are called dimensionless constants. For example, π, e etc. 

 

Application of dimensional analysis: 

 

The method of studying a physical phenomenon on the basis of dimensions is called dimensional analysis. 

 

Following are the three main uses of dimensional analysis: 

 

1. To convert a physical quantity from one system of units to other. 

2. To check the correctness of a given physical relation.

3. To derive a relationship between different physical quantities.

 

1. Conversion of one system of units to other:

 

As the magnitude of physical quantities remain same and does not depend upon our choices of units, therefore, 

                   Q = n₁u₁ = n₂u₂

where Q is the magnitude of the physical quantity, u₁ and u₂ are the units of measurement of that quantity and n₁ and n₂ are the corresponding numerical values. 

u₁ = M₁^aL₁^bT₁^c

u₂ = M₂^aL₂^b T₂^c

∴ n₁[M₁^aL₁^bT₁^c] = n₂[M₂^aL₂^b T₂^c]

⟹  n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c   

 

Q1. Convert 1 Newton into dyne.

 

Soln. Newton is the SI unit of force and dyne the CGS unit of force. Dimensional formula of force is M¹L¹T⁻²

∴ a = 1, b = 1, c = -2

In SI system; 

M₁ = 1 kg = 1000 g

L₁ = 1 m = 100 cm

T₁ = 1 s and n₁ = 1 (Newton)

In CGS system;

M₂ = 1 g ; L₂ = 1 cm ; T₂ = 1 s

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

         = 1 x [1000/1]¹ x [100/1]¹ x [1/1]⁻²

        = 1 x 10³ x 10²

        = 10⁵ 

∴ 1 N = 10⁵ dyne

 

Q2. Convert 1 erg into Joule.

 

Soln. Erg is CGS unit of energy whereas joule is SI unit of energy. Dimensional formula of energy is M¹L²T⁻².

∴ a = 1, b = 2, c = -2

In CGS system;

M₁ = 1 g ; L₁ = 1 cm ; T₁ = 1 s ; n₁ = 1

In SI system; 

M₂ = 1 kg = 1000 g

L₂ = 1 m = 100 cm

T₂ = 1 s and n₂ = ?

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

         = 1 x [1/1000]¹ x [1/100]² x [1/1]⁻²

        = 1 x 10⁻³ x 10⁻⁴

        = 10⁻⁷

∴ 1 erg =  10⁻⁷ N

 

Q3. The density of Mercury is 13.6 g/cm³ in CGS system. Find its value in SI system.

 

Soln. The dimensional formula of density is

M¹L⁻³T⁰

∴ a = 1, b = - 3, c = 0

In CGS system;

M₁ = 1 g ; L₁ = 1 cm ; T₁ = 1 s ; n₁ = 13.6

In SI system; 

M₂ = 1 kg = 1000 g

L₂ = 1 m = 100 cm

T₂ = 1 s and n₂ = ?

∴ n₂ = n₁ [M₁/M₂]^a [L₁/L₂]^b [T₁/T₂]^c 

        = 13.6 x [1/1000]¹ x [1/100]⁻³ x [1/1]⁰

        = 13.6 x 10⁻³ ⁺ ⁽⁻²⁾⁽⁻³⁾ 

        = 13.6 x 10³

∴ The density of Mercury in SI unit is 13.6 x 10³ kg/m³

 

Q4. If the value of atmospheric pressure is 10⁶ dyne / cm², find its value in SI units.

 

Q5. If the value of universal gravitational constant in SI unit is 6.6 x 10⁻¹¹ N m² kg⁻², then find its value in CGS unit.

 

 

 

2. CHECKING THE DIMENSIONAL CONSISTENCY OF EQUATIONS:

 

• Principle of homogeneity of dimensions:

 

According to this principle, a physical equation will be dimensionally correct if the dimensions of all the terms occurring on both side of the equation are the same. 

 

Q6. Check the dimensional accuracy of the equation of motion s = ut + ½at².

 

Soln. Dimensions of different terms are

[s] = [L],

[ut] = [LT⁻¹] x [T] = [L],

[½at²] =  [LT⁻²] x [T²] = [L]

 

As all the terms on both sides of the equation have the same dimensions, show the given equation is dimensionally correct. 

 

Q7. Check the correctness of the equation

       FS = ½mv² - ½mu²

       Where F is a force acting on a body of mass m and S is the distance moved by the body when its velocity changes from u to v.

 

Soln. 

    [FS] = [M¹L¹T⁻²][L] = [M¹L²T⁻²]

    [½mv²] = [M][LT⁻¹]² = [M¹L²T⁻²]

    [½mu²] = [M][LT⁻¹]² = [M¹L²T⁻²]

Since the dimensions if all the terms in the given equation are same, hence the given equation is dimensionally correct. 

 

Q8. The Vander Waal's equation for a gas is

       ( P + a/V²)(V - b) = RT

Determine the dimensions of a and b. Hence write the SI units of a and b.

 

Soln. Since the dimensionally similar quantities can be added or subtracted, therefore, 

[P] = [a/V²] 

⟹ [a] = [ PV²] = [ M¹L⁻¹T⁻²] [L³]² = [M¹L⁵T⁻²]

Also, [b] = [V] = [L³]

∴ The SI unit of a is kg m⁵/s² and that of b is m³

 

3. DEDUCING RELATION AMONG THE PHYSICAL QUANTITIES:

 

By making use of the homogeneity off dimensions, we can derive an expression for a physical quantity if we know the various factors on which it depends

 

Q9. Derive an expression for the centripetal force F acting on a particle of mass m moving with velocity v in a circle of radius r.

 

Soln. Centripetal force F depends upon mass M, velocity V and radius r.

Let F ∝ mᵃ vᵇ rᶜ 

∴ F = K mᵃ vᵇ rᶜ --------(1)

where K is a dimensionless constant. 

Dimensions of the various quantities are

 [m] = [M],  [v] = [LT⁻¹],  [r] = [L]

Writing the dimensions of various quantities in equation 1, we get

 [M¹L¹T⁻²] = 1 [M]ᵃ [LT⁻¹]ᵇ [L]ᶜ

⟹  [M¹L¹T⁻²] = [M]ᵃ [L]ᵇ ⁺ ᶜ [T]⁻ᵇ

Comparing the dimensions of similar quantities on both sides, we get

      a = 1

      b + c = 1 and 

      - 2 = - b ⟹ b = 2

∴ c = 1 - b = 1 - 2 = - 1

∴ a = 1, b = 2 and c = - 1

∴ F = K m v² r⁻¹ = K mv²/r

This is the required expression for the centripetal force.

 

Q10. The velocity  v of water waves depends on the wavelength λ, density of water ρ, and the acceleration due to gravity g. Did use by the method of dimensions the relationship between these quantities. 

 

Soln. Let  v = K λᵃ ρᵇ gᶜ -------(1)

where  K = a dimensionless is constant

Dimensions of the various quantities are

[v] = [LT⁻¹],  [λ] = [L],  [ρ] =  [M¹L⁻³], [g] = [LT⁻²]

Substituting these dimensions in equation (1), we get

[LT⁻¹] = [L]ᵃ  [M¹L⁻³]ᵇ [LT⁻²]ᶜ

∴ [M⁰ L¹T⁻¹] = [Mᵇ Lᵃ⁻³ᵇ⁺ᶜ T⁻²ᶜ]

Equating the powers of M, L and T on both sides, 

b= 0 ; a - 3b + c =1 ; - 2c = - 1

On solving,  a= ½ ; b = 0, c = ½

∴ v = K √(λg)

 

Q11. The frequency "ν" off vibration of a a stretched string depends up on:

a. Its length l

b. Its mass per unit length m and

c. The tension T in the string.

Obtain dimensionally an expression for frequency ν.

 

Soln. Let the frequency of vibration of the string be given by

       ν = K lᵃ Tᵇ mᶜ ----------(1)

where K is a dimensionless constant.

Dimension of the various quantities are

[ν] = [T⁻¹] ; [l] = [L]; [T] =  [M¹L¹T⁻²] ; [m] = [M¹L⁻¹]

Substituting this dimensions in equation 1,  we get

 [T⁻¹] = [L]ᵃ  [M¹L¹T⁻²]ᵇ  [M¹L⁻¹]ᶜ

⟹ M⁰ L⁰ T⁻¹ = Mᵇ ⁺ ᶜ Lᵃ ⁺ ᵇ ⁻ ᶜ T⁻²ᵇ

Equating the dimensions of M, L and T , we get

b + c = 0;  a + b - c = 0; - 2b = - 1

On solving, a = - 1, b = ½, c = - ½

∴ ν = K l⁻¹√(T/m) = (K/l)√(T/m)

 

Q12. The period of vibration of A tuning fork depends on the length l of its prong, density d and Young's modulus Y of its material. Deduce an expression for the period of vibration on the basis of dimensions.

 

LECTURE -2 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

POLYESTER:

Polyester is a synthetic fibre in which the polymer units are linked by ester groups. Terylene is a popular polyester fibre. The chemical compounds used in making polyester fibres are made from petroleum products called petrochemicals. Both Nylon and polyester are thermoplastic polymers. Therefore, most of the properties of polyester fibres are similar to those of Nylon. There are some differences between nylon and polyester fibres. 

Properties of polyester fibres:

1. Polyester fibre is stronger than nylon fibres.

2. Polyester fibres are also softer than nylon fibres. 

3. Polyester fabric is strong, wrinkle resistant, easy to wash and dry, not attacked by moths and ordinary chemicals and has high abrasion resistance. These properties makes it suitable for making dress material. 

4. Polyester can be blended with natural fibres like cotton or wool and known as polycot (terrycot)  and polywool (terrywool) respectively. 

Uses of polyester fibres:

1. The most important use of polyester is in making fabrics for sarees, dress materials and curtains.

2. Polyester is used for making sails of sailboats. Polyester sails are light, strong, do not stretch and and do not rot in contact with water. 

3. Polyester is used for making water hoses for fire fighting operations.

4. Polyester is used for making conveyor belts.

PET:

PET is a very familiar form of polyester. PET is the abbreviation of the synthetic polymer called Poly Ethylene Terephthalate. PET can be made into a fibre or a plastic. When used as a fibre PET generally referred to as polyester, while the term PET is usually used for the plastic form. 

Properties of PET: 

1. PET as a plastic is very lightweight. 

2. It is naturally colourless with high transparency. 

3. PET is strong and impact resistant. 

4. PET is shatterproof and hence it is used to make bottles, jars, and utensils.

5. PET bottles are used for fizzy drinks and PET jars are used for sugar, salt, rice etc.

6. It is also used to make thin films.

ACRYLIC:

Acrylic is a synthetic fibre. Acrylic fibre is made from a chemical called acrylonitrile by the process of polymerization. 

Characteristics of acrylic:

1. Acrylic is lightweight, soft and warm with a wool-like feel. 

2. Acrylic retains its shape, resists shrinkage and wrinkles. 

3. It can be dyed very well. 

4. Acrylic fibres are strong and durable. 

5. Acrylic absorbs very little water so it has "quick-dry" quality. 

6. Acrylic fibres are resistant to moths and most chemicals. 

Uses of acrylic fibres:

1. Due to its wool-like feel, acrylic fibre is often used as a substitute for wool. 

2. The wool obtained from natural sources like sheep is quite expensive. Acrylic offers a less expensive alternative to natural wool. So, the clothes made from acrylic are relatively cheaper but more durable than those made from natural wool. 

3. Many of the sweaters which the the people wear in winter, and the shawls and blankets which people use, are actually not made from natural wool, though they appear to be made from wool. They are made from synthetic fibre called acrylic. 

4. Acrylic fibre is used for making sweaters shawls blankets jackets sportswear, socks, furnishing fabrics, carpets and as lining for boots and gloves.