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.

LECTURE -1 : CLASS VIII : SCIENCE : CHAPTER 3 : SYNTHETIC FIBRES & PLASTICS

CLASS VIII   |    SCIENCE    |    CHAPTER 3

      notes prepared by subhankar Karmakar

                                                                         

Definition

                                                                                  

FIBRES: A very thin, thread-like strand from which cloth is made, is called a fibre. Fabric means cloth. Fabric is made by weaving or knitting long, twisted threads called 'yarn' made from fibres. The clothes which we wear are made of fabrics. Fabrics are made from fibres obtained from 'natural' or artificial' sources (synthetic sources). Thus, all the fibres can be divided into two groups:

(1) Natural fibres, and (ii) Synthetic fibres.

NATURAL FIBRES:

The fibres obtained from plants and animals are called natural fibres. Cotton, flax, jute, wool and silk are natural fibres. Cotton, flax and jute fibres come from plants whereas wool and silk come from animals.

SYNTHETIC FIBRES: 

The synthetic fibres are made by human beings. Rayon, nylon, polyester and acrylic are synthetic fibres.

FIBRES ARE MADE OF POLYMERS: 

POLYMER: A polymer is a 'very big molecule' formed by the combination of a large number of small molecules.

The small molecules (of chemical compounds) which join together to form a polymer are called 'monomers'. 

The monomers which make a polymer may all be of the same compound' or of 'two different compounds'.

So, a polymer is made of many small 'repeating units' (of chemical compounds) called monomers.

Polymers are of two types :

Natural polymers and Synthetic polymers. 

NATURAL POLYMERS:

Cotton, wool and silk are natural polymers. For example, cotton fibre is made of a natural polymer called cellulose. Cellulose is a polymer which is made up of a large number of small glucose molecules (or glucose units) joined one after the other. The walls of all the plant cells are made up of cellulose. So, wood contains a large amount of cellulose polymer. Thus, polymers occur in nature too. 

SYNTHETIC POLYMERS:

Nylon, polyester, acrylic, polythene, polyvinyl chloride (PVC), bakelite, and melamine are synthetic polymers (or man-made polymers). For example, nylon fibre is made up of nylon polymer in which two different types of molecules (or monomer units) are combined alternately to form long chains.

                                                                                  

SYNTHETIC FIBRES: 

a. PRODUCTION OF SYNTHETIC FIBRES: 

The man-made fibres produced from chemical substances are called synthetic fibres, Synthetic fibres are made in industry by the chemical process called 'polymerisation'. A synthetic fibre is a long chain of small units joined together. Each small unit is a chemical compound (called organic compound). Many, many such small units join together one after the other to form a very large single unit called polymer. It is this man made polymer which forms synthetic fibres. Thus, a synthetic fibre is a polymer made from the molecules of a monomer (or sometimes two monomers) joined together to form very long chains. Synthetic fibres are also known as man-made fibres or artificial fibres.

b. TYPES OF SYNTHETIC FIBRES:

Depending upon the type of chemicals used for manufacturing synthetic fibres, there are four major types of synthetic fibres (or man-made fibres). These are :

1. Rayon, 

2. Nylon, 

3. Polyester, and 

4. Acrylic.

c. RAYON IS NOT FULLY SYNTHETIC:

Rayon is a man-made fibre made from a natural material called cellulose (obtained from wood pulp).

WOOD PULP: 

Wood pulp is a soft, wet mass of fibres obtained from wood. Wood pulp contains a large amount of natural polymer called 'cellulose'.

d. FULLY SYNTHETIC FIBRES:

Nylon, polyester and acrylic are fully synthetic fibres which do not require a natural material (like cellulose) for their manufacture. These fully synthetic fibres are prepared by a number of processes by using raw materials (or chemical compounds) of petroleum origin, called petrochemicals.

RAYON & ITS CHARACTERISTICS:

Rayon is often regarded as artificial silk. It is a man-made fibre prepared from a natural raw material (called cellulose) by chemical treatment. The cellulose required for making rayon is obtained from 'wood pulp'. So, we can also say that is obtained by the chemical treatment of wood pulp (which contains cellulose). 

• PRODUCTION OF RAYON: 

Rayon is produced as follows:

(i) Wood pulp is dissolved in an alkaline solution (sodium hydroxide solution) to form a sticky liquid called 'viscose'.

(ii) Viscose is forced to pass through the tiny holes of a metal cylinder (called spinneret) into a solution of sulphuric acid when a silk like thread of rayon is formed.

• RAYON IS NOT FULLY SYNTHETIC FIBRE:

Since rayon is made from naturally occurring polymer (cellulose) present in wood pulp, therefore, rayon is neither a fully synthetic fibre nor a fully natural fibre. It is a semi-synthetic fibre. Rayon is different from truly synthetic fibres because it is obtained from a natural material (wood pulp).

• RAYON, THE ARTIFICIAL SILK:

Although rayon is obtained from a natural resource called wood pulp, yet it is said to be a man-made fibre. This is because it is obtained by the chemical treatment of wood pulp in factories. Rayon fibre is chemically identical to cotton but it has shine like silk. Since rayon resembles silk in appearance, therefore, rayon is also called artificial silk. 

• ADVANTAGES OF RAYON:

Rayon is cheaper than natural silk and can be woven like silk fibres. Rayon can also be dyed in a variety of colours.

• USES OF RAYON:

1. Rayon is used in textile industry for making clothing like sarees, blouses, dresses, socks, etc.

2. Rayon (mixed with cotton) is used to make furnishings such as bed-sheets, curtains, blankets, etc.

3. Rayon (mixed with wool) is used to make carpets.

4. Rayon is used in medical field for making bandages and surgical dressings.

5.  Rayon is used in tyre industry for the manufacture of tyre cord.

NYLON & ITS CHARACTERISTICS:

Nylon is a synthetic fibre. In fact, nylon is the first fully synthetic fibre made by man without using any natural raw materials (from plants or animals). It was made in the year 1931. 

• SOURCE OF NYLON: 

The chemical compounds (or monomers) used in making nylon are now obtained from petroleum products called petrochemicals. It is made up of the repeating units of a chemical called an 'amide'. So, nylon is a polyamide (which is a polymer). 

The name NYLON comes from the fact that it was developed in New York (NY) and London (LON). 

Nylon is a thermoplastic polymer (which can be melted by heating). Molten nylon is forced through the tiny holes in a spinneret to make nylon fibres (or nylon threads), or cast into desired shapes.

• PROPERTIES OF NYLON: 

Some of the important properties of nylon fibres are as follows: 

(i) Nylon fibres are very strong fairly elastic, lightweight and lustrous. 

(ii) Nylon fibres absorb very little water, so clothes made of nylon are easy to wash and dry. 

(iii) Nylon is wrinkle resistant. 

(iv) Nylon fibres have high abrasion resistance (high wear and tear resistance), so they are very durable (long lasting). 

(v) Nylon is not attacked by moths and ordinary chemicals.

Due to all these properties, nylon fibres have become very popular for making clothes.

• USES OF NYLON:

1. Nylon is used for making textiles (fabrics) like sarees, shirts, neck-ties, tights, socks and other garments.

2. Nylon is used in making curtains, sleeping bags and tents.

3. Nylon is used in making ropes, car seat belts, fishing nets, tyre cord, strings for sports rackets and musical instruments, bristles for toothbrushes and paint brushes. 

4. Nylon is used for making parachutes and ropes for rock climbing. 

5. Nylon is used as a plastic for making machine parts.

"All these uses of nylon are due to the high strength of nylon it is actually stronger than a steel wire of similar thickness."