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Students can Download Chapter 5 Trial Balance and Rectification of Errors Notes, Plus One Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Accountancy Notes Chapter 5 Trial Balance and Rectification of Errors

Summary:
Meaning of trial balance:
A statement showing the abstract of the balance (debit/credit) of various accounts in the ledger.

Objectives of trial balance:
The main objectives of preparing the trial balance are:

• To ascertain the arithmetical accuracy of the ledger accounts;
• To help in locating errors; and
• To help in the preparation of the final accounts.

Preparation of trial balance by the balance method:
In this method, the trial balance has three columns. The first column is for the head of the account, the second column for writing the debit balance and the third for the credit balance of each account in the ledger.

Format of a Trial balance
Trial Balance of ………………… as on March 31.2005

It is normally prepared at the end of an accounting year. However, an organisation may prepare a trial balance at the end of any chosen period, which may be monthly, quarterly, half yearly or annually depending upon its requirements.
In order to prepare a trial balance following steps are taken:

• Ascertain the balances of each account in the ledger.
• List each account and place its balance in the debit or credit column as the case may be. (If an account has a zero balance, it may be included in the trial balance with zero in the column for its normal balance).
• Compute the total of debit balances column.
• Compute the total of the credit balances column.
• Verify that the sum of the debit balances equal the sum of credit balances. If they do not tally, it indicates that there are some errors. So one must check the correctness of the balances of all accounts.

It may be noted that all assets expenses and receivables account shall have debit balances whereas all liabilities, revenues and payables accounts shall have credit balances.

Illustrative Trial Balance

Various types of errors:
1. Errors of commission:
Errors caused due to wrong recording of a transaction, wrong totalling, wrong casting, wrong balancing, etc.

2. Errors of omission:
Errors caused due to omission of recording a transaction entirely or partly in the books of account.

3. Errors of principle:
Errors arising due to wrong classification of receipts and payments between revenue and capital receipts and revenue and capital expenditure.

4. Compensating errors:
Two or more errors committed in such a way that they nullify the effect of each other on the debits and credits.

Rectification of errors:
Errors affecting only one account can be rectified by giving an explanatory note or by passing a journal entry. Errors which affect two or more accounts are rectified by passing a journal entry.

Meaning and utility of suspense account:
An account in which the difference in the trial balance is put till such time that errors are located and rectified. It facilitates the preparation of financial statements even when the trial balance does not tally.

Disposal of suspense account:
When all the errors are located and rectified the suspense account stands disposed off.

Students can Download Chapter 4 Bank Reconciliation Statement Notes, Plus One Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Accountancy Notes Chapter 4 Bank Reconciliation Statement

Summary
Bank Reconciliation Satement
A statement prepared to reconcile the bank balance as per cash book with the balance as per passbook or bank statement, by showing the items of difference between the two accounts.

Causes of difference
Timing of recording the transaction Errors made by business or by the bank.

Need for Reconciliation
It is generally experienced that when a comparison is made between the bank balance as shown in the firm’s cash book, the two balances do not tally. Hence, we have to first ascertain the causes of difference thereof and then reflect them in a statement called Bank Reconciliation Statement to reconcile (tally) the two balances.

In order to prepare a bank reconciliation statement we need to have a bank balance as per the cash book and a bank statement as on a particular day along with details of both the books.

If the two balances differ, the entries in both the books are compared and the items on account of which the difference has arisen are ascertained with the respective amounts involved so that the bank reconciliation statement may be prepared.

Proforma of bank reconciliation statement

It can also be prepared with two amount columns one showing additions (+ column) and another showing deduction (- column).

Proforma of bank reconciliation statement (table form)

Correct cash balance
It may happens that some of the receipts or payments are missing from either of the books and errors, if any, need to be rectified. This arises the need to look at the entries/errors recorded in both statements and other information available and compute the correct cash balance before reconciling the statements.

Students can Download Chapter 2 Theory Base of Accounting Notes, Plus One Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Accountancy Notes Chapter 2 Theory Base of Accounting

Summary:
Generally accepted Accounting Principles (GAAP):
GAAP refers to the rules or guidelines adopted for recording and reporting to business transactions in order to bring uniformity in the preparation and presentation of financial statements. These principles are also referred to as concepts and conventions.

From the practicality viewpoint, the various terms such as principles, conventions, modifying principles, assumptions, etc., have been used interchangeably and are referred to as basic accounting concepts.

Systems of Accounting:
There are two systems of recording business transactions, i.e., double entry system and single entry system.

Basis of Accounting:
There are two broad approach of accounting are cash basis and accural basis. Under cash basis transactions are recorded only when cash are received or paid, where as under accural basis, revenue or costs are recognises when they Occur rather than when they are paid.

Accounting Standards:
Accounting standards are written statement of uniform accounting rules and guidelines in practice for preparing the uniform and consistent financial statements.

Students can Download Chapter 1 Introduction to Accounting Notes, Plus One Accountancy Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Accountancy Notes Chapter 1 Introduction to Accounting

Summary
Meaning of Accounting:
Accounting is a process of identifying, measuring, recording the business transactions and communicating thereof the required information to the interested users.

Accounting as a source of information:
Accounting as a source of information system is the process of identifying, measuring, recording and communicating the economic events of an organization to interested users of the information.

Users of accounting information:
Accounting plays a significant role in society by providing information to management at all levels and to those having a direct financial interest in the enterprise, such as present and potential investors and creditors.

Accounting information is also important to those having an indirect financial interests, such as regulatory agencies, tax authorities, customers, labour unions, trade associations, stock exchanges, and others.

Qualitative characteristics of Accounting:
To make accounting information decision-useful, it should possess the following qualitative characteristics.

• Reliability
• Understandability
• Relevance
• Comparability

The objective of accounting:
The primary objectives of accounting are to:

• Maintain records of business;
• Calculate profit or loss;
• Depict the financial position: and
• Make information available to various groups and users.

Role of accounting:
Accounting is not an end in itself. It is a means to an end. It plays the role of a:

• Language of a business
• Historical record
• Current economic reality
• Information system
• Service to users

Students can Download Chapter 8 Excretory Products and their Elimination Notes, Plus One Zoology Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Zoology Notes Chapter 8 Excretory Products and their Elimination

What is excretion?
Excretion is the elimination common nitrogenous wastes. Ammonia, urea and uric acid are the major forms of nitrogenous wastes excreted by the animals.

Ammonotelic.ureotelic and urecotelic animals and their excretion:
1. The process of excreting ammonia is Ammonotelism. eg: Many bony fishes, aquatic amphibians and aquatic insects are ammonotelic.

2. Mammals, many terrestrial amphibians and marine fishes mainly excrete urea and are called ureotelic animals.

3. Reptiles, birds, land snails and insects excrete nitrogenous wastes as uric acid in the form of pellet or paste with a minimum loss of water and are called uricotelic animals.

Where is urea produced?
Ammonia produced by metabolism is converted into urea in the liver and released into the blood which is filtered and excreted out by the kidneys.

Excretion in lower organisms:

Human Excretory System
In humans, the excretory system consists of

1. A pair of kidneys
2. One pair of ureters
3. A urinary bladder
4. A urethra.

Each kidney of an adult human measures 10 – 12 cm in length, 5 – 7 cm in width, 2 – 3 cm in thickness with an average weight of 120 – 170 g.

1. Centre of the inner concave surface of the kidney is a notch called hilum through which ureter, blood vessels and nerves enter.

2. Innerto the hilum is a broad funnel shaped space called the renal pelvis with projections called calyces.

3. Inside the kidney, there are two zones, an outer cortex and an inner medulla.

4. The medulla is divided medullary pyramids projecting into the calyces.

5. The cortex extends in between the medullary pyramids as renal columns called Columns of Bertini.

6. Each kidney has nearly one million complex tubular structures called nephrons, which are the functional units. A diagrammatic representation of a nephron showing blood vessels, duct and tubule Each nephron has two parts -1 The glomerulus & 2 Renal tubule.

7. Glomerulus is a tuft of capillaries formed by the afferent arteriole – a fine branch of renal artery.

8. Blood from the glomerulus is carried away by an efferent arteriole.

9. The renal tubule begins with a double walled cup-like structure calle Bowman’s capsule, which encloses the glomerulus.

10. Glomerulus alongwith Bowman’s capsule, is called the malpighian body or renal corpuscle

11. The tubule continues further to form a highly coiled network proximal convoluted tubule (PCT).

12. A hairpin shaped Henle’s loop is the next part of the tubule which has a descending and an ascending limb.

13. The ascending limb continues as another highly coiled tubular region called distal convoluted tubule (DCT).

14. The DCTs open into a straight tube called collecting duct, many of which open into the renal pelvis through medullary pyramids in the calyces.

15. The Malpighian corpuscle, PCT and DCT of the nephron are situated in the cortical region of the kidney whereas the loop of Henle dips into the medulla.

Types of Nephrons:
1. In majority of nephrons, the loop of Henle is too short and extends only very little into the medulla. Such nephrons are called cortical nephrons.

2. In some of the nephrons, the loop of Henle is very long and runs deep into the medulla.These nephrons are called juxta medullary nephrons. A minute vessel of capillary network runs parallel to the Henle’s loop forming a ‘LT shaped vasa recta. Vasa recta is absent or highly reduced in cortical nephrons.

Urine Formation
1. It involves three main processes namely, glomerular filtration, reabsorption and secretion

2. The first step in urine formation is the filtration of blood, which is carried out by the glomerulus and is called glomerular filtration.

3. 1100 – 1200 ml of blood is filtered by the kidneys per minute. The glomerular capillary blood pressure causes filtration of blood through 3 layers,

4. The epithelial cells of Bowman’s capsule called podocytes which possess minute spaces called filtration slits.

5. Blood is filtered through these membranes, that almost all the constituents of the plasma except the proteins pass onto the lumen of the Bowman’s capsule. It is the process of ultra filtration.

6. The kidneys have efficient mechanism for the regulation of glomerular filtration rate. It is carried out by juxta glomerular apparatus (JGA).JGA is found in the distal convoluted tubule .

7. A fall in GFR can activate the JG cells to release renin which can stimulate the glomerular blood flow and thereby the GFR back to normal.

8. Nearly 99 percent of the filtrate is reabsorbed by the renal tubules. This process is called reabsorption.

9. For example, substances like glucose, amino acids, Na⁺, etc., in the filtrate are reabsorbed actively whereas the nitrogenous wastes and water are absorbed by passive transport.

10. During urine formation, the tubular cells secrete substances like H⁺, K⁺ and ammonia into the filtrate.

11. Tubular secretion is helpful in the maintenance of ionic and acid-base balance of body fluids.

Function Of The Tubules
Proximal Convoluted Ttubule (PCT):
1. PCT consists of simple cuboidal brush border epithelium which increases the surface area for reabsorption.

2. All essential nutrients, and 70 – 80 per cent of electrolytes and water are reabsorbed by this segment.

3. PCT also helps to maintain the pH and ionic balance of the body fluids. It occurs through selective secretion of hydrogen ions, ammonia and potassium ions into the filtrate and by absorption of HCO₃^– from it.

Henle’s Loop
Function:
It helps to maintain high osmolarity of medullary interstitial fluid. The descending limb of loop of Henle is permeable to water but impermeable to electrolytes. This concentrates the filtrate as it moves down.

The ascending limb is impermeable to water but allows transport of electrolytes actively or passively. Therefore, as the concentrated filtrate pass upward, it gets diluted due to the passage of electrolytes to the medullary fluid.

Distal Convoluted Tubule (DCT):
The reabsorption of Na⁺, HCO₃^– and water takes place in this segment. In this segment selective secretion of hydrogen and potassium ions and NH₃ to maintain the pH and sodium-potassium balance in blood.

Collecting Duct:
It extends from the cortex of the kidney to the inner parts of the medulla. Large amounts of water is reabsorbed from this region to produce a concentrated urine. This part maintains pH and ionic balance of blood by the selective secretion of H⁺ and K⁺ ions.

Mechanism Of Concentration Of The Filtrate
The flow of filtrate in the two limbs of Henle’s loop is in opposite directions and thus forms a counter current. The flow of blood through the two limbs of vasa recta is also in a counter current pattern.

The proximity between the Henle’s loop and vasa recta, as well as the counter current in them help in maintaining an increasing osmolarity towards the inner medullary interstitium, i.e. from 300 mOsmolL^-1 in the cortex to about 1200 mOsmolL^-1 in the inner medulla. This gradient is mainly caused by NaCl and urea.

NaCl is transported by the ascending limb of Henle’s loop which is exchanged with the descending limb of vasa recta. NaCl is returned to the interstitium by the ascending portion of vasa recta. Similarly, small amounts of urea enterthe thin segment of the ascending limb of Henle’s loop which is transported back to the interstitium by the collecting tubule.

This transport is facilitated by the counter current mechanism This mechanism helps to maintain a concentration gradient in the medullary interstitium. Presence of such interstitial gradient helps in an easy passage of water from the collecting tubule thereby concentrating the filtrate (urine).

Regulation Of Kidney Function
The functioning of the kidneys is influenced by hypothalamus, JGA and to a certain extent, the heart in the body are activated by changes in blood volume.

How can diuresis is prevented in body?
The decrease of fluid from the body can activate these Osmoreceptors which stimulate the hypothalamus to release antidiuretic hormone (ADH) or vasopressin from the neurohypophysis. ADH facilitates water reabsorption Hence it prevents diuresis.

An increase in body fluid volume switch off the osmoreceptors and suppress the ADH release. ADH can also affect the kidney function by its constrictory effects on blood vessels. This causes an increase in blood pressure. An increase in blood pressure increase the glomerular blood flow and the GFR.

Micturition
It is the voluntary signal from the central nervous system (CNS). This signal is initiated by the stretching of the urinary bladder in response to the stretch receptors on the walls of the bladder.

The urine is slightly acidic (pH – 6.0) and has a characterestic odour. 25 – 30 gm of urea is excreted out per day. Analysis of urine helps to know malfunctioning of the kidney.

Role Of Other Organs In Excretion
Lungs remove large amounts of CO₂ (18 litres/day) and water every day. Liver secretes bile-containing substances like bilirubin, biliverdin, cholesterol, degraded steroid hormones, vitamins and drugs.

The sweat and sebaceous glands in the skin eliminates watery fluid containing NaCl, small amounts of urea, lactic acid, etc. Sebaceous glands eliminates sterols, hydrocarbons and waxes through sebum. This secretion provides a protective oily covering for the skin.

Disorders Of The Excretory System
Uemia:
It leads to the accumulation of urea in blood. It lead to kidney failure. In such patients, urea can be removed by a process called hemodialysis.

Procedure of Dialysis:
Blood drained from a artery is pumped into a dialysing unit after adding an anticoagulant like heparin.The unit contains a coiled cellophane tube surrounded by a fluid (dialysing fluid).

The porous cellophane membrance allows the passage of molecules based on concentration gradient.
As nitrogenous wastes are absent in the dialysing fluid, the cleared blood is pumped back to the body through a vein after adding anti-heparin to it.

Remedy for kidney damage:
Kidney transplantation is the ultimate method in the correction of acute renal failures (kidney failure). A functioning kidney is used in transplantation from a donor to minimise its chances of rejection by the immune system of the host.

Renal calculi:
Stone or insoluble mass of crystallised salts (oxalates, etc.) formed within the kidney.

Glomerulonephritis:
Inflammation of glomeruli of kidney

NCERT SUPPLEMENTARY SYLLABUS
Diabetes Insipidus:
Antidiuretic hormone (ADH) is released from the posterior pituitary, prevents dehydration. It helps in the reabsorption of water by the distal parts of the kidney tubules and prevents diuresis. Deficiency of ADH leads to diabetes insipedus, that means huge amounts of dilute urine is formed followed by intense thirst.

The name itself (diabetes = overflow; insipidus = tasteless) distinguishes it from diabetes .mellitus (mel = honey), in which insulin deficiency causes large amounts of blood sugar to be lost in the urine

Artificial kidney:
Hemodialysis machine is known as the artificial kidney. Hemodialysis is an artificial process of removing toxic substances from the blood in patients of kidney failure.

Students can Download Chapter 14 Oscillations Notes, Plus One Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Physics Notes Chapter 14 Oscillations

Introduction
In this chapter, we study oscillatory motion. The description of an oscillatory motion requires some fundamental concept like period, frequency, displacement, amplitude and phase.

Periodic And Oscillatory Motions
Periodic Motion:
A motion that repeats itself at regular intervals of time is called periodic motion.
Example:

• The orbital motion of planets in the solar system.
• The piston in a steam engine going back and forth

Oscillations or Vibrations:
A body executes to and fro motion at regular intervals of time is called oscillatory (or) vibratory motion.

Note:
(1) When the frequency is small, we call it oscillation. When the frequency is high, we call it vibration.
Period and frequency:
Period (T):
Time taken to complete one oscillation is called period

Frequency (n):
The number of oscillations per second is called frequency.
frequency v = \(\frac{1}{T}\)Hz

Displacement:
Displacement of oscillation means the change of any physical property with time under consideration.

Explanation:
For example, consider the oscillation of block attached to a spring. In this case the displacement of block with time is referred to as displacement.

In the case if oscillation of simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable. The voltage across a capacitor, changing with time in an a.c. circuit is also a displacement variable

Note:
The displacement variable may take both positive and negative values.

Mathematical expression for displacement:
The displacement of a periodic function can be written as

Simple Harmonic Motion (S.H.M.)
Simple harmonic motion is the simplest form of oscillatory motion.

The oscillatory motion is said to be simple harmonic motion if the displacement ‘x’ of the particle from the origin varies with time as

Where
x(t) = displacement x as a function of time t
A = amplitude
ω = angular frequency
(ω t+ Φ) = phase (time-dependent)
Φ = phase constant or initial phase

Graphical Variation of S.H.M.

The above graph shows the graphical representation of x(t) = A coswt with time.
(Initial phase Φ = 0)

Amplitude of S.H.M.
The maximum displacement of S.H.M. from mean position is called the amplitude of S.H.M.
Note:
1.

Two simple harmonic motions having, same w and Φ but different amplitudes A and B as shown in the above figure.

2.

Two simple harmonic motions having the same A and w but different phase angle Φ as shown in the above figure.

Simple Harmonic Motion And Uniform Circular Motion

Question 1.
Show that the projection of uniform circular motion on any diameter of the circle is S.H.M.

Answer:
Consider a particle moving along the circumference of a circle of radius ‘a’ and centre O, with uniform angular velocity w. AB and CD are two mutually perpendicular diameters along X and Y axis. At time t = 0. let the particle be at P₀ so that ∠P₀OB = Φ.

After time Y second, let the particle reach P so that ∠POP₀ = ωt. N is the foot of the perpendicular drawn from P on the diameter CD.

Similarly M is the foot of the perpendicular drawn from P to the diameter AB. When the particle moves along the circumference of the circle, the foot of the perpendicular executes to and fro motion along the diameter CD or AB with O as the mean position. From the right angle triangle O MP, we get
cos (ωt + Φ) = \(\frac{\mathrm{OM}}{\mathrm{OP}}\)
∴ OM = OP cos (ωt + Φ)
X = a cos (ωt + Φ) ………………. (1)
Similarly, we get
sin (ωt + Φ) = \(\frac{y}{a}\) (or)
Y = a sin (ωt + Φ) ……………… (2)
Equation (1) and (2) are similar to equations of S.H.M. The equation(1) and (2) shows that the projection of uniform circular motion on any diameter is S.H.M.
At t = 0, if the particle is at B, then Φ = 0. Then equations (1) and (2) reduce to
x = a cos ωt ………………….. (3)
y = a sin ωt …………………….(4)

Velocity And Acceleration In Simple Harmonic Motion B
Velocity Of S.H.M.
The y displacement of S.H.M. is given by
y = a sin ω t
∴ velocity in y direction

Case – 1
At the mean position y=0, therefore velocity is maximum. The maximum velocity is given by

Case – 2
At the extreme position, y = a
∴ V_minimum = \(\sqrt{a^{2}-a^{2}}\) = 0
So the velocity of a S.H.M. varies between o and wa

Acceleration of S.H.M
We know y = a sin ω t
Velocity v = \(\frac{d y}{d t}\) = a ω cos ω t
Acceleration a = \(\frac{d^{2} y}{d t^{2}}\) = -aω² sin ωt
a = -aω² sin ωt

This equation shows that acceleration of a SHM is directly proportional to the displacement and opposite to the displacement.

Variation of displacement Y with time t:

Variation of velocity (v) with time:

Variation of acceleration with time:

Force Law For Simple Harmonic Motion
According to Newton’s second law of motion F = ma
But a = -ω²y(t)
ie. force acting on the S.H.M. in Y direction
F = -mω²y(t)
(or) F = -ky(t)
Where k= mω²
From the above equation (1), we can take an alternative definition of simple harmonic motion.

Statement:
Simple harmonic motion is the motion executed by a particle subject to a force, which is proportional to the displacement of the particle and is directed towards the mean position.

Energy In Simple Harmonic Motion
A simple harmonically moving particle possesses both potential energy and kinetic energy. Potential energy is due to its displacement against restoring force. Kinetic energy is due to its motion.

Total energy of the S.H.M. is the sum of the kinetic energy and potential energy. Total energy remains a constant throughout its motion.

Expression for Kinetic energy:
Let m be the mass of the particle executing SHM. Let V be the velocity at any instant,

Expression for potential energy:
Potential energy is work required to take the particle against the restoring force.

Work done to displace the particle through a small distance dy, dw = force × displacement
= mω²y × dy [force = ω²y ]. Therefore total work done to take the particle from o to y.


This work done is stored in the particle as its potential energy.

At extreme position y = a

At equilibrium position y = 0
∴ PE = 0
Total energy of a S.H.M.
Total energy = PE + kE

∴ Total energy = maximum KE = maximum PE

Graphical variation of PE, KE and TE of S.H.M.

Some Systems Executing Simple Harmonic Motion
Oscillations due to a spring Hooks Law:
The force acting simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
F α – x (or) F= kx
where k is called spring constant

Period of oscillation of a spring:

Consider a body of mass m attached to a massless spring of spring constant K. The other end of spring is connected to a rigid support as shown in figure. The body is placed on a frictionless horizontal surface.

If the body be displaced towards right through a small distance ‘x’, a restoring force will be developed.

The Simple Pendulum:

Consider a mass m suspended from one end of a string of length L fixed at the other end as shown in figure. Suppose P is the instantaneous position of the pendulum. At this instant its string makes an angle θ with the vertical.

The forces acting on the bob are (1) weight of bob F_g (mg) acting vertically downward. (2) Tension T in the string.

The gravitational force F_g can be divided into a radial component F_gCos θ and tangential component F_gSin θ. The radial component is cancelled by the – tension T. But the tangential component F_gSin θ produces a restoring torque.

Restoring torque τ = – F_g sin θ . L.
τ = -mg sin θ.L …………….. (1)
-ve sign shown that the torque and angular displacement θ are oppositely directed. For rotational motion of bob,.
τ = Iα …………. (2).
Where I is a moment of inertia about the point of suspension and α is angular acceleration. From eq (1) and eq (2).
Iα = -mg sin θ.L
If we assume that the displacement θ is small, sin θ ≈ θ.
∴ Iα = -mg θ.L
Iα + mg θ.L = 0

Damped Simple Harmonic Motion
Periodic oscillations of decreasing amplitude due to the presence of resistive forces of the medium are called damped oscillations.

Question 2.
Derive a differential equation for a damped simple harmonic oscillation.

Answer:
Consider a block of mass ‘m’ connected to one end of a massless spring of spring constant K. The other end of spring is connected to rigid support. The block is connected to a vane through a rod (The vane and rod are massless). The vane is submerged in a liquid.

Let the equilibrium position of block be ‘O’. If this block is moved along downward direction through a distance x, a damping force will be developed on vane due to liquid. This damping force is proportional to velocity of vane ie; damping force F_d α – v (or) F_d = -bv

where b is called damping constant. The value of b depends on the characteristics of the liquid and the vane.

The restoring force on the block due to spring. F_s = -kx. where x is the displacement of the mass from its equilibrium position.
∴ Total force on the block, F = -bv + -kx
ma = -bv + -kx
ma + bv + kx = 0
\(m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=0\)
This is the differential equation of S.H.M.

The motion of damped harmonic oscillator:
The solution of the above differential equation of damped harmonic oscillator is

Case – 1
b = 0 (There is no damping force). In this case, we get

The above result shows that, if there is no damping force (b = 0). The oscillator behaves like a undamped oscillator.

Case – 2
If b is small, the amplitude of the oscillator decreases continuously with time. The motion is approximately a periodic. The Variation of displacement x (t) with time T is shown below.

Case – 3
If damping constant b is large, the amplitude of the oscillator decreases to zero very quickly. The motion is not a periodic motion. The variation of displacement x (t) with time T is shown below.

The Energy variation of damped oscillator:
The energy of an undamped oscillator, E = \(\frac{1}{2}\)kA².
where A is the amplitude of an undamped oscillator. But for the damped oscillator, amplitude = Ae^-bt/2m.

The above equation shows that the energy of a damped oscillator decreases exponentially with time, which is shown below

Note:
Small damping means that the dimensionless ratio
\((b / \sqrt{k m})\) is much less than 1.

Forced oscillations and resonance
Free oscillation:
When a body oscillates in the absence of external forces (eg. friction etc.) the oscillations are said to be free oscillations.

Forced oscillation:
When an external periodic force is applied to a damped harmonic oscillator, the oscillator will vibrate with a constant amplitude and frequency of vibration will be that of the applied periodic force. This type of oscillation is called forced oscillation.

The differential equation of forced oscillator:
Let F(t) = F₀ cosω_dt is an external force applied to a damped oscillator.

The total force acting on the damped oscillator,
F = – bv – kx + F₀ cosω_dt
where -bv is the damping force and -kx is the linear restoring force.
F + bv + kx = F₀ cosω_dt
\(m \frac{d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=F_{0} \cos \omega_{d} t\)
This is the differential equation of an oscillator of mass m on which a periodic force of (angular) frequency ω_d is applied. The oscillator initially oscillates with its natural frequency w. When we apply the external periodic force, the oscillation with the natural frequency die out, and the body oscillates with the (angular) frequency of the external periodic force.

The motion of forced oscillator:
The solution (displacement) of the above differential equation of forced oscillator can be written as
x(t) = Acos(ω_dt + Φ)

where m is the mass of the particle v₀ and x₀ are the velocity and the displacement of the particle at time t = 0, (which is the moment when we apply the periodic force)

Case -1
When b = 0 (damping force = 0) and ω = ω_d,
we get A = \(\frac{F_{0}}{0}\)
A = ∞
This is an ideal case. This case never arises in a real situation as the damping is never perfectly zero.

Case – 2
(Small damping, driving frequency far from natural frequency).
In this ω_db << m(ω² – ω_d²). Hence we can neglect ω_db. Hence amplitude of oscillation can be written as.

Case – 3
(Small damping, driving frequency close to natural frequency).
In this case m(ω² – ω_a²) << ω_db. Hence we can neglect m (ω² – ω_a²). Hence amplitude of oscillation can be written as

This equation shows that the maximum amplitude fora given driving frequency is governed by the driving frequency and damping constant.

Resonant Oscillation
When the frequency of the external periodic force is varied, it is found that the amplitude of the forced vibration increases and reaches a maximum value and then decreases.

The amplitude will be maximum when the frequency of the applied periodic force is equal to the natural frequency of the vibration. Such oscillations are called resonant oscillations and the phenomenon is called resonance.

Graphical variation amplitude with driving frequency:

Examples of resonance:
All mechanical structures have one or more natural frequencies, and if a structure is subjected to a strong external periodic driving force that matches one of these frequencies, the resulting oscillations of the structure may rupture it.

The Tacoma Narrows Bridge at Puget Sound, Washington, USA was opened on July 1, 1940. Four months later winds produced a pulsating resultant force in resonance with the natural frequency of the structure.

This caused a steady increase in the amplitude of oscillations until the bridge collapsed. It is for the same reason the marching soldiers break steps while crossing a bridge. Aircraft designers make sure that none of the natural frequencies at which a wing can oscillate match the frequency of the engines in flight! Earthquakes cause vast devastation.

In an earthquake, short and tall structures remain unaffected while the medium height structures fall down. This happens because the natural frequencies of the short structures happen to be higher and those of taller structures lower than the frequency of the seismic waves.

Students can Download Chapter 7 Body Fluids and Circulation Notes, Plus One Zoology Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Zoology Notes Chapter 7 Body Fluids and Circulation

Blood
Blood is a special connective tissue contains

• Fluid matrix
• Plasma
• Formed elements.

Plasma:
It constitute nearly 55 per cent of the blood. 90 – 92 percent of plasma is water and proteins (Fibrinogen, globulins and albumins) contribute 6 – 8 percent of it. Fibrinogens are needed for clotting or coagulation of blood.

Defence mechanism and osmotic balance:
Globulins primarly are involved in defense mechanisms of the body and the albumins help in osmotic balance. Plasma also contains small amounts of minerals like Na⁺, Ca⁺⁺, Mg⁺⁺, HCO₃^–, CI^–, etc. Factors for coagulation or clotting of blood are also present in the plasma in an inactive form.

What is serum?
Plasma without the clotting factors is called serum.

Formed Elements

1. Erythrocytes
2. Leucocytes
3. Platelets

They constitute nearly 45 per cent of the blood.
1. Erythrocytes or red blood cells (RBC):
They are the most abundant and on an average, 5 millions to 5.5 millions of RBCs mm-3 of blood. RBCs are formed in the red bone marrow in the adults. RBCs are devoid of nucleus in most of the mammals and are biconcave in shape. They have a red coloured, iron containing complex protein called haemoglobin.

A healthy individual has 12 – 16 gms of haemoglobin in every 100 ml of blood. These molecules play a significant role in transport of respiratory gases. RBCs have an average life span of 120 days after which they are destroyed in the spleen (graveyard of RBCs).

2. Leucocytes:
They are also known as white blood cells, colourless, nucleated and are relatively lesser in number which averages 6000 – 8000 mm-3 of blood. Leucocytes are generally short lived. The two main categories of WBCs granulocytes and agranulocytes.

• Neutrophils are the most abundant cells (60 – 65 percent) of the total WBCs and basophils are the least (0.5 – 1 percent) among them.
• Neutrophils and monocytes (6 – 8 per cent) are phagocytic cells which destroy foreign organisms entering the body.
• Basophils secrete histamine, serotonin, heparin, etc., and are involved in inflammatory reactions.
• Eosinophils (2 – 3 per cent) resist infections and are also associated with allergic reactions.

Lymphocytes (20-25 percent) are of two major types – ‘B’ and ‘T’ forms. Both B and T lymphocytes are responsible for immune responses of the body.

3. Platelets:
They are also called thrombocytes, are cell fragments produced from megakaryocytes (special cells in the bone marrow). Blood normally contains 1,500,00 – 3,500,00 platelets mm-3. Platelets are involved in the coagulation or clotting of blood. A reduction in their number can lead to clotting disorders which will lead to excessive loss of blood from the body.

Blood Groups
The ABO and Rh- are widely used all over the world.

ABO grouping
It is based on the presence or absence of two surface antigens on the RBCs namely A and B. The plasma of different individuals contain two natural antibodies (proteins produced in response to antigens). There are four groups of blood, A, B, AB and O.

Who are called as universal donor and receipient?
The group ‘O’ blood can be donated to persons with any other blood groupand hence ‘O’group individuals are called ‘universal donors’. Persons with ‘AB’ group can accept blood from persons with AB as well as the other grbups of blood, such persons are called ‘universal recipients’.

Rh grouping
Rh antigen is observed on the surface of RBCs of majority (nearly 80 percent) of humans. Such individuals are called Rh positive (Rh+ve) and without antigen are called Rh negative (Rh-ve). Rh group should also be matched before transfusions.

What is Rh incompatibility?
It is the mismatching of blood between the Rh-ve blood of a pregnant mother with Rh+ve blood of the foetus. Rh antigens of the foetus do not get exposed to the Rh-ve blood of the mother in the first pregnancy as the two bloods are well separated by the placenta.

But during the delivery of the first child, there is a possibility of mixing of the maternal blood to small amounts of the Rh+ve blood from the foetus. In such cases, the mother starts preparing antibodies against Rh in her blood. In subsequent pregnancies, the

Solving of Rh incompatibility:
This can be avoided by administering anti-Rh antibodies to the mother immediately after the delivery of the first child.

Coagulation of Blood
Blood coagulates or clots in response to an injury or trauma to prevent excessive loss of blood from the body. The network of threads called fibrins in which dead and damaged formed elements of blood are trapped.

Fibrins are formed by the conversion of inactive fibrinogens in the plasma by the enzyme thrombin. Thrombins are formed from another inactive substance present in the plasma called prothrombin.

An enzyme complex, thrombokinase, is required forthe above reaction. An injury stimulates the platelets in the blood to release certain factors which activate the mechanism of coagulation .Calcium ions play a very important role in clotting.

Lymph (tissue fluid)
As the blood passes through the capillaries in tissues, fluid released out is called the interstitial fluid or tissue fluid. It has the same mineral distribution as that in plasma. An elaborate network of vessels called the lymphatic system collects this fluid and drains it back to the major veins.

The fluid present in the lymphatic system is called the lymph. Lymph is a colourless fluid which are responsible forthe immune responses of the body. Lymph is also an important carrier for nutrients, hormones, etc. Fats are absorbed through lymph in the lacteals present in the intestinal villi.

Circulatory Pathways
The circulatory patterns are of two types – open or closed.

Open circulatory system:
It is present in arthropods and molluscs in which blood pumped by the heart passes through large vessels into open spaces or body cavities called sinuses.

Closed circulatory system:
It is present in Annelids and chordates in which the blood is pumped by the heart circulated through a closed network of blood vessels.
All vertebrates possess a muscular chambered heart.
1. Fishes have a 2-chambered heart with an atrium and a ventricle.

2. Amphibians and the reptiles (except crocodiles) have a 3-chambered heart with two atria and a single ventricle.

3. Crocodiles, birds and mammals possess a 4-chambered heart with two atria and two ventricles. In fishes the heart pumps out deoxygenated blood which is oxygenated by the gills and supplied to the body parts from where deoxygenated blood is returned to the heart (single circulation).

In amphibians and reptiles, the left atrium receives oxygenated blood from the gills/lungs/skin and the right atrium gets the deoxygenated blood from other body parts. However, they get mixed up in the single ventricle which pumps out mixed blood (incomplete double circulation).

In birds and mammals, oxygenated and deoxygenated blood received by the left and right atria respectively passes on to the ventricles of the same sides. The two separate circulatory pathways are present in these organisms, hence the animals have double circulation.

Human Circulatory System
Heart:
It is situated in the thoracic cavity, in between the two lungs. It is protected by a double walled membranous bag,pericardium, enclosing the pericardial fluid. It has four chambers, two small upper chambers called atria and two larger lower chambers called ventricles.

A thin, muscular wall called the inter atrial septum separates the right and the left atria, whereas a thick- walled, the inter-ventricular septum, separates the left and the right ventricles. The atrium and the ventricle of the same side are separated by a thick fibrous tissue called the atrioventricular septum.

Tricuspid and bicuspid valve:
The opening between the right atrium and the right ventricle is guarded by a valve formed of three muscularflaps or cusps, the tricuspid valve. Bicuspid or mitral valve guards the opening between the left atrium and the left ventricle.

Semilunar valve:
The openings of the right and the left ventricles into the pulmonary artery and the aorta respectively are provided with the semilunar valves.

Function of semilunar valve:
It allows the flow of blood only in one direction, i.e., from the atria to the ventricles and from the ventricles to the pulmonary artery or aorta. These valves prevent any backward flow.

SAN & AVN:
A specialised cardiac musculature called the nodal tissue is also distributed in the right upper corner of the right atrium called the sino-atrial node (SAN). Another mass of this tissue is seen in the lower left comer of the right atrium close to the atrio-ventricular septum called the atrio-ventricular node (AVN).

A bundle of nodal fibres, atrioventricular bundle (AV bundle) continues from the AVN which passes through the atrio-ventricular septa to emerge on the top of the interventrical sepyum and divides into a right and left bundle. These branches give rise to minute fibres throughout the ventricular musculature, they are called purkinje fibres.

These fibres alongwith right and left bundles are known as bundle of HIS. The nodal musculature has the ability to generate action potentials without any external stimuli.

What is the pacemaker of heart?
The SAN can generate the maximum number of action potentials, i.e., 70 – 75 min^-1, and is responsible for initiating and maintaining the rhythmic contractile activity of the heart. Hence it is called the pacemaker. Our heart normally beats 70 – 75 times in a minute (average 72 beats min^-1).

Cardiac Cycle
As the tricuspid and bicuspid valves are open, blood from the pulmonary veins and vena cava flows into the left and the right ventricle respectively. The semilunar valves are closed at this stage. The action potential is conducted to the ventricular side by the AVN and AV bundle from where the bundle of HIS transmits it through the entire ventricular musculature.

This causes the ventricular muscles to contract, (ventricular systole), the atria undergoes relaxation (diastole). As the ventricular pressure increases the semilunar valves open, allowing the blood in the ventricles to flow through these vessels into the circulatory pathways.

The ventricles relax (ventricular diastole) and the ventricular pressure falls causing the closure of semilunar valves which prevents the backflow of blood into the ventricles.

This sequential event in the heart which is cyclically repeated is called the cardiac cycle and it consists of systole and diastole of both the atria and ventricles. The heart beats 72 times per minute,i.e., that many cardiac cycles are performed per minute.

Cardiac output is the volume of blood pumped out by each ventricle per minute and averages 5000 mL or 5 litres in a healthy individual.

Sound produced in heart:
For example, the cardiac output of an athlete will be much higherthanthat of an ordinary man. During each cardiac cycle sounds are produced, the first heart sound (lub) is associated with the closure of the tricuspid and bicuspid valves whereas the second heart sound (dub) is associated with the closure of the semilunar valves. These sounds are of clinical diagnostic significance.

Electrocardiograph (ECG)
ECG is a graphical representation of the electrical activity of the heart during a cardiac cycle. Each peak in the ECG is identified with a letter from P to T that corresponds to a specific electrical activity of the heart.
1. The P-wave represents the electrical excitation Diagrammatic presentation of a standard ECG (or depolarisation) of the atria, which leads to the
contraction of both the atria.

2. The QRS complex represents the depolarisation of the ventricles, which initiates the ventricular contraction. The contraction starts shortly after Q and marks the beginning of the systole.

3. The T-wave represents the return of the ventricles from excited to normal state (repolarisation). The end of the T-wave marks the end of systole.

Since the ECGs obtained from different individuals have the same shape any deviation from this shape indicates abnormality or disease.

Doube Circulation
It involves two types of circulation,
1. Pulmonary circulation:
The deoxygenated blood pumped into the pulmonary artery is passed on to the lungs from where the oxygenated blood is carried by the pulmonary veins into the left atrium. This pathway constitutes the pulmonary circulation.

2. Systemic circulation:
The oxygenated blood entering the aorta is carried by a network of arteries, arterioles and capillaries to the tissues from where the deoxygenated blood is collected by a system of venules, veins and vena cava and emptied into the right atrium. This is the systemic circulation.

The systemic circulation provides nutrients, O₂ and other essential substances to the tissues and takes CO₂ and other harmful substances away for elimination. A vascular connection between the digestive tract and liver called hepatic portal system.

The hepatic portal vein carries blood from intestine to the liver before it is delivered to the systemic circulation. In Coronary system of blood vessels is present in our body exclusively for the circulation of blood to and from the cardiac musculature.

Regulation Of Cardiac Activity
Normal activities of the heart are auto regulated by specialised muscles (nodal tissue), hence the heart is called myogenic. Medulla oblangata control the cardiac function through autonomic nervous system (ANS).

Neural signals through the sympathetic nerves (part of ANS) increase the rate of heart beat. On the other hand, parasympathetic neural signals decrease the rate of heart beat. Adrenal medullary hormones can also increase the cardiac output.

Disorders Of Circulatory System
High Blood Pressure (Hypertension):
Hypertension is the term for blood pressure that is higherthan normal (120/80). In this measurement 120 mm Hg (millimetres of mercury pressure) is the systolic, or pumping, pressure and 80 mm Hg is the diastolic, or resting, pressure. High blood pressure leads to heart diseases and also affects vital organs like brain and kidney.

Coronary Artery Disease (CAD):
Coronary Artery Disease, often referred to as atherosclerosis, affects the vessels that supply blood to the heart muscle. It is caused by deposits of calcium, fat, cholesterol and fibrous tissues, which makes the lumen of arteries narrower.

Angina:
It is also called ‘angina pectoris’. A symptom of acute chest pain appears when no enough oxygen is reaching the heart muscle. It occurs due to conditions that affect the blood flow.

Heart Failure:
It is the state of heart when it is not pumping blood effectively enough to meet the needs of the body. Heart failure is not the same as cardiac arrest (when the heart stops beating) or a heart attack (when the heart muscle is suddenly damaged by an inadequate blood supply).

Students can Download Chapter 15 Waves Notes, Plus One Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Physics Notes Chapter 15 Waves

Summary
The wave is the propagation of disturbance that carries energy from one point to another point, without translatory motion of particles in the medium. There are three types of wave.
1. Mechanical Wave: Requires medium for propagation.
Eg: Sound wave, Matter wave, Seismic wave, etc.

2. Electromagnetic Wave: No medium for propagation.
Eg: Light, X-rays, UV ray, etc.

3. Matter Wave: Wave associated with moving particles (microscopic particle).
Eg: Wave of moving electron, proton, etc.

Generation of longitudinal waves by tuning for k
When prongs of tuning fork moves outward, it compresses the surrounding air and a region of increased pressure is formed. This region is called condensation. When the prongs move inward a region of low pressure called rarefraction is formed. Thus condensations and rarefractions are alternately produced.

Expression for progressive wave (Displacement relation)
A plane progressive wave propagating along positive direction of ‘x’ is given by

The wave propagating along negative ‘x’ direction is given by y (x, t) = a sin (kx + ωt + Φ).
y(x, t) gives the transverse displacement of element at position x at time t.

Crust and Trough:
Crust is the point of maximum positive displacement of wave. Trough is the point of maximum negative displacement of wave.

Transverse and Longitudinal Wave
Based on the direction of propogation and vibration wave can be of two types.

Parameters of wave
Amplitude:
The magnitude of maximum displacement of element of the wave from initial position is called amplitude (a).
Phase and initial phase:
The value (kx + ωt + Φ) is called phase and f is the initial phase. The phase gives the state of motion of wave at position ‘x’ and at time V. Initial phase gives initial state of wave.
Wave length (λ):
The linear distance travelled by the wave in one complete oscillation(or vibration). Or it can be defined as distance between two conT secutive crusts or troughs. It is the distance travelled during time period T.
Wave number/Angular wave number/propagation constant:
Wave number ‘k’ is defined as

Its unit is radian/m
Time period (T):
Time for one complete oscillation/vibration is called time period.
Frequency (ν):
The number of oscillations/vibrations in one second is called frequency. Its unit is S^-1 or Hz (Hetz).
ν = \(\frac{1}{T}\)
Angular velocity or angular frequency (ω):
Angular displacement per unit time is called angular velocity or angular frequency.

The speed of travelling wave [Relation connecting V, ν and λ]

The wave propagating along x direction is represented by y = A sin (kx – ωt + Φ). As wave moves, each point on the wave from (like A) retains its displacement. This is possible only if (kx – ωt) is constant. As the wave moves both x and t are changing to keep (kx – ωt) as constant (x increase with t).

The velocity depends on wavelength and frequency. The wavelength and frequency of wave depends on the properties of medium, i.e. velocity of wave in a medium is determined by

• linear mass density
• Elastic properties.

Speed of wave in stretched string (Transverse wave)
The velocity of wave in stretched string depends on

• linear mass density (µ)
• The tension (T)

Speed of sound wave (Longitudinal wave)
The speed of sound in medium depends on

• density of medium (ρ)
• Modulus of elasticity

Case: 1( In solid )
If solid has Young’s modulus ‘Y’

Case: 2( In liquid )
If liquid has Bulk modulus ‘B’

Case: 3 In gas
The speed of sound waves in gas was determined by Newton. According to Newton, condensations and rare fractions are isothermal processes. Hence modulus of elasticity is equal to pressure.

This is called Newton’s formula.
Correction in Newton’s formula.

Question 1.
Find velocity of sound in air using Newton’s formula (P = 1.013 × 10⁵ ρ =1.239 kg m^-3)
Answer:

Note: The velocity of sound at STP is found to be 332 ms^-1
Laplace’s Formula:
Laplace corrected Newton’s formula taking condensation and rare fraction as adiabatic process. The
modulus of elasticity is now ‘γP‘ where γ = \(\frac{C_{p}}{C_{v}}\). C_p is specific heat capacity at constant pressure and C_v is specific heat capacity at constant volume.

The principle of superposition of waves
It states that when two or more waves pass through a media the net displacement of particle at any time is the algebraic sum of displacements due to each wave.

(or)

The overlapping waves algebraically add to produce a resultant wave.

Reflection of waves
Reflection from rigid boundary:
When a travelling wave is reflected by a rigid boundary, phase reversal (phase difference of π or 180°) will take place.

Reflection from open boundary:
When a travelling wave is reflected by an open boundary, no phase change will happen. The incident and reflected wave superimpose to give maximum displacement at boundary.

Standing waves

When two waves of same amplitude and frequency travelling in opposite direction superimpose the resulting wave pattern does not move to either sides. This pattern is called standing wave.
The wave travelling in positive direction of x axis y₁(x, t) = a sin(kx – ωt)
The wave travelling in negative direction of x axis y₂(x, t) = a sin(kx + ωt)
According to superposition Principle, the combined wave is
y(x, t) = y₁(x, t) + y₂(x, t)
y(x, t) = a sin(kx – ωt) + a sin(kx + ωt)
But sin A + sin B = 2 sin \(\frac{(A+B)}{2} \cos \frac{(A-B)}{2}\)
Hence we get,combined wave as

This wave has an amplitude of ‘2asinkx’, and it is not a moving wave.
Nodes & Antinodes:
The position of maximum amplitude in a standing wave is termed as anti node and position of minimum amplitude (zero) is termed as node.
Node: The amplitude of standing wave is ‘2 a sin kx’. It is zero when kx = 0, π, 2π…. etc.
ie kx = nπ.

Antinode:
The amplitude has maximum value 2a when (2a sin kx = 2a)
sinkx = 1;
ie; kx = π/2, 3π/2, 5π/2 …… etc

But k = \(\frac{2 \pi}{\lambda}\)
Hence we get

n = 0, 1, 2, 3 etc.

1. Standing Waves In Stretched String & Modes Of Vibration Of String:
A string of length L is fixed at two ends. The position of one end is chosen as x = 0, then the position of other end will be x = L. At x = 0, there will be node. To occur node at x = L, it must satisfy

The frequency of vibrations of stretched string of length L is

n = 1,2, 3…etc.
This set of frequencies at which the string can vibrate are called natural frequencies or modes of vibration or harmonics. The above equation shows that the modes of vibration (natural frequencies) of string are integral multiple of lowest frequency
n = \(\frac{V}{2 L}\) (for n = 1)
Fundamental mode(or) First harmonics:
For n = 1
ν₁ = \(\frac{V}{2 L}\)
This is the lowest frequency with which string vibrates. This is called fundamental mode or first harmonic of vibration.
Relation between I and L for first harmonics:

The string vibrate in a single segment as shown in figure.

Second harmonic:

For n = 2

Relation between I and L for second harmonics

Third harmonic:

For n = 3, there is 3^rd harmonic. Thus collection of all possible mode is called harmonic series and n is called harmonic number.

2. Vibrational modes of an air column:
(a) In closed tube:
In closed tube one end is closed and other end is open. Air column in a glass tube partially filled with water is an example of closed system. The air column in tube can be set into vibrations with the help of air excited by tuning fork.

The longitudinal waves thus generated is reflected at the closed end and a node is formed there [reflected and incident wave are out of phase and at the closed end, they superimpose to give minimum displacement]. At the open end, the displacement is maximum and antinode is formed. If L is the length of air column, anti node occurs at x = L.
We know the condition for antinode x = (n + 1/2) λ/2
Therefore L = (n + 1/2) λ/2
for n = 0, 1, 2, …….etc.
The wavelength, λ = \(\frac{2 \mathrm{L}}{(\mathrm{n}+1 / 2)}\) _____(1)
for n = 0, 1, 2, …. etc.
The frequency ν = (n + 1/2) \(\frac{V}{2 L}\) ____(2)
for n = 0, 1, 2, …….etc.
From this equation, it is clear that the air column can vibrate with different modes of frequencies (normal modes or harmonics)
Fundamental Mode(or) First harmonic:
We get fundamental mode when n=0 Substitute this in eq(2), we get
ν₁ = \(\frac{V}{4 L}\)
This is the fundamental frequency. The higher frequencies are odd harmonics of fundamental frequency ie;

(b) Open tube:
In open tube, both ends are open. At both ends, antinodes are formed.
The condition to get antinode x = n λ/2

The frequency (fundamental frequency):
We will get fundamental frequency in open tube,when n = 1. Substitute this in eq(4)
ν₁ = \(\frac{V}{2 L}\)
In open pipe all harmonies are generated whereas in closed pipe only odd harmonies are generated.
For n = 2, ν₂ = 2\(\frac{V}{2 L}\). This is second harmonic.
For n = 3, ν₃ = 3\(\frac{V}{2 L}\). This is 3rd harmonic.
ν₂ = 2 ν₁ & ν₃ = 3 .ν₁. Thus both odd and even harmonics are generated.

Beats:
When two sound waves of nearly same frequency and amplitude travelling in same direction super imposed and periodic variation of sound intensity (wavering of sound or waxing and waning of sound) is produced. This is called beats.
Explanation:
If two tuning forks of slightly different frequencies are sounded together, a regular rise and fall of sound can be heard. The sound travels in the form of condensation and rarefactions.

When two condensations due to two notes reach our ear at the same time, they superimpose to get maximum intensity (waxing of sound). If two rarefactions are reached simultaneously, they superimpose to get minimum intensity (waning of sound).

Analytical treatment of beats
Beats frequency
Suppose two sound waves of u₁ and u₂ propagating in the same direction through a medium. For simplicity let the listener be situated at x = 0 and the amplitudes of waves to be equal ie. a₁ = a₂ = a.
The displacements y₁ and y₂ due to each wave are given by
y₁ = a sin2pu₁t and y₁ = a sin2pu₂t
According to super position principle, the resultant displacement at the same time t is
y = y₁ + y₂
= a sin2pu₁t + a sin2pu₂t
y = a [sin2pu₁t + sin2pu₂t]

It is clear that, the amplitude of resultant wave (A) changes with time. It shows maxim and minima.
The resultant amplitude will be maximum, if,

Hence the amplitude of the resultant wave will be maximum at times

Time interval between successive maxima = \(\frac{1}{v_{1}-v_{2}}\)
resultant amplitude will be minimum if

Time internval between two consecutive minima = \(\frac{1}{v_{1}-v_{2}}\)
Frequency of minima = ν₁ – ν₂
Graphical representation of beats

Doppler Effect
The apparent change in the frequency of sound wave due to the relative motion of source or listener or both is called Doppler effect. It was proposed by John Christian Doppler and it was experimentally tested by Buys Ballot.

Considers source is producing sound of frequency n. Let V be the velocity of sound in the medium and I the wavelength of sound when the source and the listener are at rest. The frequency of sound heard by the listener is
ν = \(\frac{v}{\lambda}\)
Let the source and listener be moving with velocities v_s and v_l in the direction of propogation of sound from source to listener. (The direction S to L is taken as positive)
The relative velocity of sound wave with respect to the source = V – V_s.
Apparent wavelength of sound,
λ¹ = \(\frac{V-V_{s}}{v}\) ____(1)
Since the listener is moving with velocity v f, the rela¬tive velocity of sound with respect to the listener,
V¹ = V – V_l _____(2)

Question 2.
A string fixed one end is suddenly brought in to up and down motion.

1. What is the nature of the wane produced in the string and name the wave?
2. A brass wire 1 m long has a mass 6 × 10^-3 kg. If it is kept at a tension 60N, What is the speed of the wave on the wire?

Answer:
1. Transverse wave

2.

Students can Download Chapter 6 Breathing and Exchange of Gases Notes, Plus One Zoology Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus One Zoology Notes Chapter 6 Breathing and Exchange of Gases

What is respiration?
The process of exchange of O₂ from the atmosphere with CO₂ produced by the cells is called breathing, commonly known as respiration.

RESPIRATORY ORGANS:

Human Respiratory System:
The nostrils leads to a nasal chamber through the nasal passage. The nasal chamber opens into nasopharynx the common passage for food and air. Nasopharynx opens through glottis of the larynx region into the trachea.

Sound box in respiratory system:
Larynx is a cartilaginous box which helps in sound production called as the sound box. Glottis is covered by a thin elastic cartilaginous flap called epiglottis to prevent the entry of food into the larynx. Trachea is a straight tube which divides into a right and left primary bronchi.

Each bronchi undergoes repeated divisions to form the secondary and tertiary bronchi and bronchioles ending up in very thin terminal bronchioles.
Each terminal bronchiole gives rise to vascularised bag-like structures called alveoli. The branching network of bronchi, bronchioles and alveoli comprise the lungs.

The two lungs which are covered by a double layered pleura, with pleural fluid between them. It reduces friction on the lung surface. The conducting part transports the atmospheric air to the alveoli. Exchange part is the site of diffusion of O₂ and CO₂ between blood and atmospheric air.

Where is lung fitted in human body?
The lungs are situated in the thoracic chamber which is formed dorsally by the vertebral column, ventrally by the sternum, laterally by the ribs and on the lower side by the dome-shaped diaphragm. The anatomical setup of lungs in thorax is the arrangement essential for breathing, directly alterthe pulmonary volume.

Respiration involves the following steps:

• Breathing or pulmonary ventilation by which atmospheric air is drawn in and CO₂ rich alveolar air is released out.
• Diffusion of gases (O₂ and CO₂) across alveolar membrane.
• Transport of gases by the blood.
• Diffusion of O₂ and.CO₂ between blood and tissues.
• Utilisation of O₂ by the cells for catabolic reactions and resultant release of CO₂

MECHANISM OF BREATHING:
Breathing involves two stages:

1. Inspiration during which atmospheric air is drawn in and
2. Expiration by which the alveolar air is released out.

How does increase or decrease of pulmonary volume occur?
Inspiration is initiated by the contraction of diaphragm which increases the volume of thoracic chamber .The contraction of external inter-costal muscles lifts up the ribs and the sternum causing an increase in the volume of the thoracic chamber. It increase the pulmonary volume.

An increase in pulmonary volume decreases the intra-pulmonary pressure to less than the atmospheric pressure which forces the airfrom outside to move into the lungs,i.e., inspiration Relaxation of the diaphragm and the inter-costal muscles returns the diaphragm and sternum to their normal positions and reduce the thoracic volume.

It reduce the pulmonary volume. This leads to an increase in intra-pulmonary pressure to slightly above the atmospheric pressure causing the expulsion of airfrom the lungs, i.e., expiration.

Instrument used for measuring breathing movements:
A healthy human breathes 12 – 16 times/minute. The volume of air involved in breathing movements can be estimated by using a spirometer.

Respiratory Volumes and Capacities:

EXCHANGE OF GASES:
Alveoli are the sites of exchange of gases. O₂ and CO₂ are exchanged in these sites by simple diffusion.
Rate of diffusion:
Solubility of the gases and the thickness of the membranes are the important factors that affect the rate of diffusion.

Partial pressure of gases:
Pressure contributed by an individual gas in a mixture of gases is called partial pressure and is represented as pO₂ for oxygen and pCO₂ for carbon dioxide. As the solubility of CO₂ is 20 – 25 times higher than that of O₂, the amount of CO₂ that can diffuse through the diffusion membrane.

As the solubility of CO₂ is 20 – 25 times higher than that of O₂ the amount of CO₂ that can diffuse through the diffusion membrane.

The diffusion membrane is made up of three major layers such as

Its total thickness is much less than a millimetre.

TRANSPORT OF GASES:
Blood is the medium of transport for 02 and C02.

1. About 97 per cent of O₂ is transported by RBCs in the blood.
2. The remaining 3 per cent of O₂ is carried in a dissolved state through the plasma.
3. Nearly 20 – 25 per cent of CO₂ is transported by RBCs whereas 70 per cent of it is carried as bicarbonate.
4. About 7 per cent of CO₂ is carried in a dissolved state through plasma.

Transport of Oxygen:
O₂ bind with haemoglobin to form oxyhaemoglobin. Each haemoglobin molecule can carry a maximum of four molecules of O₂. Binding of oxygen with haemoglobin is related to partial pressure of O₂.

Oxygen dissociation curve is useful in studying the effect of factors like pCO₂, H⁺ concentration, etc., on binding of O₂ with haemoglobin.

Therefore O₂ gets bound to haemoglobin in the lung surface and gets dissociated at the tissues.

Transport of Carbon dioxide:
CO₂ is carried by haemoglobin as carbamino-haemoglobin (about 20 – 25 per cent). This binding is related to the partial pressure of CO₂. When pCO₂ is high and pO₂ is low as in the tissues, more binding of carbon dioxide occurs whereas, when the PCO₂ is low and pCO₂ is high as in the alveoli, dissociation of CO₂ from carbamino-haemoglobin takes place. RBCs contain a very high concentration of the enzyme, carbonic anhydrase which facilitates the reaction in both directions.

At the tissue site where partial pressure of CO₂ is high due to catabolism, CO₂ diffuses into blood (RBCs and plasma) and forms HCO₂ and H⁺. At the alveolar site where pCO₂ is low, the reaction proceeds in the opposite direction leading to the formation 0f CO₂ and H₂O. Every 100 ml of deoxygenated blood delivers approximately 4 ml of CO₂ to the alveoli.