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CBSE · Class 9 · 📘 Social_Economics · Chapter 7

Work, Energy, and Simple Machines

Work done by a constant forceKinetic energyPotential energyWork-energy theoremConservation of mechanical energyPower

Chapter 7, 'Work, Energy, and Simple Machines', introduces fundamental physics concepts. Students learn about the scientific definition of work, different forms of energy (kinetic and potential), the work-energy theorem, and the conservation of mechanical energy. The chapter also explores the concept of power and the working principles of simple machines like pulleys, inclined planes, and levers, explaining how they make tasks easier by altering force magnitude or direction.

Work Done by a Constant Force: Definition, Calculation, and Units

Work ek scientific term hai jiska matlab hai jab koi force kisi object ko uski direction mein displace karta hai.

Work ki Definition

  • Work Done (W): Jab ek constant force (F) kisi object par act karta hai aur usko force ki direction mein displacement (s) deta hai, toh work done hota hai.
  • Work done = Force × Displacement
  • Formula: \(W = F \times s\)

Important Points

  • Work ek scalar quantity hai, iski koi direction nahi hoti.
  • Work done ki value positive, negative ya zero ho sakti hai.

SI Unit of Work

  • Joule (J): Work ki SI unit Joule hai.
  • 1 Joule = 1 Newton × 1 meter (\(1 J = 1 N \times 1 m\))
  • 1 Joule work tab hota hai jab 1 Newton ka force kisi object ko 1 meter tak force ki direction mein displace karta hai.
  • Dimensionally: \(1 J = 1 kg \ m^2 \ s^{-2}\)

Force-Displacement Graph

  • Agar force constant ho, toh force-displacement graph (force y-axis par, displacement x-axis par) ke under ka area work done ko represent karta hai.
  • Area under F-s graph = Work Done
  • Even non-constant force ke liye bhi, area under F-s graph work done deta hai.
🧮Formula

Work Done: \(W = F \times s\)

Units: Joule (J)

⭐Remember

Work done ki calculation mein, force aur displacement ki direction same honi chahiye. Agar force aur displacement ki direction different ho, toh hum force ka component lete hain jo displacement ki direction mein ho.

Conditions for Zero, Positive, and Negative Work Done

Work done ki value force aur displacement ke beech ke angle par depend karti hai.

Zero Work Done

Work done zero hota hai jab:

  1. Force zero ho (F=0): Agar koi force nahi lag raha, toh work bhi zero hoga. Example: Space mein constant velocity se move karta hua object.
  2. Displacement zero ho (s=0): Agar force lagane ke baad bhi object displace na ho. Example: Wall ko push karna. Aap thak jaoge but scientifically work zero hai.
  3. Force aur Displacement perpendicular ho (angle \(90^\circ\)): Jab force ki direction displacement ke perpendicular ho. Example:
  • Ek coolie jab apne sar par load uthakar horizontal platform par chalta hai. Gravitational force vertically downward lagta hai, aur displacement horizontally hota hai. Is case mein gravitational force dwara kiya gaya work zero hota hai.
  • Circular motion mein centripetal force dwara kiya gaya work zero hota hai, kyunki centripetal force hamesha velocity (displacement ki direction) ke perpendicular hota hai.

Positive Work Done

  • Jab force aur displacement same direction mein ho (angle \(0^\circ\)).
  • Example:
  • Ek ball ko niche ki taraf fekna. Gravitational force aur displacement dono niche ki taraf hain.
  • Trolley ko push karna. Force aur displacement dono aage ki taraf hain.

Negative Work Done

  • Jab force aur displacement opposite direction mein ho (angle \(180^\circ\)).
  • Example:
  • Braking car mein friction force dwara kiya gaya work. Friction force motion ke opposite lagta hai.
  • Ek ball ko upar ki taraf fekna. Gravitational force niche lagta hai, aur displacement upar ki taraf hota hai. Is case mein gravitational force dwara kiya gaya work negative hota hai.
  • Goalie ka ball ko rokna. Goalie ball par motion ke opposite force lagata hai.
🚧Misconception

Students often confuse 'feeling tired' with 'work done'. Scientific work done ke liye displacement hona zaroori hai. Wall ko dhakka dene mein, body ki internal energy use hoti hai, isliye thakan hoti hai, but wall par work zero hai.

Energy: Concept, Forms, and Work-Energy Theorem

Energy ka Concept

  • Energy: Kaam karne ki capacity ko energy kehte hain. Jis object mein work karne ki capacity hoti hai, usmein energy hoti hai.
  • Jab kisi object par positive work kiya jata hai, toh uski energy increase hoti hai.
  • Energy ek scalar quantity hai.

SI Unit of Energy

  • Energy ki SI unit bhi Joule (J) hai, jo work ki unit ke same hai.
  • Bigger units: kilojoule (kJ), megajoule (MJ).

Work-Energy Theorem

  • Work-Energy Theorem ke according, kisi object par kiya gaya total work uski energy mein change ke barabar hota hai.
  • Formula: \(W = \Delta E\) (Change in Energy)
  • Ye theorem bahut powerful hai, especially jab forces non-constant hon ya complex situations mein.

Forms of Energy

Energy kayi forms mein exist karti hai aur ek form se dusre form mein convert ho sakti hai:

  • Mechanical Energy: Motion (kinetic) ya position (potential) ke karan hoti hai.
  • Heat Energy (Thermal Energy): Temperature difference ke karan transfer hoti hai.
  • Light Energy: Electromagnetic radiation ka form.
  • Sound Energy: Vibrations se produce hoti hai.
  • Electrical Energy: Charges ke flow se associated.
  • Chemical Energy: Chemical bonds mein stored energy.
  • Nuclear Energy: Atomic nuclei mein stored energy.

Energy Conversion Examples

  • Electric bulb: Electrical energy \(\rightarrow\) Light energy + Heat energy.
  • Electric heater: Electrical energy \(\rightarrow\) Heat energy.
  • Food digestion: Chemical energy \(\rightarrow\) Muscular (Mechanical) energy.
  • Ringing bell: Mechanical energy \(\rightarrow\) Sound energy.
  • Solar panel: Light energy \(\rightarrow\) Electrical energy.
❗Important

Energy aur work dono ki unit Joule (J) hai. Ye isliye hai kyunki work energy transfer ka ek tareeka hai.

📖Definition

Work-Energy Theorem: Work done on an object = Change in its energy.

Kinetic Energy: Definition and Formula

Kinetic Energy ki Definition

  • Kinetic Energy (KE): Kisi object mein uski motion ke karan jo energy hoti hai, use kinetic energy kehte hain.
  • Har moving object mein kinetic energy hoti hai.
  • Agar object rest par hai (velocity = 0), toh uski kinetic energy zero hoti hai.

Kinetic Energy ka Formula Derivation

  • Consider ek object jiska mass 'm' hai aur initial velocity 'u' hai.
  • Ek constant force 'F' lagane par, object 's' distance displace hota hai aur uski final velocity 'v' ho jaati hai.
  • Newton's second law se: \(F = ma\)
  • Kinematic equation se: \(v^2 = u^2 + 2as \Rightarrow s = \frac{v^2 - u^2}{2a}\)
  • Work done \(W = F \times s = ma \times \frac{v^2 - u^2}{2a} = \frac{1}{2}m(v^2 - u^2)\)
  • Work-Energy Theorem ke according, ye work done energy mein change ke barabar hai.
  • Agar object rest se start kare (u=0), toh work done uski final kinetic energy ban jaati hai.
  • So, Kinetic Energy (KE) = \(\frac{1}{2}mv^2\)

Important Points

  • Kinetic energy hamesha positive hoti hai, kyunki mass (m) aur velocity ka square (\(v^2\)) hamesha positive hote hain.
  • Kinetic energy bhi ek scalar quantity hai.
  • Agar positive work done hota hai, toh velocity aur KE badhti hai. Agar negative work done hota hai, toh velocity aur KE ghat-ti hai.
  • Agar work done zero hai, toh velocity aur KE constant rehti hai.
🧮Formula

Kinetic Energy: \(K = \frac{1}{2}mv^2\)

Variables: m = mass (kg), v = velocity (m/s)

Units: Joule (J)

💡Tip

Kinetic energy ke numericals mein units ka dhyaan rakho. Mass kg mein aur velocity m/s mein honi chahiye.

Potential Energy: Concept and Examples

Potential Energy ki Definition

  • Potential Energy (PE): Kisi object mein uski position ya configuration (shape) ke karan jo energy stored hoti hai, use potential energy kehte hain.
  • Ye stored energy work karne ki capacity rakhti hai.

Types of Potential Energy

  1. Elastic Potential Energy:
  • Object ki shape ya size mein change (deformation) ke karan stored energy.
  • Examples: Stretched rubber band, compressed spring, bent bow. Jab inko release karte hain, toh ye stored energy kinetic energy mein convert ho jaati hai.
  1. Gravitational Potential Energy:
  • Object ki height ya position ke karan stored energy, Earth ke gravitational field mein.
  • Examples: Height par rakha hua patthar, dam mein store kiya hua paani.
  1. Electrostatic Potential Energy: Charges ki relative position ke karan.
  2. Magnetic Potential Energy: Magnets ki relative position ke karan.

Potential Energy ka Source

  • Potential energy tab store hoti hai jab kisi external force dwara work kiya jata hai object ki configuration ya position ko change karne ke liye, internal conservative forces ke against.
  • Example: Spring ko compress karne mein kiya gaya work spring mein potential energy ke roop mein store ho jata hai.
  • Friction ke against kiya gaya work potential energy mein store nahi hota, balki heat mein convert ho jata hai.

Activity 7.1 se Learning

  • Ek ball ko jitni zyada height se giraya jayega, utna hi gehra depression sand mein banega.
  • Iska matlab hai ki zyada height par ball mein zyada potential energy hoti hai.
⭐Remember

Potential energy hamesha ek system ki property hoti hai (e.g., Earth-ball system, spring-object system).

Gravitational Potential Energy: Derivation and Formula

Gravitational Potential Energy ki Definition

  • Gravitational Potential Energy (U): Kisi object mein Earth ke gravitational field mein uski height ke karan jo energy stored hoti hai.
  • Reference point (usually ground level) par potential energy ko zero mana jata hai.

Gravitational Potential Energy ka Formula Derivation

  • Consider ek object jiska mass 'm' hai, ground par rakha hai.
  • Isko 'h' height tak uthane ke liye, humein gravitational force (mg) ke against force lagana padega.
  • Applied force \(F_{applied} = mg\) (object ko gradually lift karne ke liye, acceleration zero).
  • Displacement \(s = h\)
  • Work done by applied force \(W = F_{applied} \times s = mg \times h = mgh\)
  • Work-Energy Theorem ke according, ye work done object ki potential energy mein change ke barabar hota hai.
  • Agar ground par PE = 0, toh height 'h' par PE = \(mgh\).
  • Formula: \(U = mgh\)

Important Points

  • Gravitational potential energy ki unit bhi Joule (J) hai.
  • Ye formula sirf Earth ki surface ke paas valid hai, jahan 'g' constant mana ja sakta hai.
  • Potential energy path par depend nahi karti, sirf initial aur final position par depend karti hai. (Iska proof NCERT exercise Q5 mein hai).

Activity 7.1 ka Connection

  • Ball ko jitni zyada height se giraya, utna zyada work done hua usko uthane mein, aur utni zyada potential energy store hui. Isliye, sand mein gehra depression bana.
🧮Formula

Gravitational Potential Energy: \(U = mgh\)

Variables: m = mass (kg), g = acceleration due to gravity (\(m/s^2\)), h = height (m)

Units: Joule (J)

💡Tip

Potential energy ke numericals mein 'h' hamesha reference level se height hoti hai. Reference level ko hum apni convenience ke according choose kar sakte hain, but ek baar choose karne ke baad use consistent rakhna chahiye.

Law of Conservation of Mechanical Energy

Mechanical Energy

  • Mechanical Energy (ME): Kisi object ki kinetic energy aur potential energy ka sum uski mechanical energy kehlati hai.
  • Formula: \(ME = KE + PE = \frac{1}{2}mv^2 + mgh\)

Law of Conservation of Mechanical Energy

  • Statement: Agar kisi system par koi external non-conservative force (jaise friction, air resistance) act nahi kar raha hai, toh us system ki total mechanical energy (KE + PE) hamesha constant rehti hai.
  • Iska matlab hai ki energy ek form se dusre form mein convert ho sakti hai (KE \(\leftrightarrow\) PE), but total sum constant rehta hai.

Free Fall ka Example

  • Consider ek object 'h' height se free fall kar raha hai (air resistance negligible).
  • Highest Point (A): \(v=0\), so \(KE_A = 0\). \(PE_A = mgh\). \(ME_A = mgh\).
  • Intermediate Point (B): Object niche aa raha hai, velocity 'v' hai, height 'h'' hai. \(KE_B = \frac{1}{2}mv^2\). \(PE_B = mgh'\). \(ME_B = \frac{1}{2}mv^2 + mgh'\).
  • Ground Level (C): Just before hitting ground, \(h=0\), so \(PE_C = 0\). Velocity maximum 'V' hai. \(KE_C = \frac{1}{2}mV^2\). \(ME_C = \frac{1}{2}mV^2\).
  • Conservation of Mechanical Energy ke according: \(ME_A = ME_B = ME_C\)
  • \(mgh = \frac{1}{2}mv^2 + mgh' = \frac{1}{2}mV^2\)
  • Jaise-jaise object niche aata hai, uski potential energy decrease hoti hai aur kinetic energy increase hoti hai, but unka sum constant rehta hai.

Simple Pendulum ka Example (Activity 7.2)

  • Extreme Position (P/R): Velocity zero hoti hai, so KE = 0. Height maximum hoti hai, so PE maximum. Total ME = PE_max.
  • Mean Position (Q): Height minimum (zero) hoti hai, so PE = 0. Velocity maximum hoti hai, so KE maximum. Total ME = KE_max.
  • Pendulum swing karte waqt, PE KE mein aur KE PE mein continuously convert hoti rehti hai, but total mechanical energy conserved rehti hai (friction aur air resistance ignore karne par).
  • Real life mein, air resistance aur friction ke karan mechanical energy heat energy mein convert ho jaati hai, aur pendulum eventually ruk jaata hai.
❗Important

Conservation of Mechanical Energy tabhi apply hota hai jab non-conservative forces (friction, air resistance) absent hon ya negligible hon. Real-world scenarios mein, energy hamesha conserve rehti hai, but mechanical energy nahi, kyunki kuch energy dusre forms (heat, sound) mein convert ho jaati hai.

Power: Definition, Formula, and Units

Power ki Definition

  • Power (P): Work karne ki rate ko power kehte hain. Yani, kitna work kitne time mein kiya gaya.
  • Formula: \(P = \frac{W}{t}\)
  • Jitni zyada power, utna zyada work same time mein, ya utna hi work kam time mein.

SI Unit of Power

  • Watt (W): Power ki SI unit Watt hai.
  • \(1 \text{ Watt} = 1 \text{ Joule/second} \ (1 W = 1 J/s)\)
  • 1 Watt power tab hoti hai jab 1 Joule work 1 second mein kiya jata hai.

Other Units of Power

  • Kilowatt (kW): \(1 kW = 1000 W\)
  • Horsepower (hp): Ye ek older unit hai, jo engines ki power measure karne ke liye use hoti thi.
  • \(1 hp = 746 W\)

Power aur Velocity ka Relation

  • Hum jaante hain \(W = F \times s\)
  • Toh \(P = \frac{F \times s}{t}\)
  • Aur \(\frac{s}{t} = v\) (velocity)
  • So, \(P = F \times v\)
  • Power ko force aur velocity ke product ke roop mein bhi define kiya ja sakta hai.

Example

  • Stairs par daud kar upar jaana aur dheere chal kar upar jaana. Dono cases mein work done same hai (gravitational potential energy mein change), but daudne mein time kam lagta hai, isliye power zyada use hoti hai.
🧮Formula

Power: \(P = \frac{W}{t}\) or \(P = F \times v\)

Variables: W = work done (J), t = time (s), F = force (N), v = velocity (m/s)

Units: Watt (W)

❗Important

James Watt ke naam par power ki unit 'Watt' rakhi gayi hai, jinhone steam engine ko develop kiya tha.

Introduction to Simple Machines and Mechanical Advantage

Simple Machines kya hain?

  • Simple Machines: Ye aise devices hain jo work ko easier banate hain, force ki magnitude ya direction ko change karke.
  • Simple machines total work done ko kam nahi karte, balki effort ko kam karke ya convenient direction mein effort lagane ki facility provide karte hain.
  • Examples: Lever, Pulley, Inclined Plane, Wheel and Axle, Wedge, Screw.

Key Terms

  • Effort (E): Woh force jo hum machine par apply karte hain.
  • Load (L): Woh force jise machine overcome karti hai ya jise uthana hai. (e.g., object ka weight)

Mechanical Advantage (MA)

  • Mechanical Advantage: Ye load aur effort ka ratio hota hai. Ye batata hai ki machine kitna force multiply kar sakti hai.
  • Formula: \(MA = \frac{\text{Load}}{\text{Effort}} = \frac{L}{E}\)
  • Agar MA > 1, toh machine effort ko multiply karti hai (kam effort se zyada load utha sakte hain).
  • Agar MA = 1, toh machine sirf force ki direction change karti hai.
  • Agar MA < 1, toh machine effort ko increase karti hai (speed ya distance gain ke liye).

Ideal Machine

  • Ideal machine mein energy loss (friction ke karan) zero hota hai.
  • Ideal machine ke liye: Work done by effort = Work done on load
  • \(E \times d_E = L \times d_L\), jahan \(d_E\) effort displacement aur \(d_L\) load displacement hai.
  • Isse \(MA = \frac{L}{E} = \frac{d_E}{d_L}\) bhi derive hota hai.
📖Definition

Simple Machine: A device that makes work easier by changing the magnitude or direction of the applied force.

🧮Formula

Mechanical Advantage: \(MA = \frac{\text{Load}}{\text{Effort}}\)

Pulleys as Simple Machines

Pulley kya hai?

  • Pulley: Ek wheel hota hai jismein groove hota hai, jiske through rope ya cable pass hoti hai. Ye force ki direction change karne ya force ko multiply karne ke liye use hota hai.

Types of Pulleys

  1. Fixed Pulley:
  • Ek fixed support se attach hoti hai.
  • Ye sirf force ki direction change karti hai. Load ko upar uthane ke liye hum rope ko niche khichte hain, jo zyada convenient hota hai.
  • Magnitude of effort = Magnitude of load.
  • Mechanical Advantage (MA) = 1 (kyunki L/E = 1).
  • Examples: Flag hoisting, well se paani nikalna.
  1. Movable Pulley:
  • Load pulley se attach hota hai aur pulley load ke saath move karti hai.
  • Ye effort ko kam karti hai (force multiply karti hai) but force ki direction change nahi karti.
  • MA > 1 hota hai.
  1. System of Pulleys (Block and Tackle System):
  • Fixed aur movable pulleys ka combination hota hai.
  • Ye bahut heavy loads ko kam effort se uthane ke liye use hota hai.
  • MA > 1 hota hai, aur MA rope segments ki counting par depend karta hai jo load ko support karte hain.
  • Examples: Cranes, elevators.

Pulley ka Use

  • Pulleys ka main use convenience provide karna hai, ya toh force ki direction change karke ya required effort ko kam karke.
  • Work done by effort = Work done on load (friction ignore karne par).
⭐Remember

Fixed pulley ka MA hamesha 1 hota hai. Movable pulley ya pulley system ka MA 1 se zyada hota hai.

Inclined Plane as a Simple Machine

Inclined Plane kya hai?

  • Inclined Plane: Ek flat surface jo horizontal se ek angle par inclined hoti hai. Ye heavy objects ko low level se high level tak move karne mein help karti hai.
  • Examples: Ramps, hills par winding roads, ladders.

Working Principle

  • Directly heavy object ko vertically lift karne mein zyada force (object ke weight ke barabar) lagana padta hai.
  • Inclined plane par object ko push karne mein kam force lagana padta hai, but zyada distance cover karni padti hai.
  • Work done (force × distance) dono cases mein same rehta hai (friction ignore karne par).

Mechanical Advantage (MA) of an Inclined Plane

  • Load (L) = Object ka weight = \(mg\)
  • Effort (E) = Inclined plane par object ko push karne ke liye required force (\(F'\))
  • Height (h) = Vertical height tak object ko uthana hai.
  • Length (L_plane) = Inclined plane ki length.
  • Ideal case mein (frictionless): Work done by effort = Work done on load
  • \(F' \times L_{plane} = mg \times h\)
  • \(\frac{mg}{F'} = \frac{L_{plane}}{h}\)
  • So, \(MA = \frac{\text{Load}}{\text{Effort}} = \frac{mg}{F'} = \frac{L_{plane}}{h}\)
  • Kyunki \(L_{plane}\) hamesha \(h\) se zyada hoti hai, isliye inclined plane ka MA hamesha 1 se zyada hota hai.
  • Jitna kam angle (shallow slope), utni zyada \(L_{plane}\) aur utna hi zyada MA (effort kam).

Real Life Applications

  • Hills par roads winding banaye jaate hain, seedhe upar jaane ki bajaye, taaki slope kam ho aur vehicles ko kam effort lagana pade.
  • Ladders bhi inclined plane ka example hain. Inclined ladder par chadhna vertical ladder se zyada easy hota hai.
🧮Formula

Mechanical Advantage of Inclined Plane: \(MA = \frac{\text{Length of inclined plane}}{\text{Vertical height}} = \frac{L_{plane}}{h}\)

💡Tip

Inclined plane ke questions mein friction ko ignore karne par hi MA ka formula apply hota hai. Real life mein friction ke karan MA thoda kam hota hai.

Levers as Simple Machines and Their Classes

Lever kya hai?

  • Lever: Ek rigid bar (jaise rod ya plank) jo ek fixed point ke around rotate kar sakti hai. Is fixed point ko Fulcrum kehte hain.
  • Levers ka use heavy objects ko lift karne ya force ki direction change karne ke liye hota hai.

Lever ke Parts

  1. Fulcrum (F): Fixed pivot point jiske around lever rotate karta hai.
  2. Load (L): Woh force jise overcome karna hai (object ka weight).
  3. Effort (E): Woh force jo hum lever par apply karte hain.

Lever ka Principle

  • Principle of Moments: Equilibrium mein, Fulcrum ke around clockwise moment = anti-clockwise moment.
  • \(Effort \times Effort \ Arm = Load \times Load \ Arm\)
  • \(E \times d_E = L \times d_L\)
  • Effort Arm (\(d_E\)): Fulcrum se effort tak ki distance.
  • Load Arm (\(d_L\)): Fulcrum se load tak ki distance.
  • Agar effort arm, load arm se zyada ho (\(d_E > d_L\)), toh kam effort se zyada load uthaya ja sakta hai (MA > 1).

Mechanical Advantage (MA) of a Lever

  • \(MA = \frac{\text{Load}}{\text{Effort}} = \frac{E \times d_E}{L \times d_L} = \frac{d_E}{d_L}\)
  • Lever ka MA effort arm aur load arm ke ratio ke barabar hota hai.

Classes of Levers

Levers ko Fulcrum, Load aur Effort ki relative positions ke basis par teen classes mein divide kiya gaya hai:

Class 1 Lever

  • Arrangement: Fulcrum (F) Load (L) aur Effort (E) ke beech mein hota hai.
  • MA > 1, MA < 1, ya MA = 1 ho sakta hai.
  • Examples: Seesaw, scissors, crowbar, pliers, beam balance.

Class 2 Lever

  • Arrangement: Load (L) Fulcrum (F) aur Effort (E) ke beech mein hota hai.
  • Effort arm hamesha load arm se zyada hota hai (\(d_E > d_L\)).
  • MA hamesha > 1 hota hai (force multiplier).
  • Examples: Wheelbarrow, nutcracker, bottle opener, lemon squeezer.

Class 3 Lever

  • Arrangement: Effort (E) Fulcrum (F) aur Load (L) ke beech mein hota hai.
  • Effort arm hamesha load arm se kam hota hai (\(d_E < d_L\)).
  • MA hamesha < 1 hota hai (speed ya distance multiplier).
  • Examples: Fishing rod, tongs, tweezers, broom, human forearm.

Machines do not create energy

  • Koi bhi machine energy create nahi karti. Wo sirf energy ko ek form se dusre form mein convert karti hai ya force ko modify karti hai.
  • Input work (effort dwara kiya gaya) hamesha output work (load par kiya gaya) se zyada hota hai, kyunki friction mein energy loss hota hai.
  • Ideal machine mein input work = output work.
🧮Formula

Lever Principle: \(E \times d_E = L \times d_L\)

Mechanical Advantage of Lever: \(MA = \frac{d_E}{d_L}\)

💡Tip

Levers ki classes aur unke examples board exams mein frequently puche jaate hain. Har class ke 2-3 examples yaad rakhna.

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