physics notes for neet chapter 6.pdf
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60 Electrostatics 253 E3 Chapter 6 Work, Energy, Power and Collision Thus work done by a force is equal to the scalar (or dot product) of the force and the displacement of the body. ID Introduction If a number of forces F 1 , F 2 , F 3...... F n are acting on a body and it shifts from position vecto...
60 Electrostatics 253 E3 Chapter 6 Work, Energy, Power and Collision Thus work done by a force is equal to the scalar (or dot product) of the force and the displacement of the body. ID Introduction If a number of forces F 1 , F 2 , F 3...... F n are acting on a body and it shifts from position vector r 1 to position vector r 2 then W (F 1 F 2 F 3 .... F n ).( r 2 r 1 ) D YG U The terms 'work', 'energy' and 'power' are frequently used in everyday language. A farmer clearing weeds in his field is said to be working hard. A woman carrying water from a well to her house is said to be working. In a drought affected region she may be required to carry it over large distances. If she can do so, she is said to have a large stamina or energy. Energy is thus the capacity to do work. The term power is usually associated with speed. In karate, a powerful punch is one delivered at great speed. In physics we shall define these terms very precisely. We shall find that there is a loose correlation between the physical definitions and the physiological pictures these terms generate in our minds. Nature of Work Done Positive work Positive work means that force (or its component) is parallel to displacement Direction of motion Work is said to be done when a force applied on the body displaces the body through a certain distance in the direction of force. F s Work Done by a Constant Force Let a constant force F be applied on the body such that it makes an angle with the horizontal and body is displaced through a distance s U By resolving force F into two components : 0 o 90 o Fig. 6.2 The positive work signifies that the external force favours the motion of the body. Example: (i) When a person lifts a body from the ground, the work done by the (upward) lifting force is positive (i) F cos in the direction of displacement of the body. ST (ii) F sin in the perpendicular direction of displacement of the body. F man F sin F s F cos s Since body is being displacedFig. in 6.1 the direction of F cos , therefore work done by the force in displacing the body through a distance s is given by W (F cos ) s Fs cos or W F.s 6.3 by applying a force along the (ii) When a lawn roller is Fig. pulled handle at an acute angle, work done by the applied force is positive. F s (iii) When a spring is stretched, work done by the external Fig. 6.4 (stretching) force is positive. s F Fig. 6.5 s F 254 Work, Energy, Power and Collision Fg Maximum work : Wmax F s s When cos maximum 1 i.e. 0 o Minimum work : Wmin F s Direction of motion F F + s + s 90 o 180 o Fig. 6.8 When cos minimum 1 i.e 180 o Fig. 6.6 Zero work Under three condition, work done becomes zero W Fs cos 0 o (iii) When a positive charge is moved towards another positive charge. The work done by electrostatic force between them is negative. U (1) If the force is perpendicular to the displacement [ F s ] It means force does minimum [maximum negative] work when angle between force and displacement is 180. ID The negative work signifies that the external force opposes the motion of the body. Example: (i) When a person lifts a body from the ground, the work done by the (downward) force of gravity is negative. E3 When a coolie travels on a horizontal platform with a load on his head, work done against gravity by the coolie is zero. s D YG Example: (i) (ii) When a body is made Fig. to 6.7 slide over a rough surface, the work done by the frictional force is negative. 60 It means force does maximum work when angle between force and displacement is zero. Negative work Negative work means that force (or its component) is opposite to displacement i.e. (ii) When a body moves in a circle the work done by the centripetal force is always zero. Fg (iii) In case of motion of a charged particle in a magnetic field as force [F q(v B)] is always perpendicular to motion, work done by this force is always zero. (2) If there is no displacement [s = 0] When a person tries to displace a wall or heavy stone by applying a force and it does not move, then work done is zero. F U Example: (i) s (ii) A weight lifter does work in lifting the weight off the ground but does not work in holding it up. 0 ST (3) If there is no force acting on the body [F = 0] Example: Motion of an isolated body in free space. Work Done by a Variable Force The total work done in going from A to B as shown in the figure is When the magnitude and direction of a force varies with position, the work done by such a force for an infinitesimal displacement is given by W A F. d s A (F cos )ds dW F. d s B ds F B B In terms of rectangular component F Fx ˆi Fy ˆj Fz kˆ d s dxˆi dyˆj dzkˆ B W A (Fx ˆi Fy ˆj Fz kˆ ).(dxˆi dyˆj dzkˆ ) x y z or W x B Fx dx y B Fy dy z B Fz dz A Fig. 6.9 A A A Work, Energy, Power and Collision 255 Work Done in Conservative and Dimension and Units of Work Dimension : As work = Force displacement Non-conservative Field (1) In conservative field, work done by the force (line integral of the [W] [MLT 2 ] [L] [ML2 T 2 ] Units : The units of work are of two types points. W AB W AB W AB Gravitational units Joule [S.I.]: Work done is said to be one Joule, when 1 Newton force displaces the body through 1 metre in kg-m [S.I.]: 1 kg-m of work is done when a force of 1kg-wt. displaces the body through 1m its own direction. in its own direction. From, W = F.s From 1 Joule = 1 Newton 1 m 1 kg-m = 1 kg-wt 1 m Path I or Relation between Joule and erg 1 Joule = 1 N 1 m (2) In conservative field work done by the force (line integral of the force i.e. F.d l ) over a closed path/loop is zero. WAB WB A 0 or F.d l 0 From W = F s 1 gm-cm = 1gm-wt 1cm. = 981 dyne 1cm = 981 erg U 7 D YG Work Done Calculation by Force Displacement Graph Let a body, whose initial position is x i , is acted upon by a variable force (whose magnitude is changing continuously) and consequently the body acquires its final position x f. Force U F O xi x xf dx Displacement ST Let F be the average value of Fig. variable 6.10 force within the interval dx from position x to (x + dx) i.e. for small displacement dx. The work done will be the area of the shaded strip of width dx. The work done on the body in displacing it from position x i to x f will be equal to the sum of areas of all the such strips dW F dx xf xi dW xf A B Fig. 6.12 Conservative force : The forces of these type of fields are known as conservative forces. Example : Electrostatic forces, gravitational forces, elastic forces, magnetic forces etc and all the central forces are conservative in nature. If a body of mass m lifted to height h from the ground level by different path as shown in the figure B = 10 dyne cm = 10 erg W E3 work is done when a force of 1gm-wt displaces the body through 1cm in its own direction. 2 7 Path III III Fig. 6.11 gm-cm [C.G.S.] : 1 gm-cm of = 10 dyne 10 cm 5 Path II ID 1 erg 1dyne 1cm B II F.d l F.d l F.d l Path I = 9.81 Joule From W = F s I A Path III W=Fs = 9.81 N 1 metre its own direction. Path II 60 Absolute units erg [C.G.S.] : Work done is said to be one erg when 1 dyne force displaces the body through 1 cm in F.d l ) is independent of the path followed between any two force i.e. F dx xi x W x f (Area of stripof widthdx ) i W Area under curve between x i and x f i.e. Area under force-displacement curve with proper algebraic sign represents work done by the force. B I II B III l A A B IV h1 h2 h h3 A A Work done through different Fig.paths 6.13 WI F. s mg h mgh h mgh sin WIII mgh1 0 mgh2 0 mgh3 0 mgh4 WII F. s mg sin l mg sin mg(h1 h2 h3 h4 ) mgh WIV F. d s mgh It is clear that WI WII WIII WIV mgh. Further if the body is brought back to its initial position A, similar amount of work (energy) is released from the system, it means WAB mgh and WBA mgh. Hence the net work done against gravity over a round trip is zero. WNet WAB WBA mgh (mgh) 0 i.e. the gravitational force is conservative in nature. Non-conservative forces : A force is said to be non-conservative if work done by or against the force in moving a body from one position to another, depends on the path followed between these two positions and for complete cycle this work done can never be zero. 256 Work, Energy, Power and Collision Example: Frictional force, Viscous force, Airdrag etc. If a body is moved from position A to another position B on a rough 1 Joule = 10 7 erg 1 eV = 1.6 10 19 Joule table, work done against frictional force shall depend on the length of the path between A and B and not only on the position A and B. 1 kWh = 3.6 10 6 Joule 1 calorie = 4.18 Joule WAB mgs (4) Mass energy equivalence : Einstein’s special theory of relativity shows that material particle itself is a form of energy. The relation between the mass of a particle m and its equivalent energy is given as Further if the body is brought back to its initial position A, work has to be done against the frictional force, which opposes the motion. Hence the net work done against the friction over a round trip is not zero. R E mc 2 where c = velocity of light in vacuum. s 60 If m 1 amu 1.67 10 27 kg F then E 931 MeV 1.5 10 10 Joule. WBA mgs. If m 1kg then E 9 1016 Joule Fig. 6.14 WNet WAB WBA mgs mgs 2mgs 0. i.e. the friction is a non-conservative force. Examples : (i) Annihilation of matter when an electron (e ) and a E3 positron (e ) combine with each other, they annihilate or destroy each other. The masses of electron and positron are converted into energy. This energy is released in the form of -rays. Work Depends on Frame of Reference With change of frame of reference (inertial), force does not change while displacement may change. So the work done by a force will be different in different frames. Examples : (1) If a porter with a suitcase on his head moves up a staircase, work done by the upward lifting force relative to him will be zero (as displacement relative to him is zero) while relative to a person on the ground will be mgh. (2) If a person is pushing a box inside a h moving train, the work done in the frame of train Fig. 6.15 e e Each photon has energy = 0.51 MeV. ID Here two photons are emitted instead of one photon to conserve the linear momentum. (ii) Pair production : This process is the reverse of annihilation of matter. In this case, a photon ( ) having energy equal to 1.02 MeV interacts will F.s D YG U with a nucleus and give rise to electron (e ) and positron (e ). Thus energy is converted into matter. while in the frame of earth will be F. (s s 0 ) where s 0 is the displacement of the train relative to the ground. (Photon) e– + e+ Fig.nucleus 6.16 (iii) Nuclear bomb : When the is split up due to mass defect (The difference in the mass of nucleons and the nucleus), energy is released in the form of -radiations and heat. ST U (5) Various forms of energy (i) Mechanical energy (Kinetic and Potential) Energy (ii) Chemical energy The energy of a body is defined as its capacity for doing work. (iii) Electrical energy (1) Since energy of a body is the total quantity of work done, (iv) Magnetic energy therefore it is a scalar quantity. (v) Nuclear energy (2) Dimension: [ML2T 2 ] it is same as that of work or torque. (vi) Sound energy (3) Units : Joule [S.I.], erg [C.G.S.] (vii) Light energy Practical units : electron volt (eV), Kilowatt hour (KWh), Calories (viii) Heat energy (cal) (6) Transformation of energy : Conversion of energy from one form Relation between different units: to another is possible through various devices and processes. Table : 6.1 Various devices for energy conversion from one form to another Mechanical electrical N Dynamo Light Electrical Chemical electrical Light Anode S – Photoelectric A cell + – Primary cell Cathode + Work, Energy, Power and Collision 257 Chemical heat Sound Electrical Heat electrical Fe Hot Cold G Coal Burning Microphone Electrical Mechanical Electrical Heat Engine E3 60 Heat Mechanical Cu Thermo-couple Motor Heater Electrical Chemical Anode + Speaker Voltameter Electrical Light ID Electrical Sound Cathode – Bulb U Electrolyte Kinetic Energy v 2 0 2as s D YG The energy possessed by a body by virtue of its motion, is called kinetic energy. Examples : (i) Flowing water possesses kinetic energy which is used to run the water mills. (ii) Moving vehicle possesses kinetic energy. (iii) Moving air (i.e. wind) possesses kinetic energy which is used to run wind mills. (iv) The hammer possesses kinetic energy which is used to drive the nails in wood. U (v) A bullet fired from the gun has kinetic energy and due to this energy the bullet penetrates into a target. u=0 ST F Since the displacement of the body is in the direction of the applied force, then work done by the force is W F s ma Fig. 6.17 m = mass of the body, u = Initial velocity of the body (= 0) 1 2 This work done appears as the kinetic energy of the body 1 KE W mv 2 2 (2) Calculus method : Let a body is initially at rest and force F is applied on the body to displace it through small displacement d s along its own direction then small work done dW F.d s F ds dW m a ds dW m F = Force acting on the body, a = Acceleration of the body, s = Distance travelled by the body, v = Final velocity of the body From v 2 u 2 2as v2 2a W mv 2 s (1) Expression for kinetic energy : Let v v2 2a dW mdv. dv ds dt [As F = ma] dv As a dt ds dt dW m v dv …(i) 258 Work, Energy, Power and Collision ds As dt v Therefore work done on the body in order to increase its velocity from zero to v is given by v v 1 v v W 0 mv dv m 0 v dv m mv 2 2 2 0 Work done = change in kinetic energy W E This is work energy theorem, it states that work done by a force acting on a body is equal to the change in the kinetic energy of the body. 1 m (v. v ) 2 As m and v.v are always positive, kinetic energy is always positive scalar i.e. kinetic energy can never be negative. (3) Kinetic energy depends on frame of reference : The kinetic energy of a person of mass m, sitting in a train moving with speed v, is zero 1 in the frame of train but mv 2 in the frame of the earth. 2 If kinetic energy of the body increases, work is positive i.e. body moves in the direction of the force (or field) and if kinetic energy decreases, work will be negative and object will move opposite to the force (or field). Examples : (i) In case of vertical motion of body under gravity when the body is projected up, force of gravity is opposite to motion and so kinetic energy of the body decreases and when it falls down, force of gravity is in the direction of motion so kinetic energy increases. (ii) When a body moves on a rough horizontal surface, as force of friction acts opposite to motion, kinetic energy will decrease and the decrease in kinetic energy is equal to the work done against friction. (6) Relation of kinetic energy with linear momentum: As we know E 1 1 P mv 2 v 2 2 2 v ID (4) Kinetic energy according to relativity : As we know This theorem is valid for a system in presence of all types of forces (external or internal, conservative or non-conservative). 60 This work done appears as the kinetic energy of the body 1 KE mv 2. 2 E3 2 In vector form KE 1 E mv 2. 2 E 1 Pv 2 or E P2 2m But this formula is valid only for (v
