The product of force and displacement ( in the direction of force ), during which the force is acting, is defined as work.

Transcription

1 5 WORK, ENERGY ND POWER Page 5. Work The product of force and displaceent ( in the direction of force ), during which the force is acting, is defined as work. When N force is applied on a particle and the resulting displaceent of the partic e, in the direction of the force, is, the work done is defined as J ( joule ). The diensional forula of work is M L T. The displaceent ay not be in the direction of the applied force in all cases. In the figure shown, the displaceent, d, akes an angle with the applied force, F. ccording to the definition of force, Work, W force displaceent in the direction of the force F ( d cos ) ( F cos ) ( d ) ( the coponent of force in the direc ion of displaceent ) ( displaceent ) ( i ) For π /, work W 0, een en if F nd d are both non-zero. In unifor circular otion, the centripetal force acti on a particle is perpendicular to its displaceent. Hence, the work done due to centripetal force during such a otion is zero. ( ii ) If <, work done is posit e and is said to be done on the object by the force. ( iii ) If / < θ, work d ne is negatie and is said to be done by the object against the force. 5. Scalar product of two ectors The scalar product of two ectors, putting a dot ( ) between the two ectors and is defined as: B and B, also known as the dot product, is written by B l l l B l cos B cos, where θ is the angle between the two ectors. To obtain the scalar product of and B, they are to be drawn fro a coon point, O, with the sae agnitudes and directions as shown in the figure. w.ex ce.co M is the foot of perpendicular fro the head of t o B. OM ( cos θ is the agnitude of projection of o n B.

2 5 WORK, ENERGY ND POWER Page Siilarly, N is the foot of perpendicular fro the head of agnitude of projection of B o n. or B B cos B ( cos ) ( B ) ( OM ) ( agnitude of B ) ( agnitude of projection of B B cos ( B cos ) ( ) ( ON ) ( agnitude of ) ( agnitude of projection of B t o and ON ( B cos is the o n B ) B o n ) Thus, scalar product of two ectors is equal to the product of agnitude of one ector with the agnitude of projection of second ector on the direction f the first ector. The scalar product of ectors is zero if the angle between the ectors /, positie if 0 < and negatie if /. Properties of scalar product ( ) Coutatie law: B B cos θ B cos B Thus, scalar product of two ectors s c utatie. ( ) Distributie law: OP, OQ B and as shown in the f gure. Now, ( B + C ) l l l projection of OR C B + C on l l ( ON ) l l ( OM + MN ) l l l ( OM ) + l l ( MN l l l proj. of B on l + l l l proj. of B + C ww l Thus, scalar product of two ectors is distributie with respect to suation. ( 3 ) If l l ll B, 0, C B B cos 0 B and l l l l l l Thus, agnitude of a ector is equal to the square root of scalar product of the ector with itself. on l

3 5 WORK, ENERGY ND POWER Page 3 ( 4 ) If B, 90 B B cos 90 0 Thus, the scalar products of two utually perpendicular ectors is zero. ( 5 ) Scalar products of unit ectors in Cartesian co-ordinate syste: i i j j k k and i j j k k i 0 ( 6 ) Scalar product in ters of Cartesian coponents of ectors: ( 7 ) If x i + y j + z k and B B x i + B y j + B z k, then B ( x i + y j + z k ) ( B x i + B y j + B z k ) x B x + y B y + z B z B B cos cos B B x x B x y y B y z B x This forula is used to find the angle betwe n two ectors. 5.3 Work done by a ariable force force arying with position or tie is known as the ariable force. To find work done under a ariable force, consider a two-die sional otion of a particle oing along a cured path B as shown in the figure under the effect of a ariable force The cured path is diided into infinitesially s all line segents l, l,, l. Let F, F,, F n be the forces acting on the particle at line segents, l, l,, z B z l n respectiely. s the line segents are ery sall, the force oer each segent can be considered constant. The total work as the particle oes fro to B can be obtained as the su of the work done for different line segents as under. Total work, W F l F l F n ln B y www w.exa rac ace.co B z

4 5 WORK, ENERGY ND POWER Page 4 B F i li, i,, 3, If l li l l 0 B B, then W F d l F cos θ dl For one-diensional otion of the particle along X-axis and force acting in he direction of otion as shown in the figure, B Work, W F dx cos 0 o x x F d x where x and x are the x-coordinates of and B respectiely. It can be seen fro the figure that for a sall displaceent dx, the force F is alost cons ant and hence work done is Fdx which is the a ea of the strip. Total work for otion of the particle fro to B is the su of areas of uc st ips which is the area under the cure between x x and x x. For a particle oing under the ef ect of a constant force on a cured path fro the point r to r as shown in the figure, work don W r r F d r F ( r - r ) ( since, F is constant ) Thus, he work done by the constant force is the scalar product of the force and the displaceent ector of the pa ticle e.co 5.4 kinetic energy The capacity of a body to do work, by irtue of its otion, is known as the kinetic energy of the body. More the speed of the object, ore is its kinetic energy. The net force acting on a body produces acceleration or deceleration in its otion thereby changing its elocity and kinetic energy. lso, work is done by the net force on the body while causing its displaceent. The work, W, done by the force, F, on a particle of ass,, causing acceleration, a and

5 displaceent, d is gien by W 5 WORK, ENERGY ND POWER Page 5 F d a d ( ) Using the result, - 0 a d in the aboe equation, - 0 W - 0, where 0 and are the sp eds before and after the application of force respectiely. K - K 0 change in kinetic energy K, where K 0 and K are the initial and final kinetic energies. The unit of kinetic energy is the sae as that of work, i.e., joule. Thus, the work done by the resultant force on a body, in the absence of dissipatie forces, is equal to the change in the kinetic energy of the body. This stateent is known as the work-energy the re. When a particle perfors unifor circular otion, the centripetal force acting on it is perpendicular to its tangential instantaneous displaceent and hence no work is done by the centripetal force and the speed and kin c en rgy of the particle do not change. If the body is displaced in the directi n of the force acting on it, work is done by the force on the body and is positie. The inet c energy of the body increases in this case. If the displaceent of the body is against t e force acting on it, work is done by the body and is negatie. The kinetic energy f th body decreases in this case. lso, kinetic energy, K 5.5 Potential energy p, where p linear oentu of the body. The capacity of a body to do work, by irtue of its position in a force field or due to its configuration is known as the potential energy of the body. Gra itational potential energy: ww.e The graitational acceleration, g, due to the Earth s graitational force can be taken as constant for heights uch saller as copared to the radius of the Earth. If a body of ass oes fro height y to height y in the Earth s graitational field, the force on the body - g j and its displaceent ( y - y ) j the work done, W F d - g j (y - y ) j gy - gy ( )

6 5 WORK, ENERGY ND POWER Page 6 Here, the work depends on the initial and final positions of the body and not on the path followed. lso, there is no loss of echanical energy if friction and air resistance are ignored. In this case, the force is known as conseratie force and the field as conseratie field. If and are elocities of the body at heights y and y respectiely, then according to work-energy theore, The work done, W - gy - gy [ fro equation ( ) aboe ] Here, gy and gy are the potential energies of the body at heights y and y respectiely fro the surface of the Earth due to the Earth s graitational field. Surface of the Earth is the reference leel fro her potential energy is calculated. Besides, the Earth s surface, any other leel can be chosen as reference leel. Thus, graitational potential energy of a body of ass at height h fro the surface of the Earth is U gh. Fro the aboe equation, gy + + gy Thus, in a conseratie e field, he echanical energy, E, which is the su of kinetic energy ( K ) and potent al energy ( U gh ) reains constant. E K + U. Thus, the echani al energy of an isolated syste of bodies is consered in a conseratie field. This stateent is known as the law of conseration of echanical energy. 5.6 Elastic potential energy: ( Potential energy due to the configuration of a syste ) On end of an elastic spring of negligible ass and obeying Hooke s law is tied with a rigid wall and to the other end a block of ass is tied. The block is at x 0 when the spring is not extended or copressed. On displacing the block, the restoring force is produced in the spring which tries to restore the block to its original position. The restoring force is directly proportional to the displaceent of the block and is in a direction opposite to the displaceent.

7 5 WORK, ENERGY ND POWER Page 7 thew F - x or F - kx, where k is the force constant of the spring which is defined as the force required to pull or copress the spring by unit displaceent. Its unit is N and its diensional forula is M L 0 T. Work done by the applied force on the spring is, W x x kxdx k xdx k 0 0 x x 0 kx This work done on the spring is stored in the for of elastic potential energy of the spring. Taking the potential energy to be zero for x 0, for hange i length equal to x, the potential stored in the spring will be U kx Relation between force and potential energy The change in kinetic energy, K, of the par icle is equal to work, W, done on it by force, F, in displacing it through a sall distance, x. K W F x But, by the law of conseration of echanical energy, U K + U 0 F x + U 0 F - x U du instantaneous alue of F - li - x 0 x dx In the case of a spring, the potential energy, U kx du F - - k ( x ) - kx dx 5.7 Power This equation holds only in the case of conseratie forces. w.exarace.co Th tie rate of doing work is known as power. or Power is defined as the work done per unit tie. www If W is the work done in tie interal t, the aerage power in tie interal t is < P > P W t li t 0 and W t instantaneous power at tie t is dw dt If dw is the work done by a force F during the displaceent dr, then

8 5 WORK, ENERGY ND POWER Page 8 dw dr P F F, where is the instantaneous elocity at tie t. dt dt Power is a scalar quantity like work and energy. Its unit is watt ( W ). W J /s. Its diensional forula is M L T 3. kilowatt ( kw ) 0 3 watt ( W ) and egawatt ( MW ) 0 6 W. In British syste, unit of power is horsepower ( hp ). hp 746 W. kilowatt-hour (kwh) is the electrical unit of energy (work) as the p oduct f power and tie. unit of electrical energy kwh J. 5.8 Elastic and inelastic collisions In elastic collision, total linear oentu and tot l kinetic energy of the colliding bodies are consered. In inelastic collision, total linear oentu of the colliding bodies is consered, but part or whole of the kinetic energy is lost in her fors of energy. In both, elastic as well as inelastic coll sions, total energy ( energy of all fors, echanical, internal, sound, etc. ) and total linear oentu are consered. Inelastic collision in one diension: Partly inelastic collision: Suppose a sphere f a s oing with elocity in X-direction collides with another sphere of ass oing in the sae direction with elocity. ( > ) Let and be their elocities in the sae ( X ) direction after the collision which we w nt to find. cording to the law of conseration of linear oentu, + + or, ( - ) ( - ) ( ) Now, a paraeter known as coefficient of restitution, e, is defined as under: ' ' Co-efficient of restitution, e - elocity of separation after the collision - elocity of approach before the collision

9 5 WORK, ENERGY ND POWER Page 9 For partly inelastic collision, 0 < e <. ' ' e ( ) ( ) Soling equations ( ) and ( ) for and, we get ' e ( e ) and Thus, and can be calculated fro the aboe equations. Copletely inelastic collision: ( e ) ' e Putting e 0 for copletely inelastic collision in the aboe equations, ' ' ( the coon elocity the spheres after collision ) Thus, in copletely inelastic collision, the colliding bodies oe jointly with a coon elocity. Here, the kinetic energy before collision, K i + and the kinetic energy after collision, K f ( ) ( ) K f - K i - - ( ) ( ) The negatie sign indicates that the k netic energy decreases during inelastic collision. Elastic collision in one d ension: Putting e for ela tic c llision in equation ( ), we get + + ( 3 ) Multiplying equations ( ) and ( 3 ), we get ' ' - ( - ) - ) ( - ( ). Multiplying by and rearranging, w.exa exarace.co exa(ra co This shows that the kinetic energy is consered for e. Hence, elastic collision can also be defined as the one in which e. Special cases of elastic collision: Taking e for elastic collision, the elocities of the spheres after the collision are gien by ' - and ' - -

10 5 WORK, ENERGY ND POWER Page 0 ( ) For >>, neglecting as copared to in the aboe equations, we get and - This shows that the larger sphere continues to oe with the sae elocity, whereas the elocity of the saller sphere increases. If the saller sphere was at rest, it oes with twice the elocity of the larger sphere after the collision. ( ) For >>, neglecting as copared to in the aboe equatio s, we get and - In this case also, the larger sphere continues to oe with he sae elocity, but the elocity of the saller sphere changes. ( i ) If the saller sphere were oing with twice he speed of the larger sphere, it becoes stationary after the collision. ( ii ) If the saller sphere were oing with elocity less than twice the elocity of the larger sphere, it continues to oe in he sae direction, but with decreased speed. ( iii ) If the saller sphere were oing with elocity ore than twice the elocity of the larger sphere, it rebounds and starts oing in the opposite direction with the elocity gien as aboe. If the larger sphere was stati nary, it reains stationary. In this case, the saller sphere rebounds and oe with the sae speed in the reerse direction. Suarizing the aboe t o cases, when one of the two spheres colliding is uch ore assie than the other then he elocity of the larger sphere reains alost the sae after the collision, whereas the elocity of the saller sphere after the collision is alost twice the elocity of larger sphere less the elocity of the saller sphere before the collision. ( 3 ) If then and. Thus, in elastic collision of two spheres of equal ass their elocities get exchanged. Elastic collisi n in two diensions Suppose a sphere of ass oing in X direction with elocity collides elastically with ww.exarace.co a stationary ( 0 ) sphere of ass as shown in the figure. fter the collision, they oe in the directions aking angles θ and with the X-axis with elocities ' and ' respectiely. ccording to the law of conseration of oentu, ' + '

11 5 WORK, ENERGY ND POWER Page Equating the X- and Y-coponents of the oenta ' cos + ' cos ( ) and 0 ' sin - ' sin ( ) s the collision is elastic, kinetic energy is consered, + ( 3 ) Using the aboe three equations, any three unknown quantities can be deterined. 5.9 Different fors of energy Besides echanical energy, soe exaples of other fors of nergy are as under. Internal Energy: The su of ibrational kinetic energy and potential energy due to utual attraction and repulsion between constituent particles of a substa ce is known as internal energy of the body. Due to work done against friction, interna energy and hence teperature of the body increases. Heat or theral energy: The kinetic energy of the constituen particles of a body due to their rando otion is known as heat or theral energy of th body. The difference between internal nal energy and heat is analogous to echanical energy and work. When work is done by body o body B, echanical energy of body decreases and that of body B increases. Siilar y, when heat is transferred fro body to body B, internal energy of body decr ases and that of body B increases. Cheical energy: stable copound has energy less than its constituent eleents in free state. This difference is known as c eical energy or cheical binding energy. In a cheical reaction, if the cheical energy of the products is less than that of the reactants, heat is eoled and the reaction is called exotheric. Here, cheical energy got conerted into heat energy. Siilarly, if the cheical energy of the products is ore than that of the reactants, heat is absorbed and the rea tion is called endotheric. Here, heat energy got conerted into cheical energy. Elec rical energy: The energy associated with electric current is known as electrical energy. When electric current wcu is used in a heater, electrical energy is conerted into heat energy. Siilarly, in a lap, it is conerted into heat and light energy. Nuclear energy: The ass of a nucleus is less than the su of the asses of its constituent protons and neutrons in free state. The energy equialent to this ass difference is known as nuclear energy or nuclear binding energy. In a nuclear fission reaction, when heay nuclei like uraniu are bobarded by neutrons, they break up into saller nuclei releasing huge aount of nuclear energy. Such reactions are used in nuclear reactors for producing power and in

12 5 WORK, ENERGY ND POWER Page atoic weapons. In the Sun and stars, nuclear fusion reaction occurs in which lighter nuclei like protons, deuterons fuse at high teperature to for a heliu nucleus releasing huge aount of energy. t the icroscopic leel, all different fors of energy are in the for of potential and or kinetic energy. Equialence of ass and energy lbert Einstein gae an equation for inter-conersion of ass and energy as under E c, where c s is the elocity of light in acuu. Conseration of energy The total energy of an isolated syste reains constant. This i the stateent of law of conseration of energy. One for of energy ay get conerted into nother for or energy. Energy cannot be created or destroyed. The unierse is an isolated syste and so the total energy of the unierse reains constant.