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2 Work Energy Power and Collision, Direction of motion. F Direction of motion, s 90 o 180, The positive work signifies that the external force favours. the motion of the body 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 Example i When a person lifts a body from the. the work done by the upward lifting force is positive ground the work done by the downward force of. gravity is negative, ii When a body is made to slide over a rough. ii When a lawn roller is pulled by applying a force along. surface the work done by the frictional force is, the handle at an acute angle work done by the applied.

force is positive, iii When a positive charge is moved towards. iii When a spring is stretched work done by the, another positive charge The work done by. external stretching force is positive, electrostatic force between them is negative. Maximum work Wmax F s Minimum work Wmin F s, When cos maximum 1 i e 0 o When cos minimum 1 i e 180 o. It means force does maximum work when angle between It means force does minimum maximum. force and displacement is zero negative work when angle between force and. displacement is 180 o, Under three condition work done becomes zero W Fs cos 0.

1 If the force is perpendicular to the displacement F s. Example i When a coolie travels on a horizontal platform. with a load on his head work done against s, gravity by the coolie is zero Work done against. friction is ve, ii When a body moves in a circle the work done by Fg. the centripetal force is always zero, 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, Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda.

Work Energy Power and Collision 3, 2 If there is no displacement s 0. Example i When a person tries to displace a wall or heavy F. stone by applying a force then it does not move, the work done is zero s 0. ii A weight lifter does work in lifting the weight off. the ground but does not work in holding it up, 3 If there is no force acting on the body F 0. Example Motion of an isolated body in free space, Sample Problems based on work done by constant force. Problem 1 A body of mass 5 kg is placed at the origin and can move only on the x axis A force of 10 N is. acting on it in a direction making an angle of 60 o with the x axis and displaces it along the x. axis by 4 metres The work done by the force is, a 2 5 J b 7 25 J c 40 J d 20 J.

Solution d Work done F s F s cos 10 4 cos 60 20 J, Problem 2 A force F 5 i 3 j N is applied over a particle which displaces it from its origin to the point. r 2 i 1 j metres The work done on the particle is, a 7 J b 13 J c 7 J d 11 J. Solution c Work done F r 5 i 3 j 2 i j 10 3 7 J, Problem 3 A horizontal force of 5 N is required to maintain a velocity of 2 m s for a block of 10 kg mass. sliding over a rough surface The work done by this force in one minute is. a 600 J b 60 J c 6 J d 6000 J, Solution a Work done Force displacement F s F v t 5 2 60 600 J. Problem 4 A box of mass 1 kg is pulled on a horizontal plane of length 1 m by a force of 8 N then it is raised. vertically to a height of 2m the net work done is, a 28 J b 8 J c 18 J d None of above.

Solution a Work done to displace it horizontally F s 8 1 8 J. Work done to raise it vertically F s mgh 1 10 2 20 J. Net work done 8 20 28 J, Problem 5 A 10 kg satellite completes one revolution around the earth at a height of 100 km in 108. minutes The work done by the gravitational force of earth will be. 108 10 100 10, a 108 100 10 J b J c J d Zero, Solution d Work done by centripetal force in circular motion is always equal to zero. 6 4 Work Done by a Variable Force, 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 dW F d s. The total work done in going from A to B as shown in the figure is. Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda. 4 Work Energy Power and Collision, In terms of rectangular component F Fx i Fy j Fz k ds. d s dx i dy j dz k, W F i F j F k dx i dy j dz k, Sample Problems based on work done by variable force.

Problem 6 A position dependent force F 7 2 x 3 x 2 N acts on a small abject of mass 2 kg to displace. it from x 0 to x 5m The work done in joule is CBSE PMT 1994. a 70 J b 270 J c 35 J d 135 J, Solution d Work done. F dx 7 2 x 3 x dx 7 x x 2 x 3 50 35 25 125 135 J, Problem 7 A particle moves under the effect of a force F Cx from x 0 to x x1 The work done in the. process is, a Cx 12 b Cx 12 c Cx 1 d Zero, Solution b Work done F dx Cx dx C C x 12. Problem 8 The vessels A and B of equal volume and weight are immersed in water to a depth h The vessel. A has an opening at the bottom through which water can enter If the work done in immersing. A and B are W A and WB respectively then, a WA WB b WA WB c WA WB d WA WB. Solution b When the vessels are immersed in water work has to be done against up thrust force but due to. opening at the bottom in vessel A up thrust force goes on decreasing So work done will be less. in this case, Problem 9 Work done in time t on a body of mass m which is accelerated from rest to a speed v in time t1.

as a function of time t is given by, 1 v 2 v 2 1 mv 2 1 v2 2. a m t b m t c t d m t, 2 t1 t1 2 t1 2 t12, 1 1 1 v 2 v. Solution d Work done F s ma a t 2 m a 2 t 2 m t As accelerati on a given. 2 2 2 t1 t1, 6 5 Dimension and Units of Work, Dimension As work Force displacement. W Force Displacement, MLT 2 L ML 2 T 2, Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda. Work Energy Power and Collision 5, Units The units of work are of two types.

Absolute units Gravitational units, Joule S I Work done is said to be one Joule when 1 kg m S I 1 Kg m of work is done when a force. Newton force displaces the body through 1 meter in its own of 1kg wt displaces the body through 1m in its. direction own direction, From W F s From W F s, 1 Joule 1 Newton 1 metre 1 kg m 1 kg wt 1 metre. 9 81 N 1 metre 9 81 Joule, Erg C G S Work done is said to be one erg when 1 gm cm C G S 1 gm cm of work is done when. dyne force displaces the body through 1 cm in its own a force of 1gm wt displaces the body through 1cm. direction in its own direction, From W F s From W F s. 1 Erg 1Dyne 1cm 1 gm cm 1gm wt 1cm 981 dyne, Relation between Joule and erg.

1 Joule 1 N 1 m 105 dyne 102 cm, 107 dyne cm 107 Erg. 6 6 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. Let F be the average value of variable 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 Force, Area of strip of width dx. O xi xf Displacement, W Area under curve Between x i and x f dx. i e Area under force displacement curve with proper algebraic sign represents work done by the. Sample problems based on force displacement graph, Problem 10 A 10 kg mass moves along x axis Its acceleration as a function of its position is shown in the. figure What is the total work done on the mass by the force as the mass moves from x 0 to. x 8 cm AMU Med 2000, a 8 10 2 J b 16 10 2 J, c 4 10 4 J d 1 6 10 3 J 20.

Solution a Work done on the mass mass covered area between the 10. graph and displacement axis on a t graph 5, 10 8 10 2 20 10 2 8 10 2 J 0 2 4 6 8 x cm. Problem 11 The relationship between force and position is shown in the figure given in one dimensional. case The work done by the force in displacing a body from x 1 cm to x 5 cm is. Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda. 6 Work Energy Power and Collision, a 20 ergs 20, 1 2 3 4 5 6. c 70 ergs 10, d 700 ergs, Solution a Work done Covered area on force displacement graph 1 10 1 20 1 20 1 10 20. Problem 12 The graph between the resistive force F acting on a body and the distance covered by the body is. shown in the figure The mass of the body is 25 kg and initial velocity is 2 m s When the. distance covered by the body is 5 m its kinetic energy would be. 0 1 2 3 4 x m, Solution d Initial kinetic energy of the body mu 2 25 2 2 50 J. Final kinetic energy Initial energy work done against resistive force Area between graph and. displacement axis 50 4 20 50 40 10 J, 6 7 Work Done in Conservative and Non Conservative Field.

1 In conservative field work done by the force line integral of the force i e F d l is. independent of the path followed between any two points. W A B W A B W A B I, Path I Path II Path III II, or F d l F d l F d l III. Path I Path II Path III, 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, W A B WB A 0, or F dl 0 A B, 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 man m lifted to height h from the ground level by different path as shown in the. I II II I h, Work done through different paths A A A A.

WI F s mg h mgh, WII F s mg sin l mg sin mgh, WIII mgh 1 0 mgh 2 0 mgh 3 0 mgh 4 mg h1 h2 h3 h4 mgh. WIV F d s mgh, Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda. Work Energy Power and Collision 7, 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 strip is zero. WNet WAB WBA, 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 a zero. Example Frictional force Viscous force Airdrag etc. If a body is moved from position A to another position B on a rough table work done against. frictional force shall depends on the length of the path between A and B and not only on the position. Further if the body is brought back to its initial position A work has to be done against the. frictional force which always opposes the motion Hence the net work done. against the friction over a round trip is not zero R. WNet WAB WBA mgs mgs 2 mgs 0 F, i e the friction is a non conservative force. Sample problems based on work done in conservative and non conservative field. Problem 13 If W1 W2 and W3 represent the work done in moving a particle from A to B along three. different paths 1 2 and 3 respectively as shown in the gravitational field of a point mass m. find the correct relation, a W1 W2 W3 b W1 W2 W3, c W1 W2 W3 d W2 W1 W3 A. Solution b As gravitational field is conservative in nature So work done in moving a particle from A to B. does not depends upon the path followed by the body It always remains same. Problem 14 A particle of mass 0 01 kg travels along a curve with velocity given by 4 i 16k ms 1 After some. time its velocity becomes 8 i 20 j ms 1 due to the action of a conservative force The work. done on particle during this interval of time is, a 0 32 J b 6 9 J c 9 6 J d 0 96 J. Solution d v1 4 2 16 2 272 and v 2 8 2 20 2 464, Work done Increase in kinetic energy m v 22 v 12 0 01 464 272 0 96 J. 6 8 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. Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda. 8 Work Energy Power and Collision, 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 moving train the work done in the frame of train will. F s 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, 6 9 Energy, The energy of a body is defined as its capacity for doing work. 1 Since energy of a body is the total quantity of work done therefore it is a scalar quantity. 2 Dimension ML 2 T 2 it is same as that of work or torque. 3 Units Joule S I erg C G S, Practical units electron volt eV Kilowatt hour KWh Calories Cal. Relation between different units 1 Joule 10 7 erg, 1 eV 1 6 10 19 Joule. 1 KWh 3 6 10 6 Joule, 1 Calorie 4 18 Joule, 4 Mass energy equivalence Einstein s special theory of.

Collection by Pradeep Kshetrapal for students at genius academy Maxwell classes Korba Gupta classes Kusmunda Work Energy Power and Collision 1