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Work Energy Power

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Published in: NEET | Physics
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Physica Class XI Notes On Work-Energy & Power.

Akhilesh K / Lucknow

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Qualification: M.Sc (NIT Rourkela - 2019)

Teaches: All Subjects, English, Mathematics, Science, Chemistry, Physics, Algebra, IIT JEE Mains, AIPMT, NEET

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  1. WORK, ENERGY AND POWER Work done by a constant force: an object undergoes a displacement S along a straight line while acted on a force F that makes an angle e with S as shown. The work done W by the agent is the product of the component of force in the direction of displacement and the magnitude of displacement W = FScosO Work done is a scalar quantity and S.l. unit is Nm or Joule (J). We can also write; work done as a scalar product of force and displacement From this definition, we conclude the following points: (i) (ii) (iii) (iv) Work done by a force is zero, if point of application of force does not move ( s=o) Work done by a force is zero if displacement is perpendicular to the force (9=900) If angle between force and displacement is acute (0 900), we say that work done by the force is negative or work is done by the object Work done by Variable force B 2 02 _X Consider a particle being displaced along the curved path under the action of a varying force, as shown in figure. In such situation, we cannot use W— (Fcos8)S to calculate the work done by the force because this relationship applies when F is constant in magnitude and direction However if we imagine that the particle undergoes a very small displacement All shown in figure(a). then F is approximately constant over 'this interval and we can express the work done by the force for this small displacement as WI FRA Il In order to calculate work done. the whole curved path is assumed to be divided in small segments A 11, A 12, A Let Fie F2. F3. Fn be the force at respective segments. The force over each SUCh segment can be considered as constant because the segments are very small.
  2. Total work done 11 + 12 + 13 + Fn•A In If we take I Al I + O , the above summation gets converted into an integral F Fcosadl Work done by a spring k x x A common physical system for which the force varies with position is a spring-block as shown in figure. If the spring is stretched or compressed by a small distance from its unstitched or compressed by a small distance from its unscratched configuration. the spring will exert a force on the block given by F = •kx, where x is compression or elongation in spring, k is a constant called spring constant whose value depends inversely on un-stretched length and the nature of material of spring. Negative sign in above equation indicates that the direction of the spring force is opposite to x , the displacement of the free end. Consider a spring block system as shown in figure and let us calculate work done by the spring when block is displaced by Xo At any moment if elongation is x, then force on block by spring is kx towards left. Therefore, work done by the spring when block further displaced by dx dw = - kxdx ( Negative sign indicates displacement is opposite to spring 'force) Total work done by the spring xo kx dx 1 — — kX2 2 Similarly, work done by the spring when it is given a compression xo is 1 — kX2 2 We can also say that work done by external agent 1 2 Power If external force is applied to an point like object and if the work done by this force is A'" in the time interval At, then the average power during this interval is defined as At The work done on the object contributes to increasing energy of the object. A more general definition of power is the time rate of energy transfer. This instantaneous power is the limiting value of the average power as At approaches zero dt
  3. dw = F Therefore the instantaneous power can be written as dt dt The Sl unit of power is Joule per second (J/s), also called watt (W) Energy A body is said to possess energy if it has the capacity to do work. When a body possessing energy does some work, part of its energy is used up. Conversely if some work is done upon an object, the object will be given some energy. Energy and work are mutually convertible. Kinetic energy Kinetic energy (K.E.) is the capacity of a body to do work by virtue of its motion If a body of mass m has velocity v its kinetic energy is equivalent to the work, which an external force would have to do to bring the body from rest to its velocity v. The numerical value of the kinetic energy can be calculated from the formula 1 K. E. -ntv2 2 Consider a constant force F which acting on a mass m initially at rest, particle accelerate with constant velocity and attend velocity v after displacement of S For the formula v2 — u2 = 2as Initial velocity is zero v2 = 2as Multiply both the sides by m mv2 = 2mas mv2 = 2W [ As work = FS = masl W (1/2)mv2 But Kinetic energy of body is equivalent to the work done in giving the velocity to the body Hence K.E = (1/2)rnv2 Since both m and v2 are always positive KAE is always positive and does not depend up on the direction of motion of body. Another equation for kinetic energy 1 m2v2 1 E = —mv2 2 Potential energy Ip2 Potential energy is the energy due to position. _lf a body is in a position such that if it were released it would begin to move, it has potential energy There are two common forms of potential energy, gravitational and elastic
  4. Gravitational potential energy It is possessed by virtue of height When an object is allowed to fall from one level to a lower level it gains speed due to gravitational pull, i.e. it gains kinetic energy. Therefore. in possessing height, a body has the ability to convert its gravitational potential energy into kinetic energy. P.E. =mgh mg h The gravitational potential energy is equivalent to the negative of the amount of work done by the weight of the body in causing the descent. If a mass m is at a height h above a lower level, the P.E. possessed by the mass is (mg) (h) Since h is the height of an object above a specific level, an object below the specified level has negative potential energy Elastic potential Energy It is a property of stretched or compressed springs. The end of a stretched elastic spring will begin to move if it is released. The spring therefore possesses potential energy due to its elasticity (i.e. due to change in its configuration) The amount of elastic potential energy stored in a spring of natural length a and spring constant k when it is extended by a length x is equal to 'the amount of work necessary to produce the extension Work done = (1/2)kx2 so Elastic Potential energy = (1/2) kx2 Elastic potential energy is never negative whether the spring is extended or compressed Work energy theorem When a body is acted upon by force acceleration is produced in it. Thus velocity of the body changes and hence the kinetic energy of the body also changes. Also force acting on a body displaces the body and so work is said to be done on the body by force. These facts indicate that there should be some relation between the work done on body and change in its kinetic energy. The work done by the force F W-FS W = ma W = mas Also v2 1.12 = 2as Multiplying both sides by m m(v2 = 2ams 1 ntv 2 1 2 2 1 2 2 1 2 mas 2
  5. Work energy theorem for variable force Work-energy theorem is valid from variable force Suppose position dependent force F(x) acts on a body of mass m Work done under the influence of force F(x)dx dv m — dx dt dx m — dv dt mvdv I W vdv W = m vdv Vi 1 2 2 mvf - m 2 1 ntVi 2 Conservative force A conservative force may be defined as on for which work done in moving between two points A and B is independent of the path taken between two points. Work done to move particles through stairs is equal to moving particle vertically . The implication of "conservative" in this context is that you could move it from A and B by one path and returns to A by another path with no net loss of energy any close return path A takes net work zero. Or mechanical energy is conserved
  6. Non-conservative force Consider a body moving on a rough surface from A to B and then back from B to A. Work done against frictional forces only add up because in both the displacement work is done against frictional force only. Hence frictional force cannot be considered as a conservative force. It is non-conservative force Conservation of mechanical energy Kinetic and potential energy both are forms of mechanical energy. The total mechanical energy of a body or system of bodies will be changed in values if (a) An external force other than weight causes work to be done( work done by weight is potential energy and is therefore already included in the total mechanical energy) (b) Some mechanical energy is converted into another form of energy ( e.g. sound, heat , light) such a conversion of energy usually takes place when a sudden change in the motion of the system occurs. For instance, when two moving objects collide some mechanical energy is converted into sound energy, which is heard as a bang at the impact. If neither (a) nor (b) occurs, then the total mechanical energy of a system remains constant. This is the principle of Conservation of Mechanical Energy and can be expressed as The total mechanical energy of a system remains constant provided that no external work is done and no mechanical energy is converted into another form of energy When this principle is used in solving problems, a careful appraisal must be made of any external forces, which are acting. Some external forces do work and hence cause a change in the total energy of the system.

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