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Transcript
Work Readings: Chapter 11 1 Newton’s Second Law: Net Force is zero – acceleration is zero – velocity is constant – kinetic energy is constant Fnet 0 a 0 v const 1 K mv 2 const 2 Newton’s Second Law: If Net Force is not zero then: What is the relation between the change of Kinetic energy and the Net Force? v Fnet a Where v Fnet ma m t v 1 Fnet vt m vt mvv mv 2 t 2 1 Fnet s mv 2 K 2 s vt - displacement 2 A B v Fnet a s xB x A If Fnet const then 1 1 2 Fnet ( xB x A ) K mvB mv A2 2 2 This equation gives the relation between the displacement and velocity. It is the same equation as v B2 v A2 2a( xB x A ) for the motion with constant acceleration (constant net force) 3 A B v a Fnet s xB x A If Fnet const then we have integral xB 1 1 2 2 F dx K mv mv B A x net 2 2 A If the net force depends also on the velocity then the relation becomes more complicated xB Fnet s or F net dx is the Work done by net force xA W xB F net xA dx 4 Then Work is W Fs s F cos s This combination is called the dot product (or scalar product) of two vectors F coss F s 5 Dot Product a Dot product is a scalar: a b ab cos b If we know the components of the vectors then the dot product can be calculated as a b a x bx a y by y ay ax by If vectors are orthogonal then dot product is zero 90 cos 0 0 ab 0 a bx x b a 900 b 6 a b ab cos Dot Product: properties (ca ) b c(a b ) cab cos a (a1 a2 ) b a1 b a2 b Dot product is positive if 900 cos 0 Dot product is negative if 900 cos 0 The magnitude of angle is 600 a is 5, the magnitude of What is the dot product of a and b b is 2, the b 1 a b 5 2cos 60 5 2 5 2 0 7 Work produced by a force, acting on an object, is the dot product of the force and displacement final point W F s W initial point Work has the same units as the energy: F s kg m kg m 2 1J 1 N m 1 2 m 1 s s2 Example: What is the work produced by the tension force? T 10 N s 100m 450 W T s T s cos s 10 100cos 450 707 N m 707 J8 K K final K initial W F s W F s F s 0 (a 0) K final K initial v f vi W F s 0 K final K initial (a 0) v f vi W F s 0 K final K initial (a 0) v f vi W F s 0 K final K initial (a 0) v f vi 9 Work produced by a NET FORCE is equal to a change of KINETIC ENERGY (it follows from the second Newton’s law) Wnet Fnet s K K final K initial The NET FORCE is equal to (vector) sum of all forces acting on the object Fnet F1 F2 ... then Wnet ( F1 F2 ...) s F1 s F2 s ... W1 W2 ... K so the sum of the works produced by all forces acting on the object is equal to the change of kinetic energy, W1 W2 ... K 10 Example: What is the change of kinetic energy? Or what is the final velocity of the block if the initial velocity is 5 m/s? The mass of the block is 5 kg. Forces: normal force n gravitational force tension w T 10 N s n w T 450 s 100m Work produced by the net force is equal to the change of kinetic energy. Wnet 1 1 2 Fnet s K mv f mvi2 2 2 Since Fnet n w T then Wnet Wn Ww WT n s w s T s T s cos Wn n s 0 Ww w s 0 WT T s T s cos 11 Example: What is the change of kinetic energy? Or what is the final velocity of the block if the initial velocity is 5 m/s? The mass of the block is 5 kg. Forces: normal force n gravitational force tension w T 10 N s n w T 450 1 1 1 2 2 mv f mvi Wnet mvi2 T s cos 2 2 2 T cos 2 2 v f vi 2 s m s 100m This problem can be also solved by finding acceleration (this is the motion with constant acceleration). Then v v 2as 2 f 2 i T cos a m 12 Example: Free fall motion. Forces: yi gravitational force w yf s w Work produced by the net force (gravitational force) is equal to the change of kinetic energy. Wnet 1 1 2 Fnet s K mv f mvi2 2 2 Wnet Ww w s mgs mg( yi y f ) then 1 1 mv 2f mvi2 2 2 1 1 mgyi mvi2 mgy f mv 2f 2 2 Conservation of mechanical energy Work produced by gravitational force can be written as the change of gravitational potential energy 13 Not for all forces the work can be written as the difference between the potential energy between two points. Only if the force does not depend on velocity then we can introduce potential energy as W F s U initial U final Potential energy depends only on the position of the object It means also that the work done by the force does not depend on the trajectory (path) of the object: W path1 W path 2 U A U B Such forces are called conservative forces 14 Conservative Forces Example 1: Gravitational Force U mgh Example 2: Elastic Force xf W xi xf Fdx ( kx )dx k xi 2 x 2 xf xi x 2f xi2 k k U initial U final 2 2 x2 Uk 2 Nonconservative (dissipative) forces Example: Friction Direction of friction force is always opposite to the direction of velocity, so the friction force depends on velocity. The work done by friction force depends on path. 15 If we know force we can find the corresponding potential energy W F s U (U final U initial ) We assume that the potential (for example) at point A is 0, then the potential energy at point B is B B A A U B U B F ds F ds U s mg y Fy mg y If we know potential we can find the force as Example: Gravitational force: Example: Elastic force: U mgy x2 Uk 2 The exact relation between U ( x , y, z ) F the force and potential: x x Fx Fs k x 2 / 2 x y kx U ( x16 , y, z ) U ( x , y , z ) Fz Fy y z Nonconservative (dissipative) forces Example 1: Friction Direction of friction force is always opposite to the direction of velocity, so the friction force depends on velocity. The work done by friction force depends on path. W path1 f k s1 f k s2 f k ( s1 s2 ) fk path 1 B s2 s1 fk path 2 s3 fk A Example 2: External Force W path 2 f k s3 s3 ( s1 s2 ) W path1 W path 2 17 The work done by the net force is equal to the change of kinetic energy. The net force is the sum of conservative forces (gravitational force, elastic force, …) and nonconservative force (friction force ..). The work done by conservative forces can be written as the change of potential energy. Then Wnet Fnet s Fconservative s Fnonconservative s U initial U final Fnonconservative s K final K initial Fnonconservative s (U final K final ) (U initial K initial ) Emech, final Emech,initial Work done by nonconservative forces is equal to the change of mechanical energy of the system (sum of kinetic energy, 18 gravitational potential energy, elastic potential energy, and so on). Example vinitial 0 h v final 0 Work done by friction force is equal to the change of mechanical energy: W friction fk ds Emech, final Emech,initial mgh 19 Nonconservative forces: friction (dissipative) forces and external forces. Friction force decreases the mechanical energy of the system but increases the TEMPERATURE of the system – increases thermal energy of the system. Then F friction s Ethermal ,initial Ethermal , final Fnonconservative s Fexternal s Ethermal , initial Ethermal , final Emech, final Emech,initial Fexternal s ( Emech, final Ethermal , final ) ( Emech,initial Ethermal ,initial ) Work done by external forces is equal to the change of the total energy of the system (mechanical + thermal) 20 Work done by external forces is equal to the change of the total energy of the system (mechanical + thermal) Thermal energy is also a mechanical energy – this is the energy of motion of atoms or molecules inside the objects. Fexternal s ( Emech, final Ethermal , final ) ( Emech,initial Ethermal ,initial ) 21