Wor Wo k r Readings: C hapter Chapter

Transcription

1 Work Readings: Chapter 11 1

2 Newton s Second Law: Net Force is zero acceleration is zero velocity is constant kinetic energy is constant 1 Fnet a v K mv const const 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 F net a Δv Fnet ma m Δ t Δv 1 FnetvΔ t m vδ t mvδ v mδv Δt 1 FnetΔ s mδ v ΔK Δ s vδt Where - displacement

3 A v B F net a F If then net Δ s const F ( 1 1 ) Δ K mv mv B net B A B A This equation gives the relation between the displacement and velocity. It is the same equation as A v v a( ) B A B A for the motion with constant acceleration (constant net force) 3

4 A v B F net a F Δ s If then we have integral net const B A B A 1 1 F d Δ K mv mv net B A If the net force depends also on the velocity then the relation becomes more complicated Fnet Δs B A F net d or is the Work done by net force B W Fnetd A 4

5 Then Work is W F Δ s FcosθΔs s This combination is called the dot product (or scalar product) of two vectors F cos θδ s F Δ s 5

6 Dot Product a θ If we know the components of the vectors then the dot product can be calculated as b a b a b + a b y y Dot product is a scalar: a b abcosθ y a b y y a a b b If vectors are orthogonal then dot product is zero a θ 9 cosθ b 9 6 a b

7 Dot Product: properties ( ca) b c( a b) cabcosθ ( a + a ) b a b + a b 1 1 Dot product is positive if θ < 9 cosθ > a b abcosθ a θ b Dot product is negative if θ > 9 cosθ < The magnitude of angle is θ 6 What is the dot product of and a b is 5, the magnitude of is, the a b 1 a b 5 cos

8 Work produced by a force, acting on an object, is the dot product of the force and displacement W F Δs final point W F Δs initial point Work has the same units as the energy: kg m kg m 1 J 1 N m 1 m 1 s s Eample: What is the work produced by the tension force? T 1N Δ s 1m θ 45 W T Δ s TΔ scosθ 1 1cos 45 Δs 77N m 77J 8

9 Δ K K final Kinitial W F Δs W F Δ s FΔ s > ( a > ) K final > Kinitial v f > v i W F Δ s > ( a > ) K final > Kinitial vf > vi W F Δ s K final K initial W F Δ s < K final ( a ) v v < Kinitial v f < vi f ( a < ) i 9

10 Work produced by a NET FORCE is equal to a change of KINETIC ENERGY (it follows from the second Newton s law) W F Δ s Δ K K K net net final initial The NET FORCE is equal to (vector) sum of all forces acting on the object then Fnet F1 + F +... W ( F + F +...) Δ s F Δ s + F Δ s +... W + W +... ΔK net so the sum of the works produced by all forces acting on the object is equal to the change of kinetic energy, W1 + W +... ΔK 1

11 Eample: 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. n w Δ s Forces: normal force gravitational force tension Δs T 1N 1m n w T θ 45 Work produced by the net force is equal to the change of kinetic energy. 1 1 Wnet Fnet Δ s Δ K mvf mvi Fnet n+ w+ T Wnet Wn + Ww + WT n Δ s + w Δ s + T Δ s TΔ s cos θ W n Δ s W w Δ s W T Δ s TΔscosθ n w Since then T 11

12 Eample: 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. n w Δ s Forces: normal force gravitational force tension Δs T 1N 1m n w T θ mv f mvi + Wnet mvi + TΔscosθ T cos θ vf vi + Δs m This problem can be also solved by finding acceleration (this is the motion with constant acceleration). Then v v + aδs f i T cosθ a m 1

13 Eample: Free fall motion. y i y f w Δs Forces: gravitational force Work produced by the net force (gravitational force) is equal to the change of kinetic energy. 1 1 W F Δ s Δ K mv mv net net f i w then 1 1 W ( ) net Ww w Δ s mgδ s mg yi yf mv f mvi 1 1 mgy + mv mgy + mv i i f f Conservation of mechanical energy Work produced by gravitational force can be written as the change of gravitational potential energy 13

14 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 Δ s U U initial 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 W U U path1 path A B Such forces are called conservative forces 14

15 Conservative Forces Eample 1: Gravitational Force U mgh Eample : Elastic Force f f W Fd ( k) d k i i i i f k k Uinitial U final Nonconservative (dissipative) forces Eample: Friction f U k 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

16 If we know force we can find the corresponding potential energy W F Δ s Δ U ( U U ) final initial We assume that the potential (for eample) at point A is, then the potential energy at point B is B B U U F ds F ds B B A A If we know potential we can find the force as ΔU U Fs Δ s mgδy Eample: Gravitational force: U mgy F y mg Δy Eample: Elastic force: The eact relation between the force and potential: F U kδ ( /) k F U(, y, z) k Δ F y U(, y, z) y F z y U(, y, z) 16 z

17 Nonconservative (dissipative) forces Eample 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. f k path 1 f f k s s 1 s 3 f k B path W f s f s f s s W path1 k 1 ( ) k k 1 + f s path k 3 s ( s + s ) 3 1 A Wpath1 Wpath Eample : Eternal Force 17

18 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 Δ s Fconservative Δ s + Fnonconservative Δ s U U + F Δ s K K F Δ s ( U + K ) ( U + K ) initial final nonconservative final initial nonconservative final final initial initial E E mech, final mech, 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).

19 Eample v initial h v final Work done by friction force is equal to the change of mechanical energy: Wfiti friction fk ds Emech, final Emech hi, initial itil mgh 19

20 Nonconservative forces: friction (dissipative) forces and eternal 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 Δ s E E friction thermal, initial thermal, final F Δ s F Δ s + E E nonconservative eternal thermal, initial thermal, final E E mech, final mech, initial F Δ s ( E + E ) ( E + E ) eternal mech, final thermal, final mech, initial thermal, initial Work done by eternal forces is equal to the change of the total energy of the system (mechanical + thermal)

21 Work done by eternal 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. F Δ s ( E + E ) ( E + E ) eternal mech, final thermal, final mech, initial thermal, initial 1