1. CONSERVATION LAWS PHY1012F WORK Gregor Leigh gregor.leigh@uct.ac.za

2. CONSERVATION LAWS WORKPHY1012F 2 WORK Learning outcomes: At the end of this chapter you should be able to… Extend the law of conservation of energy to include the thermal energy of isolated systems. Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems. Distinguish between conservative and nonconservative forces. Calculate the rate of energy transfer (power).

3. CONSERVATION LAWS WORKPHY1012F 3 THERMAL ENERGY An object as a whole has: Kinetic energy, K (due to movement) Potential energy, U (due to position) Mechanical energy, E mech Particles within an object (i.e. atoms or molecules) have: Kinetic energy (associated with the substance’s temperature) Potential energy (associated with the substance’s phase) Thermal energy, E th

4. CONSERVATION LAWS WORKPHY1012F 4 SYSTEM ENERGY The sum of a system’s mechanical energy and the thermal energy of its internal particles is called the system energy, E sys. E sys = E mech + E th = K + U + E th Conversions between energy types within the system are called energy transformations. Energy exchanges between the system and its environment are called energy transfers.

5. CONSERVATION LAWS WORKPHY1012F 5 ENERGY TRANSFORMATIONS Isolated system  no energy enters or leaves the system. Transformations are indicated with arrows: e.g. K  E th. Conversions between K and U are easily reversible, but we say that E mech is dissipated when it is transformed into E th since it is extremely difficult to transform E th back into E mech. Friction is a common cause of the dissipation of mechanical energy. SYSTEM K U E th E sys E mech + E th

6. CONSERVATION LAWS WORKPHY1012F 6  E sys = W ENERGY TRANSFERS The exchange of energy between a system and its environment by mechanical means (i.e. through the agency of forces) is called work, W. Energy can also be transferred by the non-mechanical process of heat. (Thermodynamics is not covered by this course.) Work is regarded as a system asset: work done on the system by the environ- ment increases the system’s energy: W > 0. work done by the system on the environ- ment decreases the system’s energy: W < 0. SYSTEM =  K +  U +  E th K U E th ENVIRONMENT W < 0 Q < 0 W > 0 Q > 0 workheat workheat

7. CONSERVATION LAWS WORKPHY1012F 7 WORK and KINETIC ENERGY Consider a body sliding on a frictionless surface, under the action of some (possibly varying) force… …as it moves from an initial position, s i, to a final position, s f … s s i, v is s f, v fs FsFs FsFs Newton II: (chain rule)

8. CONSERVATION LAWS WORKPHY1012F 8 WORK and KINETIC ENERGY … mv s dv s = F s ds  ½ mv f 2 – ½ mv i 2 Work done by moving the object from s i to s f. Hence  K = W No work is done if s f = s i. I.e. To do work, the force must cause the body to undergo displacement. Units: [N m = (kg m/s 2 ) m = kg m 2 /s 2 = joule, J] Notes:

9. CONSERVATION LAWS WORKPHY1012F 9 WORK DONE ON A SYSTEM Work-Kinetic energy theorem: When one or more forces act on a particle as it is displaced from an initial position to a final position, the net work done on the particle by these forces causes the particle’s kinetic energy to change by  K = W net. FsFs s Force curve The work, W, done on a system is given by the area under a F- vs -s graph. (cf. Impulse, J, and F- vs -t graphs.)  K   p. I.e. you cannot change one without changing the other, since… displacement

10. CONSERVATION LAWS WORKPHY1012F 10 WORK DONE BY A CONSTANT FORCE In the special case of a constant force… s sisi sfsf FsFs  ss  W Energy transfer 0° to < 90° 90° 90° to 180° F(  s)…F(  s) cos  E sys incr; K (and v) incr. F(  s) cos  …–F(  s) 0E sys, K (and v) constant. E sys decr; K (and v) decr. and

11. CONSERVATION LAWS WORKPHY1012F 11 THE DOT PRODUCT The quantity F(  s) cos  is the product of the two vectors, force,, and displacement,, and is more elegantly written as the dot product of the two vectors,. y x 1 1 Note first: and: I.e.the dot product is the sum of the products of the components.

12. CONSERVATION LAWS WORKPHY1012F 12 THE DOT PRODUCT  is the angle between the two vectors. Since it is a scalar quantity, the dot product is also known as the scalar product. Vectors can also be multiplied using a different procedure (the cross product) to produce a vector product (q.v.). Notes:

13. CONSERVATION LAWS WORKPHY1012F 13 Instead… If the force varies in a simple way, we can calculate the work geometrically, by plotting and determining the area under a F- vs -s graph. Otherwise the integral must evaluated mathematically. WORK DONE BY A VARIABLE FORCE If the force applied to a system varies during the course of the motion, we cannot take F s out of the integral… F net s (N) s (m) 2460 4 8 0

14. CONSERVATION LAWS WORKPHY1012F 14 WORK DONE BY GRAVITY y s  dy = –cos  ds ds Consider an object sliding down an arbitrarily-shaped frictionless surface as it moves a short distance ds.  Work done by gravity is thus path-independent. Gravity is therefore a conservative force. Notes:  W grav = –mg  y

15. CONSERVATION LAWS WORKPHY1012F 15 CONSERVATIVE FORCES The work done by a conservative force on a particle moving between two points does not depend on the path. The net work done by a conservative force on a particle moving around any closed path is zero. Conservative forces transform mechanical energy losslessly between the two forms, kinetic and potential. Any conservative force has associated with it its own form of potential energy: the work done by a conservative force in moving a particle from an initial position i to a final position f, denoted W c (i  f), changes the potential energy of the particle according to:  U = –W c (i  f)

16. CONSERVATION LAWS WORKPHY1012F 16 Potential energy…Work done by… WORK DONE BY CONSERVATIVE FORCES and POTENTIAL ENERGY Force of gravity, :  U = –W c Spring force, : W grav = –mg  y W sp = –½ k(  s) 2 Gravitational, U g : Elastic, U sp : U sp = ½ k(  s) 2 U g = mgy

17. CONSERVATION LAWS WORKPHY1012F 17 NONCONSERVATIVE FORCES The work done by nonconservative forces is path-dependent. s W fric = f k (  s)cos180° = –  k mg  s Whether the block slides directly to point A, or via point B, makes a difference to  s and hence to W fric. B A All kinetic frictional forces and drag forces are nonconservative forces. E.g. The work done by friction is

18. CONSERVATION LAWS WORKPHY1012F 18 NONCONSERVATIVE FORCES A nonconservative force has no associated form of potential energy. Instead, the work done by a nonconservative force increases the thermal energy, E th, of the system – a form of energy which has no “potential” for being reconverted to mechanical energy. A nonconservative force is consequently known as a dissipative force. Thus:  E th = –W diss. W diss is always negative since the force opposes motion. Thus  E th is always positive. Hence dissipative forces always increase the thermal energy of a system, and never decrease it.

19. CONSERVATION LAWSWORKPHY1012F 19 CONSERVATION OF ENERGY (work-kinetic energy theorem)  K = W net  K = W c + W nc  K = –  U + W nc (i.e.  K +  U =  E mech = W nc )  K = –  U + W diss + W ext  K = –  U –  E th + W ext  K +  U +  E th = W ext  E sys = W ext (energy equation of the system) (choose system carefully to include all dissipative forces )

20. CONSERVATION LAWS WORKPHY1012F 20 LAW OF CONSERVATION OF ENERGY The total energy E sys = E mech + E th of an isolated system is a constant. The kinetic, potential and thermal energies within the system can be transformed into each other, but their sum cannot change. Further, the mechanical energy E mech = K + U is conserved if the system is both isolated and non­ dissipative.

21. CONSERVATION LAWS WORKPHY1012F 21 POWER Power is the rate at which energy is transformed or transferred: Units: [J/s = watt, W] Power is also the rate at which work is done: Hence: P = Fv cos 

22. CONSERVATION LAWS WORKPHY1012F 22 WORK Learning outcomes: At the end of this chapter you should be able to… Extend the law of conservation of energy to include the thermal energy of isolated systems. Calculate the work done on and by systems and apply the work-kinetic energy theorem to the solution of problems. Distinguish between conservative and nonconservative forces. Calculate the rate of energy transfer (power).