1. PHY1012FWORK Gregor Leighgregor.leigh@uct.ac.za

2. WORK WORK • 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). Learning outcomes:At the end of this chapter you should be able to…

3. WORK THERMAL ENERGY An object as a whole has: • Kinetic energy, K (due to movement) • Potential energy, U (due to position) Mechanical energy, Emech 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, Eth

4. WORK SYSTEM ENERGY The sum of a system’s mechanical energy and the thermal energy of its internal particles is called the system energy, Esys. Esys = Emech + Eth = K + U + Eth Conversions between energy types within the system are called energy transformations. Energy exchanges between the system and its environment are called energy transfers.

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

6. K Eth U WORK ENERGY TRANSFERS ENVIRONMENT • The exchange of energy between a system and its environment by mechanicalmeans (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 systemasset: • 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. W > 0 work heat Q > 0 SYSTEM Esys = W = K + U +  Eth W < 0 work heat Q < 0 ENVIRONMENT

7. s WORK WORK and KINETIC ENERGY Consider a body sliding on a frictionless surface, under the action of some (possibly varying) force… Fs Fs si , vis sf , vfs …as it moves from an initial position, si, to a final position, sf … Newton II: (chain rule)

8. WORK WORK and KINETIC ENERGY … mvs dvs = Fs ds ½mvf2–½mvi2 HenceK = W Work done by moving the object from si to sf. Notes: • No work is done if sf = si. I.e. To do work, the force must cause the body to undergo displacement. • Units: [Nm = (kgm/s2)m = kgm2/s2=joule, J]

9. Fs displacement • s WORK 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 = Wnet. • 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…

10. WORK WORK DONE BY A CONSTANT FORCE In the special case of a constant force…  Fs s • s si sf  W Energy transfer and 0° to < 90° F(s)…F(s)cos Esys incr; K (and v) incr. 90° 0 Esys, K (and v) constant. 90° to 180° F(s)cos…–F(s) Esys decr; K (and v) decr.

11. WORK 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 Note first: 1 and: • x 1 I.e. the dot product is the sum of the products of the components.

12. WORK THE DOT PRODUCT Notes: • 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.).

13. Fnet s(N) 8 4 s (m) 0 0 2 4 6 WORK WORK DONE BY A VARIABLE FORCE If the force applied to a system varies during the course of the motion, we cannot take Fs out of the integral… 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.

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

15. WORK 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 Wc(if) , changes the potential energy of the particle according to: U = –Wc(if)

16. WORK WORK DONE BY CONSERVATIVE FORCES and POTENTIAL ENERGY Force of gravity, : U = –Wc Gravitational, Ug: Ug = mgy Wgrav = –mgy Elastic, Usp: Spring force, : Usp = ½k(s)2 Wsp = –½k(s)2

17. WORK NONCONSERVATIVE FORCES The work done by nonconservative forces is path-dependent. • s E.g. The work done by friction is A Wfric = fk(s)cos180° = –kmgs Whether the block slides directly to point A, or via point B, makes a difference to s and hence to Wfric. B All kinetic frictional forces and drag forces are nonconservative forces.

18. WORK NONCONSERVATIVE FORCES • A nonconservative force has no associated form of potential energy. Instead, the work done by a nonconservative force increases the thermal energy, Eth, 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: Eth = –Wdiss. • Wdiss is alwaysnegative since the force opposes motion. Thus Ethis always positive. Hence dissipative forces always increase the thermal energy of a system, and never decrease it.

19. CONSERVATION OF ENERGY (work-kinetic energy theorem) PHY1012F WORK K = Wnet K = Wc + Wnc (i.e. K + U=Emech =Wnc) K = –U + Wnc K = –U + Wdiss + Wext (choose system carefully to include all dissipative forces) K = –U – Eth + Wext K + U + Eth =Wext Esys =Wext (energy equation of the system) 19

20. WORK LAW OF CONSERVATION OF ENERGY The total energy Esys =Emech + Eth 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 Emech = K + Uis conserved if the system is both isolated and non­dissipative.

21. WORK 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 = Fvcos

22. WORK WORK • 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). Learning outcomes:At the end of this chapter you should be able to…