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Work and Energy in Physics

Explore the concepts of work, energy, and thermal energy in isolated systems, including energy transformations and transfers. Learn to calculate work done by systems and the work-kinetic energy theorem.

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Work and Energy in Physics

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  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…

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