Physics II Today’s Agenda

1. Physics IIToday’s Agenda • Work & Energy. • Discussion. • Definition. • Work of a constant force. • Power • Work kinetic-energy theorem. • Work of a sum of constant forces. • Work for a sum of displacements with constant force. • Work done by a spring • Conservation of Energy • Comments.

2. See text Work & Energy • One of the most important concepts in physics. • Alternative approach to mechanics. • Many applications beyond mechanics. • Thermodynamics (movement of heat). • Quantum mechanics... • Very useful tools. • You will learn new (sometimes much easier) ways to solve problems.

3. Forms of Energy • Kinetic: Energy of motion. • A car on the highway has kinetic energy. • We have to remove this energy to stop it. • The breaks of a car get HOT! • This is an example of turning one form of energy into another. (More about this soon)...

4. Forms of Energy • Potential: Stored, “potentially” ready to use. • Gravitational. • Hydro-electric dams etc... • Electromagnetic • Atomic (springs, chemical...) • Nuclear • Sun, power stations, bombs...

5. e+ e- + 5,000,000,000 V - 5,000,000,000 V Mass = Energy • Particle Physics: E = 1010 eV (a) (b) E = MC2 M ( poof ! ) (c)

6. Energy Conservation • Energy cannot be destroyed or created. • Just changed from one form to another. • We say energy is conserved ! • True for any isolated system. • i.e when we put on the brakes, the kinetic energy of the car is turned into heat using friction in the brakes. The total energy of the “car-breaks-road-atmosphere” system is the same. • The energy of the car “alone” is not conserved... • It is reduced by the braking. • Doing “work” on a system will change it’s “energy”...

7. Definition of Work: Ingredients: Force (F), displacement (S) Work, W, of a constant force F acting through a displacement S is: W = F.S = FScos(q) = FSS F S q FS displacement “Dot Product”

8. Work: 1-D Example (constant force) • A force F= 10Npushes a box across a frictionless floor for a distance Dx = 5m. F Dx Work done byF on box : WF=F.Dx=FDx (since F is parallel to Dx) WF = (10 N)x(5m) = 50N-m. See example 7.1

9. mks cgs other BTU = 1054 J calorie = 4.184 J foot-lb = 1.356 J eV = 1.6x10-19 J Dyne-cm (erg) = 10-7 J N-m (Joule) Units: Force x Distance = Work Newton x [M][L] / [T]2 Meter = Joule [L] [M][L]2 / [T]2

10. F S Power • We have seen that W = F. S • This does not depend on time ! • Power is the “rate of doing work”: • If the force does not depend on time: W/  t = F. S/  t = F.v P = F.v • Units of power: J/sec = Nm/sec = Watts v

11. Comments: • Time interval not relevant. • Run up the stairs quickly or slowly...same W. Since W = F.S • No work is done if: • F = 0 or • S = 0 or • q= 90o

12. Comments... W = F.S • No work done if q= 90o. • No work done by T. • No work done by N. T v v N

13. v1 v2 F Dx Work & Kinetic Energy: • A force F= 10Npushes a box across a frictionlessfloor for a distance Dx = 5m. The speed of the box is v1 before the push, and v2 after the push. m i

14. v1 v2 F Dx Work & Kinetic Energy... • Since the force F is constant, acceleration a will be constant. We have shown that for constant a: • v22 - v12 = 2a(x2-x1 ) = 2aDx. • multiply by 1/2m: 1/2mv22 - 1/2mv12 = maDx • But F = ma1/2mv22 - 1/2mv12 = FDx m a i

15. v1 v2 F Dx Work & Kinetic Energy... • So we find that • 1/2mv22 - 1/2mv12 = FDx = WF • Define Kinetic Energy K: K = 1/2mv2 • K2 - K1 = WF • WF = DK (Work kinetic-energy theorem) m a i

16. Fnet K2 K1 dS Work Kinetic-Energy Theorem: {NetWork done on object} = {change in kinetic energy of object} • This is true in general:

17. F(x) x1 x2 dx Work done by Variable Force: (1D) • When the force was constant, we wrote W = FDx • area under F vs x plot: • For variable force, we find the areaby integrating: • dW = F(x) dx. F Wg x Dx

18. A simple application:Work done by gravity on a falling object • What is the speed of an object after falling a distance H, assuming it starts at rest ? • Wg = F.S = mgScos(0) = mgH Wg = mgH Work Kinetic-Energy Theorem: Wg = mgH= 1/2mv2 v0 = 0 mg j S H v

19. Conservation of Energy • If only conservative forces are present, the total energy (sum of potential and kinetic energies) of a systemis conserved (i.e. constant).E = K + U is constant!!! • Both K and U can change as long as E = K + U is constant.

20. Example: The simple pendulum. • Suppose we release a bob or mass m from rest a distance h1 above it’s lowest possible point. • What is the maximum speed of the bob and wheredoes this happen ? • To what height h2 does it rise on the other side ? m h1 h2 v See example A Pendulum

21. Example: The simple pendulum. • Energy is conserved since gravity is a conservative force (E = K + U is constant) • Choose y = 0 at the original position of the bob, and U = 0 at y = 0 (arbitrary choice).E = 1/2mv2 + mgy. y y=0 h1 h2 v See example , A Pendulum

22. Example: The simple pendulum. • E = 1/2mv2 + mgy. • Initially, y = 0 and v = 0, so E = 0. • Since E = 0 initially, E = 0 always since energyisconserved. y y=0 See example , A Pendulum

23. Example: The simple pendulum. • E = 1/2mv2 + mgy. • So at y = -h, E = 1/2mv2 - mgh= 0. 1/2mv2 = mgh • 1/2mv2will be maximum when mghis minimum. • 1/2mv2will be maximum at the bottom of the swing ! y y=0 h y=-h See example , A Pendulum

24. Example: The simple pendulum. • 1/2mv2will be maximum at the bottom of the swing ! • So at y = -h11/2mv2 = mgh1v2 = 2gh1 y y=0 h1 y=-h1 v See example , A Pendulum

25. Example: The simple pendulum. • Since 1/2mv2 - mgh = 0it is clear that the maximum height on the other side will be at y = 0 and v = 0. • The ball returns to it’s original height. y y=0 See example , A Pendulum

26. Example: The simple pendulum. • The ball will oscillate back and forth. The limits on it’s height and speed are a consequence of the sharing of energy between K and U. E = 1/2mv2 + mgy= K + U = 0. y See example A Pendulum

27. Vertical Springs and HOOKE’S LAW (b) (a) • A spring is hung vertically, it’s relaxed position at y=0(a). When a mass m is hung from it’s end, the new equilibrium position is yE(b). • Hook’s Law relates the force exerted by the spring with the elongation of the spring • Force exerted by the spring is directly proportional to it’s elongation from it’s resting position • F=-kx(negative sign shows that the force is in the opposite direction of the force) • F=mg when spring is elongated and nonmoving so that mg=kx j k x = 0 X=xf m mg

28. Vertical Springs (b) (a) • If we choose x = 0 to be at the equilibrium position of the mass hanging on the spring, we can define the potential in the simple form. • Notice that g does not appear in this expression !! • By choosing our coordinates and constants cleverly, we can hide the effects of gravity. j k x = 0 m

29. 1-D Variable Force Example: Spring • For a spring we know that Fx = -kx. F(x) x1 x2 x equilibrium -kx F= - k x1 F= - k x2

30. Spring... • The work done by the spring Wsduring a displacement from x1to x2 is the area under the F(x) vs x plot between x1and x2. F(x) x1 x2 x Ws equilibrium -kx

31. Spring... • The work done by the spring Wsduring a displacement from x1to x2 is the area under the F(x) vs x plot between x1and x2. F(x) x1 x2 x Ws -kx

32. Non-conservative Forces: • If the work done does notdepend on the path taken, the force involved is said to be conservative. • If the work done does depend on the path taken, the force involved is said to be non-conservative. • An example of a non-conservative force is friction: • Pushing a box across the floor, the amount of work that is done by friction depends on the path taken. • Work done is proportional to the length of the path !

33. Non-conservative Forces: Friction • Suppose you are pushing a box across a flat floor. The mass of the box is m and the kinetic coefficient of friction is m. • The work done in pushing it a distance D is given by:Wf = Ff.D = -mmgD. Ff = -mmg D

34. See text: 8-6 Non-conservative Forces: Friction • Since the force is constant in magnitude, and opposite in direction to the displacement, the work done in pushing the box through an arbitrary path of length L is justWf = -mmgL. • Clearly, the work done depends on the path taken. • Wpath 2 > Wpath 1. B path 1 path 2 A

35. Generalized Work Energy Theorem: • Suppose FNET = FC + FNC (sum of conservative and non-conservative forces). • The total work done is: WTOT = WC + WNC • The Work Kinetic-Energy theorem says that: WTOT = DK. • WTOT = WC + WNC = DK • But WC = -DU So WNC = DK + DU = DE or WNC = Ei - Ef

36. Generalized Work Energy Theorem: WNC = DK + DU = DE • The change in total energy of a system is equal to the work done on it by non-conservative forces. E of system not conserved ! • Or the Potential Energy + Kinetic Energy + Internal Energy is a constant equal to the Total Energy • If all the forces are conservative, we know that energy is conserved: DK + DU = DE = 0 which says that WNC = 0,which makes sense. • If some non-conservative force (like friction) does work,energy will not be conserved by an amount equal to this work, which also makes sense.