Energy and Work

1. Energy and Work Energy - the ability of a body or system of bodies to perform work. A body is given energy when a force does work on it.

2. But What is Work? • A force does work on a body (and changes its energy) when it causes a displacement. • If a force causes no displacement, it does no work.

3. Riddle Me This If a man holds a 50 kg box at arms length for 2 hours as he stands still, how much work does he do on the box? Zip ZERO Zilch Nada NONE

4. Counterintuitive Results • There is no work done by a force if it causes no displacement. • Forces perpendicular to displacement, such as the normal force, can do no work. • Likewise, centripetal forces never do work.

5. Calculating Work • Work is the dot product of force and displacement. • Work is a scalar resulting from the interaction of two vectors.

6. Vector Multiplication • There are three ways to multiply vectors: • Scalar Multiplication • Dot Product • Cross Product

7. Scalar Multiplication • Magnitude of vector changes. • Direction of vector does not change. If m = 5 kg a = 10 m·s-1 F = 50 N

8. Dot Product Note that the dot product of two vectors gives a scalar . θ

9. Dot Product Geometrically, the dot product is the projection of one vector on a second vector multiplied by the magnitude of the second vector. θ

10. Calculating Work θ

11. Which does more work? Two forces are acting on the box shown causing it to move across the floor. Which force does more work? F2 θ F1

12. F Vectors and Work

13. Vectors and Work F s W = F • s W = F s cos 0o W = F s Maximum positive work

14. F Vectors and Work

15. Vectors and Work F  s W = F • s W = F s cos  Only the component of force aligned with displacement does work. Work is less.

16. F Vectors and Work

17.  s Vectors and Work F W = F • s W = F s cos 180o W = - F s Maximum negative work.

18. F mg Gravity often does negative work. When the load goes up, gravity does negative work and the crane does positive work. When the load goes down, gravity does positive work and the crane does negative work.

19. Positive, zero, or negative work? A box is being moved with a velocity v by a force P (parallel to v) along a level floor. The normal force is FN, the frictional force is fk, and the weight of the box is mg. Decide which forces do positive, zero, or negative work.

20. Positive, zero, or negative work? v FN fk P s mg

21. Units of Work J = N·m J = kg·m2·s-2

22. F(x) xb W =  F(x) dx xa x xa xb Work and variable force The area under the curve of a graph of force vs displacement gives the work done by the force.

23. Net Work Net work (Wnet) is the sum of the work done on an object by all forces acting upon the object.

24. The Work-Energy Theorem • Consider a force applied to an object (ΣF ≠ 0). • Newton’s second law tells us that this net force will produce an acceleration. • Since the object is accelerating, its displacement will change, hence the net force does work.

25. The Work-Energy Theorem

26. Kinetic Energy • A form of mechanical energy • Energy due to motion • K = ½ m v2 • K: Kinetic Energy in Joules. • m: mass in kg • v: speed in m/s

27. The Work-Energy Theorem

28. The Work-Energy Theorem Wnet = DKE • When net work due to all forces acting upon an object is positive, the kinetic energy of the object will increase. • When net work due to all forces acting upon an object is negative, the kinetic energy of the object will decrease. • When there is no net work acting upon an object, the kinetic energy of the object will be unchanged.

29. Power • Power is the rate of which work is done. • No matter how fast we get up the stairs, our work is the same. • When we run upstairs, power demands on our body are high. • When we walk upstairs, power demands on our body are lower.

30. Power • The rate at which work is done. • Pave = W / t • P = dW/dt • P = F • v

31. Units of Power • Watt = J/s • ft lb / s • horsepower • 550 ft lb / s • 746 Watts

32. Power Problem Develop an expression for the power output of an airplane cruising at constant speed v in level flight. Assume that the aerodynamic drag force is given by FD = bv2. By what factor must the power be increased to increase airspeed by 25%?

33. How We Buy Energy… • The kilowatt-hour is a commonly used unit by the electrical power company. • Power companies charge you by the kilowatt-hour (kWh), but this not power, it is really energy consumed. • 1 kW = 1000 W • 1 h = 3600 s 1 kWh = 1000J/s • 3600s = 3.6 x 106J

34. More about force types • Conservative forces: • Work in moving an object is path independent. • Work in moving an object along a closed path is zero. • Work may be related to a change in potential energy or used in the work-energy theorem. • Ex: gravity, electrostatic, magnetostatic, springs • Non-conservative forces: • Work is path dependent. • Work along a closed path is NOT zero. • Work may be related to a change in total energy (including thermal energy). • Ex: friction, drag, magnetodynamic Conservation of Energy holds only for ALL Forces Conservation of Mechanical Energy holds only for Conservative Forces

35. Mechanical Energy:Potential energy • Energy an object possesses by virtue of its position or configuration. • Represented by the letter U. • Examples: • Gravitational potential energy. • Electrical potential energy. • Spring potential energy.

36. Gravitational Potential Energy (Ug) • For objects near the earth’s surface, the gravitational pull of the earth is constant, so • Wg = mgDx • The force necessary to lift an object at constant velocity is equal to the weight, so we can say • DUg = -Wg = mgh

37. Where is Gravitational Potential Energy Zero? • Ug has been defined to be zero when objects are infinitely far away, and becoming negative as objects get closer. • This literal definition is impractical in most problems. • It is customary to assign a point at which Ug is zero. Usually this is the lowest point an object can reach in a given situation • Then, anything above this point is a positive Ug.

38. Ideal Spring • Obeys Hooke’s Law. • Fs(x) = -kx • Fs is restoring force exerted BY the spring. • Ws =  Fs(x)dx = -k  xdx • Ws is the work done BY the spring. • Us = ½ k x2

39. Spring Problem Three identical springs (X, Y, and Z) are hung as shown. When a 5.0-kg mass is hung on X, the mass descends 4.0 cm from its initial point. When a 7.0-kg mass is hung on Z, how far does the mass descend? X Y Z

40. Boundary System System Environment

41. Isolated System Boundary allows no exchange with environment. E = U + K + Eint = Constant No mass can enter or leave! No energy can enter or leave! Energy is constant, or conserved!

42. Law of Conservation of Energy Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another. Applies to isolated systems.

43. Law of Conservation of Mechanical Energy E = U + K = C E = U + K = 0 for gravity • Ug = mghf - mghi • K = ½ mvf2 -½mvi2

44. Law of Conservation of Mechanical Energy E = U + K = C E = U + K = 0 for springs • Ug = ½ kx2 • K = ½mvf2 -½mvi2

45. Law of Conservation of Energy E = U + K + Eint= C E = U + K +  Eint = 0 Eint is thermal energy. Mechanical energy may be converted to and from heat.

46. James Prescott Joule • Father of Conservation of Energy. • Studied electrical motors. • Derived the mechanical equivalent of heat. • Measured heat of water as it fell. • Measured cooling of expanding gases.

47. h Pendulum Energy ½mvmax2 = mgh For minimum and maximum points of swing ½mv12 + mgh1 = ½mv22 + mgh2 For any points two points in the pendulum’s swing

48. -x m m x m Spring Energy 0 ½ kx2 = ½ mvmax2 For maximum and minimum displacements from equilibrium ½ kx12 + ½ mv12 = ½ kx22 + ½ mv22 For any two points in a spring’s oscillation

49. Announcements 9/11/2014 Clicker Quiz. Homework Check.

50. Total Energy U K Spring and Pendulum Energy Profile E x