Energy, Work and Power

1. Energy, Work and Power

2. Energy Energy: the currency of the universe. Just like money, it comes in many forms! Everything that is accomplished has to be “paid for” with some form of energy. Energy can’t be created or destroyed, but it can be transformed from one kind into another and it can be transferred from one object to another.

3. Doing WORK is one way to transfer energy from one object to another. Work = Force x displacement W = Fd • Unit for work is Newton x meter. One Newton-meter is also called a Joule, J.

4. Work- the transfer of energy

5. Work = Force x displacement • Work is not done unless there is a displacement. • If you hold an object a long time, you may get tired, but NO work was done. • If you push against a solid wall for hours, there is still NO work done.

6. For work to be done, the displacement of the object must be along the same direction as the applied force. They must be parallel. • If the force and the displacement are perpendicular to each other, NO work is done by the force.

7. F d • For example, in lifting a book, the force exerted by your hands is upward and the displacement is upward- work is done. • Similarly, in lowering a book, the force exerted by your hands is still upward, and the displacement is downward. • The force and the displacement are STILL parallel, so work is still done. • But since they are in opposite directions, now it is NEGATIVE work. d F

8. F • On the other hand, while carrying a book down the hallway, the force from your hands is vertical, and the displacement of the book is horizontal. • Therefore, NO work is done by your hands. • Since the book is obviously moving, what force IS doing work??? The static friction force between your hands and the book is acting parallel to the displacement and IS doing work! d

9. Work = Force x distance

10. Your Force • So,….while climbing stairs or walking up an incline, only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top. Vertical component of d Horizontal component of d

11. Example How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall? W = Force x displacement Force = weight of the cat Which is parallel to the weight- the length of the ramp or the height? d = height NOT length W = mg x h W = 5 x 10 x 3 W = 150 J 7 m 3 m

12. How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long? ZERO, because your Force is vertical, but the displacement is horizontal.

13. Example Displacement = 20 m A boy pushes a lawnmower 20 meters across the yard. If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees, how much work did he do? F cos q q F W = (F cos q )d W = (200 cos 50) 20 W = 2571 J

14. q A 5.0 kg box is pulled 6m across a rough horizontal floor (m = 0.4) with a force of 80N at an angle of 35 degrees above the horizontal. What is the work done by EACH force exerted on it? What is the NET work done? Does the gravitational force do any work? NO! It is perpendicular to the displacement. Does the Normal force do any work? No! It is perpendicular to the displacement. Does the applied Force do any work? Yes, but ONLY its horizontal component! WF = Fcosq x d = 80cos 35 x 6 = 393.19 J Does friction do any work? Yes, but first, what is the normal force? It’s NOT mg! Normal = mg – Fsinq Wf = -f x d = -mNd = -m(mg – Fsinq)d = -90.53 J What is the NET work done? 393.19 J – 90.53 J = 302.66 J Normal FA f mg

15. Watch for those “key words” NOTE: If while pushing an object, it is moving at a constant velocity, the NET force must be zero. So….. Your applied force must be exactly equal to any resistant forces like friction.

16. Energy and Work have no direction associated with them and are therefore scalar quantities, not vectors. YEAH!!

18. Power is the rate at which work is done- how fast you do work. Power = work / time P = W / t • You may be able to do a lot of work, but if it takes you a long time, you are not very powerful. • The faster you can do work, the more powerful you are.

19. The unit for power is Joule / seconds which is also called a Watt, W (just like the rating for light bulbs) In the US, we usually measure power developed in motors in “horsepower” 1 hp = 746 W

20. Example A power lifter picks up a 80 kg barbell above his head a distance of 2 meters in 0.5 seconds. How powerful was he? P = W / t W = Fd W = mg x h W = 80 kg x 10 m/s2 x 2 m = 1600 J P = 1600 J / 0.5 s P = 3200 W

21. Another way of looking at Power: Power = Force x velocity

23. Kinds of Energy

24. Kinetic Energy the energy of motion K = ½ mv2

25. Potential Energy Stored energy It is called potential energy because it has the potential to do work.

26. Different kinds of Potential (stored) Energy • Example 1: Spring potential energy, SPE, in the stretched string of a bow or spring or rubber band. SPE = ½ kx2 • Example 2: Chemical potential energy in fuels- gasoline, propane, batteries, food! • Example 3: Gravitational potential energy, GPE- stored in an object due to its position from a chosen reference point.

27. Gravitational potential energy GPE = weight x height GPE = mgh Since you can measure height from more than one reference point, it is important to specify the location from which you are measuring.

28. The GPE may benegative. For example, if your reference point is the top of a cliff and the object is at its base, its “height” would be negative, so mgh would also be negative. • The GPE only depends on the weight and the height, not on the path that it took to get to that height.

29. Many different forms of Energy… Thermal Energy Solar Energy Atomic Energy Sound Energy Electromagnetic Energy Nuclear Energy Electrical Energy E = mc2

31. Work and Energy Often, some force must do work to give an object potential or kinetic energy. “Work” is the transfer of energy!! You push a wagon and it starts moving kinetic energy. You stretch a spring and you transform your work energy  spring potential energy. Or, you lift an object to a certain height- you transfer your work energy into the object in the form of gravitational potential energy. Work = Force x distance = change in energy

32. Example of Work = change in energy How much more distance is required to stop if a car is going twice as fast (all other things remaining the same)? The work done by the forces stopping the car = the change in the kinetic energy Fd = D½ mv2 With TWICE the speed, the car has FOUR times the kinetic energy. Therefore it takes FOUR times the stopping distance. (What FORCE is doing the work??)

33. The Work-Kinetic Energy Theorem NET Work done by all forces = D Kinetic Energy Wnet = ½ mv2f – ½ mv2o

34. Example, W = Fd = DK A 500kg car moving at 15m/s skids 20m to a stop. How much kinetic energy did the car lose? DK = ½ mvf2 – ½ mvo2 (but vf = 0!) DK = -½ (500 kg)(15 m/s) 2 DK = -56250 J What force was applied to stop the car? F·d = DK F = DK / d F = -56250 J / 20 m F = -2812.5 N

35. Example W = Fd = DK A 500kg car moving at 15 m/s slows to 10m/s. How much kinetic energy did the car lose? DK = ½ mvf2 – ½ mvo2 DK = ½ (500 kg)(10 m/s)2 - ½ (500 kg)(15 m/s)2 DK = -31250 J What force was applied to slow the car if the distance moved was 12 m? F·d = DK F = DK / d F = -31250 J / 12 m F = -2604 N

36. Example W = Fd = DK A 500 kg car moving on a flat road at 15 m/s skids to a stop. How much kinetic energy did the car lose? DK = ½ mvf2 – ½ mvo2 DK = -½ (500 kg)(15 m/s)2 DK = -56250 J How far did the car skid if the effective coefficient of friction was m = 0.6? Stopping force = friction = mN = mmg F·d = DK -(mmg)·d = DK d = DK / (mmg) *be careful to group in the denominator! d = 56250 J / (0.6 · 500 kg · 9.8 m/s2) = 19.13 m

37. Back to Power… Since Power = Work / time and Net work = DK… Power = DK / time In fact, Power can be calculated in many ways since Power = Energy / time, and there are MANY forms of energy!

39. Conservation of Mechanical Energy • Draw a sketch and choose a reference point for height. • Look at the first position of your object. If it is moving, it has Kinetic energy. If it has some height above or below your reference point, it has Potential energy. • Repeat for the second location. • If there is no friction or air resistance, set the mechanical energies at each location equal. E1 = E2 mgh1 + ½mv12 = mgh2 + ½ mv22 5. If there is friction or air resistance, use E1 – E2 to find the energy lost.

40. Force, N Position, m Graphing Force vs. postion • If you graph the applied force vs. the position, you can find how much work was done by the force. Work = Fd = “area under the curve”. Total Work = 2 N x 2 m + 3N x 4m = 16 J Area UNDER the x-axis is NEGATIVE work = - 1N x 2m F Net work = 16 J – 2 J = 14 J d

41. Back to the Work-Kinetic Energy Theorem… According to that theorem, net work done = the change in the kinetic energy Wnet = DK But, if the work can be found by taking the “area under the curve”, then it is also true that Area under the curve = DK = ½ mvf2 – ½ mvo2 Therefore, the area can be used to predict the final velocity of an object given its initial velocity and its mass.

42. For example… Suppose from the previous graph (Area = Wnet = 14 J), the object upon which the forces were exerted had a mass of 3 kg and an initial velocity of 4 m/s. What would be its final velocity? Area under the curve = ½ mvf2 – ½ mvo2 14 J = ½(3 kg)vf 2 – ½(3 kg)(4 m/s)2 vf = 5.0 m/s

43. The Spring Force If you hang an object from a spring, the gravitational force pulls down on the object and the spring force pulls up.

44. The Spring Force The spring force is given by Fspring = kx Where x is the amount that the spring stretched and k is the “spring constant” which describes how stiff the spring is

45. The Spring Force If the mass is hanging at rest, then Fspring = mg Or kx = mg (this is called “Hooke’s Law) The easiest way to determine the spring constant k is to hang a known mass from the spring and measure how far the spring stretches! k = mg / x

46. Graphing the Spring Force Suppose a certain spring had a spring constant k = 30 N/m. Graphing spring force vs. displacement: On horizontal axis- the displacement of the spring: x On vertical axis- the spring force = kx = 30x What would the graph look like? Fs = kx In “function” language: f(x) = 30x

47. x2 x1 Spring Force vs. Displacement Fs = 30x Fs How could you use the graph To determine the work done by The spring from some x1 to x2? Take the AREA under the curve! x

48. Analytically… The work done by the spring is given by Ws = ½ kxf2 – ½ kxo2 where x is the distance the spring is stretched or compressed (Which would yield the same result as taking the area under the curve!)

49. I love mrs. BRown

50. Mechanical Energy Mechanical Energy = Kinetic Energy + Potential Energy E = ½ mv2 + mgh