Work equation,applications,energy conservation

Work

1. 1 CHAPTER 4: Work, Energy and Power Lecture 1.5 hours Tutorial 5 hours

2. 2 Overview : Work Energy Power Kinetic energy Gravitational potential energy Elastic potential energy Principle of conservation of energy Average and instantaneous power

3. 3 At the end of this chapter, students should be able to: State the physical meaning of dot (scalar) product for work: Define and apply work done by a constant force , determine work done from a force-displacement graph. Learning Outcome: 4.1 Work (2.5 hours)   

4. 4 4.1 Work 4.1.1 Work, W is a product of the component of the force parallel to the displacement times the displacement of a body . OR is a scalar (dot) product between force and displacement of a body . where At the end of this lesson, students should be able to: State the physical meaning of dot (scalar) product for work.

5. 5 a scalar quantity. S.I. unit of work: The joule (1 J) is a work done by a force of 1 N which results in a displacement of 1 m in the direction of the force .

6. 6 4.1.2 Force-displacement graph At the end of this lesson, students should be able to: Apply work done by a constant force and determine work done from a force-displacement graph. Force / N Displacement / m s 1 s 2 Work = Area

7. 7 4.1.3 Applications of work’s equation Case 1 : Work done by a horizontal force, F on an object. Case 2 : Work done by a vertical force, F on an object. Figure 4.1 and and Figure 4.2

8. 8 Case 3 : Work done by a horizontal forces, F 1 and F 2 on an object. Case 4 : Work done by a force, F and frictional force , f on an object. Figure 4.3 and Figure 4.4 and OR

9. 9 Caution : Work done on an object is zero when F = or s = and   = 90 .

10. 10 Sign for work. If 0  <  <90 ( acute angle ) then cos  > (positive value) therefore W > 0 (positive)  work done on the system ( by the external force) where energy is transferred to the system. If 90  <  <180 ( obtuse angle ) then cos  <0 (negative value) therefore W < 0 (negative)  work done by the system where energy is transferred from the system.

11. 11 You push your physics reference book 1.50 m along a horizontal table with a horizontal force of 5.00 N. The frictional force is 1.60 N. Calculate a. the work done by the 5.00 N force, b. the work done by the frictional force, c. the total work done on the book. Solution : a. Example 4.1 : and

12. 12 Solution : b. c. OR and

13. 13 A box of mass 20 kg moves up a rough plane which is inclined to the horizontal at 25.0. It is pulled by a horizontal force F of magnitude 250 N. The coefficient of kinetic friction between the box and the plane is 0.300. a. If the box travels 3.80 m along the plane, determine i . the work done on the box by the force F , ii. the work done on the box by the gravitational force, iii. the work done on the box by the reaction force, iv. the work done on the box by the frictional force, v. the total work done on the box. b. If the speed of the box is zero at the bottom of the plane, calculate its speed when it is travelled 3.80 m. Example 4.2 :

14. 14 Solution : a. i . and

15. 15 Solution : a. ii. iii. iv. and and and

16. 16 Solution : a. v. b. Given

17. 17 A horizontal force F is applied to a 2.0 kg radio-controlled car as it moves along a straight track. The force varies with the displacement of the car as shown in Figure 4.5. Calculate the work done by the force F when the car moves from 0 to 7 m. Solution : Example 4.3 : Figure 4.5

18. 18 Exercise 4.1 : 1. A block of mass 2.50 kg is pushed 2.20 m along a frictionless horizontal table by a constant 16.0 N force directed 25.0 below the horizontal. Determine the work done on the block by a. the applied force, b. the normal force exerted by the table, and c. the gravitational force. d. Determine the total work on the block. (Given g = 9.81 m s  2 ) ANS. : 31.9 J; (b) & (c) U think; 31.9 J

19. 19 2. Figure 4.6 shows an overhead view of three horizontal forces acting on a cargo that was initially stationary but that now moves across a frictionless floor. The force magnitudes are F 1 = 3.00 N, F 2 = 4.00 N and F 3 = 10.0 N. Determine the total work done on the cargo by the three forces during the first 4.00 m of displacement. ANS. : 15.3 J Figure 4.6

20. 20 At the end of this chapter, students should be able to: Define and use: Gravitational potential energy, Elastic potential energy for spring, Kinetic energy, State the principle of conservation of energy. Apply the principle of conservation of energy State and apply the work-energy theorem, Learning Outcome: 4.2 Energy and conservation of energy (3 hours)      

21. 21 4.2.1 Energy is a system’s ability to do work . The S.I. unit for energy is same to the unit of work ( joule, J ). is a scalar quantity . Forms of Energy Description Chemical Energy released when chemical bonds between atoms and molecules are broken. Electrical Energy that is associated with the flow of electrical charge. Heat Energy that flows from one place to another as a result of a temperature difference. Internal Total of kinetic and potential energy of atoms or molecules within a body. 4.2 Energy and conservation of energy

22. 22 Forms of Energy Description Table 4.1 Nuclear Energy released by the splitting of heavy nuclei. Mass Energy released when there is a loss of small amount of mass in a nuclear process. The amount of energy can be calculated from Einstein’s mass-energy equation, E = mc 2 Radiant Heat Energy associated with infra-red radiation. Sound Energy transmitted through the propagation of a series of compression and rarefaction in solid, liquid or gas. Mechanical a. Kinetic b. Gravitational potential c. Elastic potential Energy associated with the motion of a body. Energy associated with the position of a body in a gravitational field. Energy stored in a compressed or stretched spring.

23. 23 4.2.2 Potential Energy is an energy stored in a body or system because of its position, shape and state . Gravitational potential energy, U is an energy stored in a body or system because of its position . depends only on the height of the object above the surface of the Earth . where

24. 24 In a smooth pulley system, a force F is required to bring an object of mass 5.00 kg to the height of 20.0 m at a constant speed of 3.00 m s 1 as shown in Figure 4.7. Determine a. the force, F b. the work done by the force, F . Example 4.4 : Figure 4.7

25. 25 Solution : a. Since the object moves at the constant speed, thus b. Constant speed and

26. 26 Elastic potential energy, U s is an energy stored in in elastic materials as the result of their stretching or compressing . Hooke’s Law states “ the restoring force, F s of spring is directly proportional to the amount of stretch or compression ( extension or elongation ), x if the limit of proportionality is not exceeded ” OR Negative sign : direction of F s is always opposite to the direction of the amount extension/compression, x . where

27. 27 Case 1: The spring is hung vertically and its is stretched by a suspended object with mass, m . The spring is in equilibrium, thus Initial position Final position Figure 4.8

28. Figure 4.9 (Equilibrium position) Case 2: The spring is attached to an object and it is stretched and compressed by a force, F . 28 The spring is in equilibrium, hence

29. 29 Caution: For calculation , use : The unit of k is : From the Hooke’s law ( without “  ” sign ), a restoring force, F s against extension of the spring, x graph is shown. where Figure 4.10

30. 30 The equation of elastic potential energy, U s for compressing or stretching a spring is

31. 31 A force of magnitude 800 N caused an extension of 20 cm on a spring. Determine the elastic potential energy of the spring when a. the extension of the spring is 30 cm. b. a mass of 60 kg is suspended vertically from the spring. Solution : a. Given x =0.300 m. Example 4.5 :

32. 32 Solution : b. Given m =60 kg. When the spring in equilibrium, thus

33. 33 4.2.3 Kinetic energy, K is an energy of a body due to its motion . Work- energy theorem states “ work done by the nett force on a body equals the change in the body’s kinetic energy ”. where

34. 34 A stationary object of mass 3.0 kg is pulled to the north by a constant force of magnitude 50 N on a smooth surface. Determine the speed of the object when it is travelled 4.0 m away. Solution : The nett force acting on the object is given by B y applying the work-kinetic energy theorem, thus Example 4.6 : Top view

35. 35 An object of mass 2.0 kg moves along the x-axis and is acted on by a force F . Figure 4.11 shows how F varies with distance travelled, s . The speed of the object at s = 0 is 10 m s 1 . Determine a. the speed of the object at s = 10 m, b. the kinetic energy of the object at s = 6.0 m. Example 4.7 : Figure 4.11

36. 36 Solution : a.

37. 37 Solution : b.

38. 38 Exercise 4.2 : Use gravitational acceleration, g = 9.81 m s  2 1. A bullet of mass 15 g moves horizontally at velocity of 250 m s 1 .It strikes a wooden block of mass 400 g placed at rest on a floor. After striking the block, the bullet is embedded in the block. The block then moves through 15 m and stops. Calculate the coefficient of kinetic friction between the block and the floor. ANS. : 0.278 2. A parcel is launched at an initial speed of 3.0 m s 1 up a rough plane inclined at an angle of 35 above the horizontal. The coefficient of kinetic friction between the parcel and the plane is 0.30. Determine a. the maximum distance travelled by the parcel up the plane, b. the speed of the parcel when it slides back to the starting point. ANS. : 0.560 m; 1.90 m s 1

39. 39 4.2.4 Principle of conservation of energy states “ in an isolated (closed) system, the total energy of that system is constant ”. we get The initial of total energy = the final of total energy Conservation of mechanical energy In an isolated system, the mechanical energy of a system is the sum of its potential energy, U and the kinetic energy, K of the objects are constant. OR OR At the end of this lesson, students should be able to: State and apply the principle of conservation of energy.

40. 40 Before After Figure 4.12 A 1.5 kg sphere is dropped from a height of 30 cm onto a spring of spring constant, k = 2000 N m 1 . After the block hits the spring, the spring experiences maximum compression, x as shown in Figure 4.12. a. Describe the energy conversion occurred after the sphere is dropped onto the spring until the spring experiences maximum compression, x . b. Calculate the speed of the sphere just before strikes the spring. c. Determine the maximum compression, x . Example 4.8 :

41. 41 The spring is not stretched hence U s = . The sphere is at height h 1 above ground with speed, v just before strikes the spring. Therefore The sphere is at height h 2 above the ground after compressing the spring by x . The speed of the sphere at this moment is zero. Hence The spring is not stretched hence U s = . The sphere is at height h above ground therefore U = mgh and it is stationary hence K = 0 . (2) (3) (1) Solution : a.

42. 42 Solution : b. Applying the principle of conservation of energy involving the situation (1) and (2), and

43. 43 Solution : c. Applying the principle of conservation of energy involving the situation (2) and (3), and

44. 44 A bullet of mass, m 1 = 5.00 g is fired into a wooden block of mass, m 2 = 1.00 kg suspended from some light wires as shown in Figure 4.13. The block, initially at rest. The bullet embeds in the block, and together swing through a height, h = 5.50 cm. Calculate a. the initial speed of the bullet. b. the amount of energy lost to the surrounding. Example 4.9 : Figure 4.13

45. 45 (1) (3) (2) Solution : a. Applying the principle of conservation of energy involving the situation (2) and (3),

46. 46 Solution : Applying the principle of conservation of linear momentum involving the situation (1) and (2), b. The energy lost to the surrounding, Q is given by

47. 47 Objects P and Q of masses 2.0 kg and 4.0 kg respectively are connected by a light string and suspended as shown in Figure 4.14. Object Q is released from rest. Calculate the speed of Q at the instant just before it strikes the floor. Example 4.10 : Figure 4.14 P Q Smooth pulley

48. 48 Solution : Initial P Q Smooth pulley P Q Smooth pulley Final

49. 49 Exercise 4.3 : Use gravitational acceleration, g = 9.81 m s  2 1. If it takes 4.00 J of work to stretch a spring 10.0 cm from its initial length, determine the extra work required to stretch it an additional 10.0 cm. ANS. : 12.0 J 2. A book of mass 0.250 kg is placed on top of a light vertical spring of force constant 5000 N m 1 that is compressed by 10.0 cm. If the spring is released, calculate the height of the book rise from its initial position. ANS. : 10.2 m 3. A 60 kg bungee jumper jumps from a bridge. She is tied to a bungee cord that is 12 m long when unstretched and falls a total distance of 31 m. Calculate a. the spring constant of the bungee cord. b. the maximum acceleration experienced by the jumper. ANS. : 100 N m 1 ; 22 m s 2

50. 50 4. A 2.00 kg block is pushed against a light spring of the force constant, k = 400 N m -1 , compressing it x =0.220 m. When the block is released, it moves along a frictionless horizontal surface and then up a frictionless incline plane with slope  =37.0 as shown in Figure 4.15. Calculate a. the speed of the block as it slides along the horizontal surface after leaves the spring. b. the distance travelled by the block up the incline plane before it slides back down. ANS. : 3.11 m s 1 ; 0.81 m Figure 4.15

51. 51 5. A ball of mass 0.50 kg is at point A with initial speed, u =4 m s 1 at a height of 10 m as shown in Figure 4.16 (Ignore the frictional force). Determine a. the total energy at point A, b. the speed of the ball at point B where the height is 3 m, c. the speed of the ball at point D, d. the maximum height of point C so that the ball can pass over it. ANS. : 53.1 J; 12.4 m s 1 ; 14.6 m s 1 ; 10.8 m Figure 4.16

52. 52 At the end of this chapter, students should be able to: Define and use average power, and instantaneous power, Learning Outcome: 4.3 Power (1 hour)

53. 53 4.3 Power 4.3.1 Power, P is a rate at which work is done . OR a rate at which energy is transferred . is a scalar quantity . S.I. unit of the power: At the end of this lesson, students should be able to: Define and use average power

54. 54 Figure 4.17 At the end of this lesson, students should be able to: Use instantaneous power. OR and and where

55. 55 An elevator has a mass of 1.5 Mg and is carrying 15 passengers through a height of 20 m from the ground. If the time taken to lift the elevator to that height is 55 s. Calculate the average power required by the motor if no energy is lost. (The average mass per passenger is 55 kg) Solution : M = mass of the elevator + mass of the 15 passengers M = 1500 + (55  15) = 2325 kg Example 4.11 :

56. 56 An object of mass 2.0 kg moves at a constant speed of 5.0 m s  1 up a plane inclined at 30  to the horizontal. The constant frictional force acting on the object is 4.0 N. Determine a. the rate of work done against the gravitational force, b. the rate of work done against the frictional force, c. the power supplied to the object. Solution : Example 4.12 :

57. 57 Solution : a. the rate of work done against the gravitational force is given by and and OR

58. 58 Solution : b. The rate of work done against the frictional force is c. The power supplied to the object, P supplied = the power lost against gravitational and frictional forces, P lost and

59. 59 Exercise 4.4 : Use gravitational acceleration, g = 9.81 m s  2 1. A person of mass 50 kg runs 200 m up a straight road inclined at an angle of 20  in 50 s. Neglect friction and air resistance. Determine a. the work done, b. the average power of the person. ANS. : 3.3610 4 J; 672 W

60. 60 2. A car of mass 1500 kg moves at a constant speed v up a road with an inclination of 1 in 10 as shown in Figure 4.18. All resistances against the motion of the car can be neglected. If the engine car supplies a power of 12.5 kW, calculate the speed v . ANS. : 8.50 m s 1 Figure 4.18

61. 61 THE END. Next Chapter… CHAPTER 5 : Circular motion