TOPIC 5: Work, Energy & Power

1. TOPIC 5:Work, Energy & Power

2. WORK • Definition of Work: • When a force causes a displacement of an object • Components of the force need to be in the direction of the displacement

3. Net Work done by a Constant Net Force Work = Force (F) x Displacement (x) W = Fx W = Fx = (Fcosθ)x ** Only the component of the force in the direction of the displacement, contributes to work

4. Units of Work Work = Force x Displacement = Newtons x meters Newton x meter  Joule (J) * Joule is named after James Prescott Joule (1818-1889) who made major contributions to the understanding of energy, heat, and electricity =

5. Work • Work: • Scalar quantity • Can be positive or negative • Positive work  Exists when the force & displacement vectors point in the same direction • Negative work  Exists when the force & displacement vectors point in opposite directions

6. Problem How much work is done on a vacuum cleaner pulled 3 m by a force of 50 N at an angle of 30° above the horizontal? W = (Fcosθ)x W = ? F = 50N d = 3m θ = 30° W = (50N)(cos30°)(3m) = 130 J

7. ENERGY Kinetic Energy: * Energy associated with an object in motion * Depends on speed and mass * Scalar quantity * SI unit for all forms of energy = Joule (J) KE = ½ mv2 KE = ½ x mass x (velocity)2

8. Kinetic Energy If a bowling ball and a soccer ball are traveling at the same speed, which do you think has more kinetic energy? KE = ½ mv2 * Both are moving with identical speeds * Bowling ball has more mass than the soccer ball  Bowling ball has more kinetic energy

9. Kinetic Energy Problem A 7 kg bowling ball moves at 3 m/s. How fast must a 2.45 g tennis ball move in order to have the same kinetic energy as the bowling ball? Velocity of tennis ball = 160 m/s

10. Work-Kinetic Energy Theorem Work-kinetic Energy Theorem: • Net work done on a particle equals the change in its kinetic energy (KE) W = ΔKE

11. PROBLEM What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg.

12. PROBLEM • What is the soccer ball’s speed immediately after being kicked? Its mass is 0.42 kg. W = F ∙ Δx W = (240 N) (0.20 m) = 48 J W = ΔKE = 48 J KE = ½ mv2 = 48 J v2 = 2(48 J)/0.42 kg v = 15 m/s

13. Work-Kinetic Energy Theorem On a frozen pond, a person kicks a 10 kg sled, giving it an initial speed of 2.2 m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10? m = 10 kg vi = 2.2 m/s vf = 0 m/s μk = 0.10 d = ?

14. Work-Kinetic Energy Theorem Wnet = Fnetdcosθ * Net work done of the sled is provided by the force of kinetic friction Wnet = Fkdcosθ  Fk = μkN  N = mg Wnet = μkmgdcosθ * The force of kinetic friction is in the direction opposite of d  θ = 180° * Sled comes to rest  So, final KE = 0 Wnet = Δ KE = ½ mv2f – ½ mv2i Wnet = -1/2 mv2i

15. Work-Kinetic Energy Theorem Use the work-kinetic energy theorem, and solve for d Wnet = ΔKE - ½ mv2i = μkmgdcosθ d = 2.5 m

16. POWER POWER: * A quantity that measures the rate at which work is done or energy is transformed * Power = work / time interval P = W/Δt (W = Fx  P = Fx/Δt  v = x/Δt) * Power = Force x speed P = Fv

17. POWER SI Unit for Power: Watt (W)  Defined as 1 joule per second (J/s) Horsepower = Another unit of power 1 hp = 746 watts

18. POWER PROBLEM A 193 kg curtain needs to be raised 7.5 m, in as close to 5 s as possible. The power ratings for three motors are listed as 1 kW, 3.5 kW, and 5.5 kW. What motor is best for the job?

19. POWER PROBLEM m = 193 kg Δt = 5s d =7.5m P = ? P = W/Δt = Fx/Δt = mgx/Δt = (193kg)(9.8m/s2)(7.5m)/5s = 280 W  2.8 kW ** Best motor to use = 3.5 kW motor. The 1 kW motor will not lift the curtain fast enough, and the 5.5 kW motor will lift the curtain too fast

20. POTENTIAL ENERGY Potential Energy: * Stored energy * Associated with an object that has the potential to move because of its position relative to some other location Example: Balancing rock- Arches National Park, Utah Delicate Arch- Arches National Park, Utah

23. GRAVITATIONAL POTENTIAL ENERGY- Definition Gravitational potential energy PEg is the energy an object of mass m has by virtue of its position relative to the surface of the earth. That position is measured by the height h of the object relative to an arbitrary zero level: PEg = mgh SI Unit = Joule (J)

24. Problem What is the bucket’s gravitational potential energy?

25. Problem • What is the bucket’s gravitational potential energy? PE = mgh PE = (2.00 kg)(9.80 m/s2)(4.00 m) PE = 78.4 J

26. Gravitational Potential Energy Example: A Gymnast on a Trampoline The gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast?

27. Gravitational Potential Energy

28. Elastic Potential Energy * Energy stored in any compressed or stretched object • Spring, stretched strings of a tennis racket or guitar, rubber bands, bungee cords, trampolines, an arrow drawn into a bow, etc.

29. Springs • When an external force compresses or stretches a spring  Elastic potential energy is stored in the spring • The more stretch, the more stored energy • For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); • Constant of proportionality is known as the spring constant (k) Fspring = k * x

30. Hooke’s Law • If a spring is not stretched or compressed  no potential energy is being stored • Spring is in an Equilibrium position • Equilibrium position: Position spring naturally assumes when there is no force applied to it • Zero potential energy position

31. Hooke’s Law • Special equation for springs • Relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant PE elastic = ½kx2 k = Spring constant (N/m) Stiffer the spring  Larger the spring constant x = Amount of compression relative to the equilibrium position

32. Potential Energy Problem A 70 kg stuntman is attached to a bungee cord with an unstretched length of 15 m. He jumps off the bridge spanning a river from a height of 50m. When he finally stops, the cord has a stretched length of 44 m. Treat the stuntman as a point mass, and disregard the weight of the bungee cord. Assuming the spring constant of the bungee cord is 71.8 N/m, what is the total potential energy relative to the water when the man stops falling?

33. Potential Energy Problem * Zero level for gravitational potential energy is chosen to be the surface of the water * Total potential energy  sum of the gravitational & elastic potential energy PEtotal = PEg + PEelastic = mgh + ½ kx2 * Substitute the values into the equation PEtotal = 3.43 x 104 J

34. Potential Energy • The energy stored in an object due to its position relative to some zero position • An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height • An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position

35. Linking Work to Mechanical Energy • WORK is a force acting upon an object to cause a displacement • When work is done upon an object, that object gains energy • Energy acquired by the objects upon which work is done is known as MECHANICAL ENERGY

36. Mechanical Energy • Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position

37. Total Mechanical Energy *Total Mechanical Energy: The sum of kinetic energy & all forms of potential energy 1. Kinetic Energy (Energy of motion) KE = ½ mv2 2. Potential Energy (Stored energy of position) a. Gravitational PEg = mgh b. Elastic PEelastic = ½ kx2

38. Mechanical Energy CONSERVATION OF MECHANICAL ENERGY: * In the absence of friction, mechanical energy is conserved, so the amount of mechanical energy remains constant MEi = MEf Initial mechanical energy = final mechanical energy (in the absence of friction) PEi + KEi = PEf + KEf mghi + ½ mvi2 = mghf + ½ mvf2

39. Conservation of Energy Problem Starting from rest, a child zooms down a frictionless slide from an initial height of 3 m. What is her speed at the bottom of the slide? (Assume she has a mass of 25 kg)

40. Conservation of Energy Problem hi = 3m m = 25kg vi = 0 m/s hf = 0m vf = ? • Slide is frictionless  Mechanical energy is conserved • Kinetic energy & potential energy = only forms of energy present • KE = ½ mv2 PEg = mgh • Final gravitational potential energy = zero (Bottom of the slide)  PEgf = 0 • Initial gravitational potential energy  Top of the slide  PEgi = mghi  (25kg)(9.8m/s2)(3m) = 736 J

41. Conservation of Energy Problem hi = 3m m = 25kg vi = 0 m/s hf = 0m vf = ? • Initial Kinetic Energy = 0, because child starts at rest • KEi = 0 • Final Kinetic Energy • KEf = ½ mv2  ½ (25kg)v2f • MEi = MEf PEi + KEi = PEf + Kef 736 J + 0 J = 0 J + (1/2)(25kg)(v2f) vf = 7.67 m/s

42. Mechanical Energy  Ability to do Work • An object that possesses mechanical energy is able to do work • Its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced • Classic Example  Massive wrecking ball of a demolition machine

45. Mechanical Energy is the ability to do work… An object that possesses mechanical energy (whether it be kinetic energy or potential energy) has the ability to do work That is… its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced

46. Mechanical Energy • Work is a force acting on an object to cause a displacement • In the process of doing work  the object which is doing the work exchanges energy with the object upon which the work is done • When work is done up the object  that object gains energy

47. Mechanical Energy • A weightlifter applies a force to cause a barbell to be displaced • Barbell now possesses mechanical energy- all in the form of potential energy ** The energy acquired by the objects upon which work is done is known as mechanical energy

48. Mechanical Energy is the ability to do work… Examples on website: Massive wrecking ball of a demolition machine The wrecking ball is a massive object which is swung backwards to a high position and allowed to swing forward into a building structure or other object in order to demolish it Upon hitting the structure, the wrecking ball applies a force to it in order to cause the wall of the structure to be displaced Mechanical energy = ability to do work

49. Work- Energy Theorem Categorize forces based upon whether or not their presence is capable of changing an object’s total mechanical energy * Certain types of forces, which when present and when involved in doing work on objects, will change the total mechanical energy of the object * Other types of forces can never change the total mechanical energy of an object, but rather only transform the energy of an object from PE to KE or vice versa ** Two categories of forces  Internal & External

50. Work- Energy Theorem External Forces: Applied force, normal force, tension force, friction force and air resistance force Internal Forces: Gravity forces, spring forces, electrical forces and magnetic forces