Work, Power, & Energy

1. Work, Power, & Energy

2. Forces & Motion Throughout this semester, we have examined objects’ changing motion. Forces are applied to objects and their position or velocity may change. Similarly, a Force can alter the momentum of an object and transfer it to a different object.

3. Force Work& Motion (or no motion) However, sometimes these forces may not actually change the motion of an object. For example, if I push on a pencil with a force of 45N, it will roll forward and accelerate at certain rate. If I push on a wall with a force of 45N, it will not move.

4. Work If a Force causes a change in motion, we refer to it as Work.

5. Work = Motion Work occurs when motion occurs in the direction of a Force. If there is no change in position (distance) then no Work has occurred.

6. Work & Newton’s Laws Note that this does not mean the Force is zero, only the Work. By relating the change in position to the Forces involved and Newton’s Laws, we derive the equation for Work.

7. Work Equation W = Fd where F = Force (N) and d = displacement (m). Work is thus given the unit of N x m which is also referred to as a Joule. (1J = 1 N x m)

8. Ex. Problem Calculate the Work done by a parent pushing a 5kg baby in a 20kg stroller over a distance of 15m. The Force applied is 250N.

9. Work in One Dimension Note that this equation only works for work done in one dimension (vertical or horizontal).

10. Work in 2 Dimensions As Force is applied in two dimensions, we must separate our Work into components based on the motion of the object.

11. Work in 2 Dimensions For example, using the above problem, the baby’s little sister pushes on the same stroller with a force of 150N at an angle of 60o to the horizontal. The stroller moves 15m. What is the Work?

12. Solution Using this setup, we see that the Force is at an angle, but the resultant motion is along the horizontal. Thus, when dealing in the x-direction, this gives us this triangle:

13. Work in 2 Dimensions By taking this into account, we change our Work equation into W = Fdcosθ. Solving, we get:

14. Work in One Direction Only For our purposes, we will be solving for the work in one direction.

15. Work in One Direction Only Thus, if a force is applied at an angle but the actual displacement occurs in the x-direction, we solve for Work in the x-direction. If an object is moving upward, we will solve using only the y-direction.

16. Rate of Work While work is an application of Force and distance, the rate is not accounted for. For example, moving a stack of books in 20 seconds differs from moving the same stack in 30 seconds.

17. Power The rate at which work is done is called Power. Or put into equation form, Power is equal to the work done per unit of time.

18. Power Equation This gives us the equation: P = W/t where W = Work (Joules) and t = time (seconds).

19. Watts (and Kilowatts) The Unit for Power is the Joule/second which is given the name Watt (W). Because a Watt is such a small unit, we often measure power in Kilowatts (kW).

20. Ex. Problem If a motor lifts an elevator with a force of 1500N upward for 25m in 5 seconds, how much Power is produced?

21. Simple Machines Often, machines are used to accomplish tasks. Simple machines are those that increase the force, change the direction of a force, or alter the distance over which the Force is applied, thus making things easier for us.

22. Mechanical Advantage The advantage they give us is known as Mechanical Advantage and is a comparison of our input force (termed effort force) compared to the force of the machine (resistance force).

23. Energy Before work can occur, or a Force can be applied, there needs to be some potential for these things to occur. In Physics, energy refers to the ability of an object to produce change to itself or the surroundings.

24. Energy Types There are multiple types of energy including chemical potential energy, Thermal Energy, and Kinetic Energy. In our study of motion, we will start with Kinetic Energy.

25. Kinetic Energy Kinetic Energy is the energy of an object due to its motion. In simple terms, a moving object has the ability to change its surroundings.

26. Kinetic Energy For example, a moving baseball has the ability to apply a force to a window and smash it. A stationary baseball does not have this potential.

27. What influences KE? However, not all things have the same ability to cause change. For example, a thrown bullet vs. a fired bullet will do differing amounts of change to a wall or object. Why is this?

28. Velocity & KE The fired bullet is moving much faster than the thrown bullet. Thus, the velocity of an object affects its Kinetic Energy.

29. Mass & KE Additionally, the mass of an object will also affect its Kinetic Energy. By relating mass and velocity to Newton’s Laws, we can derive the following equation:

30. KE Equation KE = ½ mv2 where m = mass (kg); v = velocity (m/s); and KE = Kinetic Energy - measured in Joules (J).

31. Ex. Problem How much KE does a 0.025kg bullet traveling at 250m/s have?

32. Work & Energy So, when a moving object acts upon another object, it does so due to its Energy. When this object moves or changes the other object’s position, it has also done work.

33. Work - Energy Theorem Thus, there is a relationship between Energy and Work. This is known as the Work-Energy Theorem and is represented as follows:

34. Work - Energy Theorem W = ΔKE or W = KEf - KEi In simpler terms, to do more work, there must be more energy available.

35. Ex. Problem A football player pushes a 50kg sled at 2 m/s and then speeds up to 5 m/s. How much work was done on the sled?

36. Potential Energy Often Energy is referred to as the ability to cause change. This implies that the change might not actually be occurring yet or the object may not be in motion yet.

37. Potential Energy Thus, the POTENTIAL to cause change is also considered a form of Energy. We will look at two types of potential energy.

38. Gravitational Potential Energy Objects on Earth are constantly being acted upon by gravity. Thus, there is a consistent force acting on them that may cause them to move. This idea that Gravity can result in motion is known as Gravitational Potential Energy (GPE or U).

39. Gravitational Potential Energy In order for objects to move due to gravity, they must have some height above the ground. In other words, the gravity needs the opportunity to pull them down or do Work on them.

40. GPE Equation By comparing the acceleration due to gravity, the height and the mass of the object, we derive the equation for GPE (Gravitational Potential Energy).

41. GPE Equation GPE = mgh where m = mass (kg), g = acceleration due to gravity (9.8m/s2), and h = height (m) The Unit for GPE and all potential energy is still the Joule (J).

42. Ex. Problem A 5g pencil rests on a desk 0.8m above the floor. Calculate the GPE for the pencil.

43. Elastic Potential Energy Stretched and compressed objects also exhibit a form of potential energy. Because they are stretched/compressed, they may move.

44. Elastic Potential Energy How much they are stretched or compressed dictates how much potential energy they have. Examples of this include springs and rubber bands.

45. Conservation of Energy Similarly to the conservation of momentum, the total amount of Energy present in a closed system remains constant. Thus, as objects collide or move, the total amount of energy present remains constant.

46. KE + PE However, the type of energy they exhibit may change. For example, as a ball falls, it has height (and therefore GPE) and velocity (KE).

47. Mechanical Energy The combination of these types of energy is known as Mechanical Energy. This can be represented by the following equation: Energyinitial = Energyfinal or KEi + PEi = KEf + PEf

48. Mechanical Energy Equation After substituting, we get this equation: ½ mvi2 + mghi = ½ mvf2 + mghf

49. Ex. Problem 1 A bullet is shot and hits a clay pigeon that is 25m above the ground. If the bullet is traveling at 225m/s when it hits the clay pigeon, how much energy does it have?

50. Ex. Problem 2 In the diagram below, a roller coaster car is moving at 8.0m/s at point A. What is the coaster’s velocity at point B if it has a mass of 350.0kg?