Lesson 1: Basic Terminology and Concepts Work Definition and Mathematics of Work

1. Lesson 1: Basic Terminology and Concepts • Work • Definition and Mathematics of Work • Calculating the Amount of Work Done by Forces • Potential Energy • Kinetic Energy • Mechanical Energy • Power

2. Definition and Mathematics of Work • In physics, work is defined as a _________ acting upon an object to ____________ a __________________. force cause displacement Work is being done Work is not being done Work is not being done

3. Let’s practice – work or no work • A student applies a force to a wall and becomes exhausted. • A calculator falls off a table and free falls to the ground. • A waiter carries a tray full of beverages above his head by one arm across the room • A rocket accelerates through space.

4. F θ d Fy F θ Fx d Calculating the Amount of Work Done by Forces • F - is the force in Newton, which causes the displacement of the object. • d - is the displacement in meters • θ = angle between force and displacement • W - is work in N∙m or Joule (J). 1 J = 1 N∙m = 1 kg∙m2/s2 • Work is a _____________ quantity • Work is independent of time the force acts on the object. W = F∙d∙cosθ scalar Only the horizontal component of the force (Fcosθ) causes a horizontal displacement.

5. W = F∙d∙cosθ positive, negative or zero work Positive work negative work - force acts in the direction opposite the objects motion in order to slow it down. no work

6. To Do Work, Forces Must Cause Displacements W = F∙d∙cosθ = 0

7. d F The angle in work equation W = F∙d∙cosθ • The angle in the equation is the angle between the force and the displacement vectors. F & d are in the same direction, θ is 0o.

8. example • A 20.0 N force is used to push a 2.00 kg cart a distance of 5.00 meters. Determine the amount of work done on the cart by the force. 20.0 N

9. example • How much work is done in lifting a 5.0 kg box from the floor to a height of 1.2 m above the floor?

10. example • A 2.3 kg block rests on a horizontal surface. A constant force of 5.0 N is applied to the block at an angle of 30.o to the horizontal; determine the work done on the block a distance of 2.0 meters along the surface. 5.0 N 30o 2.3 kg

11. practice • Matt pulls block along a horizontal surface at constant velocity. The diagram show the components of the force exerted on the block by Matt. Determine how much work is done against friction. 6.0 N F 8.0 N 3.0 m

12. example • A neighbor pushes a lawnmower four times as far as you do but exert only half the force, which one of you does more work and by how much?

13. Force vs. displacement graph • The area under a force versus displacement graph is the work done by the force. Example: a block is pulled along a table with 10. N over a distance of 1.0 m. W = Fd = (10. N)(1.0 m) = 10. J work Force (N) Displacement (m) height base area

14. Potential energy • An object can store energy as the result of its position. ________________________ is the stored energy of position possessed by an object. • Two form: • Gravitational • Elastic Potential energy

15. Gravitational potential energy vertical position (height). • Gravitational potential energy is the energy stored in an object as the result of its _________________________ • The energy is stored as the result of the _____________ attraction of the Earth for the object. • The work done in raising an object must result in an increase in the object's _______________________ • The gravitational potential energy of an object is dependent on three variables: • The mass of the object • The height of the object • The gravitational field strength • Equation: ______________________ • m: mass, in kilograms • h: height, in meters • g: acceleration of gravity = 9.81 m/s2 gravitational gravitational potential energy PEgrav = m∙g∙h

16. GPE • GPE = mgh • The equation shows that . . . • . . . the more gravitational potential energy it’s got. • the more mass a body has • or the stronger the gravitational field it’s in • or the higher up it is

17. GPE and work done by gravity • When an object falls, gravity does positive work. Object loses GPE. • Wgrav = mg(hi – hf) • Wgrav = - mg(hf – hi) = - mg∆h hi As long as the falling height is the same, gravity did The same amount of work regardless of which path is taken. hf

18. GPE and work against gravity • When an object is raised against gravity at constant speed (no change in kinetic energy), gravity does negative work. Object gains GPE. • Work done against gravity = mg∆h hf As long as the object is raised to the same height, work done against gravity is the same regardless of which path is taken. hi

19. The increase in an object's potential energy equals the work done in raising an object Each path up to the seat top requires the same amount of work. The amount of work done by a force on any object is given by the equation W = F∙d∙cosθ where F is the force, d is the displacement and θ is the angle between the force and the displacement vector. In all three cases, θ equals to 0o

20. example • The diagram shows points A, B, and C at or near Earth’s surface. As a mass is moved from A to B, 100. joules of work are done against gravity. What is the amount of work done against gravity as an identical mass is moved from A to C?

21. Unit of energy • The unit of energy is the same as work: _______ • 1 joule = 1 (kg)∙(m/s2)∙(m) = 1 Newton ∙ meter • 1 joule = 1 (kg)∙(m2/s2) Joules Work and energy has the same unit

22. Gravitational potential energy is relative • To determine the gravitational potential energy of an object, a _______ height position must first be assigned. • Typically, the ___________ is considered to be a position of zero height. • But, it doesn’t have to be: • It could be relative to the height above the lab table. • It could be relative to the bottom of a mountain • It could be the lowest position on a roller coaster zero ground

23. example • How much potential energy is gained by an object with a mass of 2.00 kg that is lifted from the floor to the top of 0.92 m high table?

24. The graph of gravitational potential energy vs. vertical height for an object near Earth's surface gives the weight of the object. The weight of the object is the slope of the line. Weight = __________

25. Elastic potential energy • Elastic potential energy is the energy stored in ______________ materials as the result of their stretching or compressing. • Elastic potential energy can be stored in • Rubber bands • Bungee cores • Springs • trampolines elastic

26. Hooke’s Law F = kx Spring force = spring constant x displacement • F in the force needed to displace (by stretching or compressing) a spring x meters from the equilibrium (relaxed) position. The SI unit of F is Newton. • k is spring constant. It is a measure of stiffness of the spring. The greater value of k means a stiffer spring because more force is needed to stretch or compress it that spring. The Si units of k are N/m. depends on the material made up of the spring. k is in N/m • x the distance difference between the length of stretched/compressed spring and its relaxed (equilibrium) spring.

27. example • A spring has a spring constant of 25 N/m.  What is the minimum force required to stretch the spring 0.25 meter from its equilibrium position?

28. example • The graph below shows elongation as a function of the applied force for two springs, A and B. Compared to the spring constant for spring A, the spring constant for spring B is • smaller • larger • the same

29. Elastic potential energy in a spring • Elastic potential energy is the Work done on the spring. PEs = Favg∙d = Favg∙x = (½ k∙x)∙x = ½ kx2 Note: F is the average force • k: spring constant • x: amount of compression or extension relative to equilibrium position PEs = ½ k∙x2

30. Elastic potential energy is directly proportional to x2 Elastic potential energy elongation

31. example • A spring has a spring constant of 120 N/m.  How much potential energy is stored in the spring as it is stretched 0.20 meter?

32. example • The unstretched spring in the diagram has a length of 0.40 meter and a spring constant k.  A weight is hung from the spring, causing it to stretch to a length of 0.60 meter.  In terms of k, how many joules of elastic potential energy are stored in this stretched spring?

33. example • Determine the potential energy stored in the spring with a spring constant of 25.0 N/m when a force of 2.50 N is applied to it.

34. example • As shown in the diagram, a 0.50-meter-long spring is stretched from its equilibrium position to a length of 1.00 meter by a weight. If 15 joules of energy are stored in the stretched spring, what is the value of the spring constant?

35. example • A 10.-newton force is required to hold a stretched spring 0.20 meter from its rest position. What is the potential energy stored in the stretched spring?

36. A force of 0.2 N is needed to compress a spring a distance of 0.02 meter. What is the potential energy stored in this compressed spring?

37. KE = ½ mv2 Kinetic energy motion • Kinetic energy is the energy of _______. • An object which has motion - whether it be vertical or horizontal motion - has kinetic energy. • The equation for kinetic energy is: __________________ • Where KE is kinetic energy, in joules • v is the speed of the object, in m/s • m is the mass of the object, in kg

38. Kinetic Energy • KE = ½ mv2 • The equation shows that . . . • . . . the more kinetic energy it has. • the more mass a body has • or the faster it’s moving

39. KE is directly proportional to m, so doubling the mass doubles kinetic energy, and tripling the mass makes it three times greater. • KE is proportional to v2, so doubling the speed quadruples kinetic energy, and tripling the speed makes it nine times greater. Kinetic energy Kinetic energy speed mass

40. Example • A 55 kg toy sailboat is cruising at 3 m/s. What is its kinetic energy? Note: Kinetic energy (along with every other type of energy) is a scalar, not a vector!

41. example • An object moving at a constant speed of 25 meters per second possesses 450 joules of kinetic energy. What is the object's mass?

42. example • A cart of mass m traveling at a speed v has kinetic energy KE.  If the mass of the cart is doubled and its speed is halved, the kinetic energy of the cart will be • half as great • twice as great • one-fourth as great • four times as great

43. example • Which graph best represents the relationship between the kinetic energy, KE, and the velocity of an object accelerating in a straight line? a b c d

44. Mechanical Energy • Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position) or both. The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy TME = KE + PEg + PEs

45. Mechanical Energy as the Ability to Do Work • Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy - is able to do work.

46. The diagram shows the motion of Brie as she glides down the hill and makes one of her record-setting jumps. TME = TME = TME = TME = TME =

47. Power • Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation. • The standard metric unit of power is the Watt.

48. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. • The power rating of a car relates to how rapidly the car can be accelerated. • Some people are more power-full than others. That is, some people are capable of doing the same amount of work in less time or more work in the same amount of time

49. example • Ben Pumpiniron elevates his 80-kg body up the 2.0-meter stairwell in 1.8 seconds. What is his power? It can be assumed that Ben must apply an (80 kg x 9.81 m/s2) -Newton downward force upon the stairs to elevate his body.

50. Another equation for power