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Work, Energy, and Power

Work, Energy, and Power. Lesson 1: Basic Terminology and Concepts Definition and Mathematics of Work Calculating the Amount of Work Done by Forces Kinetic Energy Potential Energy Mechanical Energy Power Lesson 2 - The Work-Energy Relationship Work-Energy Principle

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Work, Energy, and Power

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  1. Work, Energy, and Power Lesson 1: Basic Terminology and Concepts • Definition and Mathematics of Work • Calculating the Amount of Work Done by Forces • Kinetic Energy • Potential Energy • Mechanical Energy • Power Lesson 2 - The Work-Energy Relationship • Work-Energy Principle • Internal vs. External Forces • Analysis of Situations Involving External Forces • Analysis of Situations in Which Mechanical Energy is Conserved • Application and Practice Questions

  2. Definition and Mathematics of Work force cause • In physics, work is defined as a _________ acting upon an object to ____________ a __________________. • In order for a force to qualify as having done work on an object, there must be a displacement and the force must ___________ the displacement displacement cause Work done Work not 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. no work work no work work

  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 scalar quantity • Work is independent of time the force acts on the object. Only the horizontal component of the force (Fcosθ) causes a horizontal displacement.

  5. Example 1 • How much work is done on a vacuum cleaner pulled 3.0 m by a force of 50.0 N at an angle of 30o above the horizontal?.

  6. Example 2 • 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? Given: d = h = 1.2 meters; m = 5.0 kg; θ = 0 Unknown: W = ? W = F∙dcosθ F = mg = (5.0 kg)(9.81 m/s2) cos0o = 49 N W = F∙d = (49 N) (1.2 m) = 59 J

  7. Example 3 • 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. • Given: F = 5.0 N; m = 2.3 kg d = 2.0 m θ= 30o 5.0 N 30o 2.3 kg • unknown: • W = ? J • Solve: • W = F∙d∙cosθ • W = (5.0 N)(2.0 m)(cos30o) = 8.7 J

  8. Example 4 • 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. • Given: Fx = 8.0 N Fy = 6.0 N dx = 3.0 m 6.0 N • unknown: W = ? J F W = Fxdx W = (8.0 N)(3.0 m) = 24 J 8.0 N 3.0 m

  9. Example 5 • 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? Wyou = Fd Wneighbor = (½ F)(4d) = 2 Fd = 2 Wyou The neighbor, twice as much

  10. The sign of work W = F∙d∙cosθ Work done – 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

  11. When F is ┴ to d, W = 0 To Do Work, Forces Must Cause Displacements W = F∙d∙cosθ = 0

  12. Only the horizontal component of the force (Fcosθ) causes a horizontal displacement.

  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. Example • A student produced various elongations of a spring by applying a series of forces to the spring. The graph below represents the relationship between the applied force and the elongation of the spring. Determine the work done between 0.0 m to 0.20 m.

  15. d F The angle in work equation • 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.

  16. What is θ in each case?

  17. practices • 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. • 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? • 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. • 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.

  18. Work and Energy are related • When work is done on a system, that system’s energy equals to the amount of work done on it. • Work and energy have the same unit: Joule • For example, if you push a cart, you do work on the cart, the cart is going to speed up and its temperature may increase, its energy is increased. If you lift a rock, you do work on the rock and you increase the rock’s energy. • There are many forms of energy. • Potential energy • Kinetic energy • Internal energy

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

  20. Gravitational potential energy • When you lift an object, you do work against gravity. As a result, its position is higher, and it has more gravitational potential energy. • Gravitational potential energy is the energy stored in an object as the result of its vertical position (height) • The energy is stored as the result of the gravitational attraction of the Earth for the object. • Gravitational depends on • m: mass, in kilograms • g: acceleration due to gravity = 9.81 m/s2 • h: height

  21. Gravitational Potential Energy is relative: To determine the gravitational potential energy of an object, a zero height position must first be assigned. Typically, the ground 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

  22. Gravitational potential energy Equation Gravitation potential energy equals to work done against gravity . • change in height, in meters • Gravitational attraction between Earth and the object: • m: mass, in kilograms • g: acceleration due to gravity = 9.81 m/s2

  23. Change in GPE only depends on change in height, not path As long as the object starts and ends at the same height, the object has the same change in GPE because gravity does the same amount of work regardless of which path is taken.

  24. 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? 100 J As long as the object starts and ends at the same height, the object has the same change in GPE because gravity does the same amount of work regardless of which path is taken.

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

  26. 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? • Known: • m = 2.00 kg • h = 0.92 m • g = 9.81 m/s2 Solve: ∆PE = mg∆h ∆PE = (2.00 kg)(9.81m/s2)(0.92 m) = 18 J • unknown: • PE = ? J

  27. GPE vs. Vertical Height Graph The graph of gravitational potential energy vs. vertical height for an object near Earth's surface is a straight line. The slope is the weight of the object.

  28. Elastic potential energy • Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing when a force is applied. • Elastic potential energy can be stored in • Rubber bands • Bungee cores • Springs • trampolines

  29. Elastic potential energy in a spring • k: spring constant • x: amount of compression or elongation relative to equilibrium position equilibrium x elongation

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

  31. 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? PE = ½ kx2 15 J = ½ k (0.50 m)2 k = 120 N/m

  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? PEs = ½ kx2 PEs = ½ k(0.20 m)2 PEs = (0.020 k) J

  33. Elongation ( or compression) depends on force • For certain springs, the amount of force (F) is directly proportional to the amount of elongation or compression (x); the constant of proportionality is known as the spring constant(k).

  34. Hooke’s Law • 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. • x the distance difference between the length of stretched/compressed spring and its relaxed (equilibrium) spring.

  35. example • Determine the x in F = kx

  36. force elongation The slope of Fsvs. x • Spring force is directly proportional to the elongation of the spring (displacement) The slope represents spring constant: k = F / x (N/m)

  37. elongation force caution • Sometimes, we might see a graph such as this: The slope represents the inverse of spring constant: Slope = 1/k

  38. Example • Given the following data table and corresponding graph, calculate the spring constant of this spring.

  39. Example • A 20.-newton weight is attached to a spring, causing it to stretch, as shown in the diagram. What is the spring constant of this spring?

  40. 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

  41. Work done stretching a spring and its Elastic Potential Energy • Elastic potential energy stored in a spring equals to the work done in stretching it. Work = Area = ½ (base)(height) Work = ½ (x)(F) Work = ½ (x)(k∙x) Work = ½ k∙x2 PEs = Work = ½ k∙x2

  42. Example • If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

  43. 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. Solve: PEs = ½ k∙x2 To find x, use Fs = kx, (2.50 N) = (25.0 N/m)(x) x = 0.100 m PEs = ½ (25.0 N/m)(0.100 m)2 PEs = 0.125 J Given: Fs = 2.50 N k = 25.0 N/m Unknown: PEs = ? J

  44. 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?

  45. KE = ½ mv2 Kinetic energy • Kinetic energy is the energy of motion. • 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 • Kinetic energy is a scalar quantity.

  46. Questions • Which of the following has kinetic energy? • a falling sky diver • a parked car • a shark chasing a fish • a calculator sitting on a desk • If a bowling ball and a volleyball are traveling at the same speed, do they have the same kinetic energy? • Car A and car B are identical and are traveling at the same speed. Car A is going north while car B is going east. Which car has greater kinetic energy?

  47. Speed has more impact on kinetic energy • 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

  48. Example • A 7.00 kg bowling ball moves at 3.00 m/s. How much kinetic energy does the bowling ball heave? How fast must a 2.45 g table-tennis ball move in order to have the same kinetic energy as the bowling ball? Is this speed reasonable for a table-tennis ball?

  49. 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? • Known: • KE = 450 J • v = 25 m/s • Unknown: • m = ? kg Solve: KE = ½ mv2 450 J = ½ (m)(25 m/s)2 m = 1.4 kg

  50. 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

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