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Work and Energy Objectives: The student will understand the basic terms associated with Work and Energy. The concepts of work and energy are closely tied to the concept of force because an applied force can do work on an object and cause a change in energy. Energy is defined as the ability to do work.

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Work The concept of work in physics is much more narrowly defined than the common use of the word. Work is done on an object when an applied force moves it through a distance. In our everyday language, work is related to expenditure of muscular effort, but this is not the case in the language of physics. A person that holds a heavy object does no physical work because the force is not moving the object through a distance. Work, according to the physics definition, is being accomplished while the heavy object is being lifted but not while the object is stationary. Another example of the absence of work is a mass on the end of a string rotating in a horizontal circle on a frictionless surface. The centripetal force is directed toward the center of the circle and, therefore, is not moving the object through a distance; that is, the force is not in the direction of motion of the object. (However, work was done to set the mass in motion.) Mathematically, work is W = F · x, where F is the applied force and x is the distance moved, that is, displacement. Work is a scalar. The SI unit for work is the joule (J), which is newton-meter or kg m/s 2.

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Work is done by a varying force W = F · x If work is done by a varying force, the above equation cannot be used. This figure shows the force-versus-displacement graph for an object that has three different successive forces acting on it. The force is increasing in segment I, is constant in segment II, and is decreasing in segment III. The work performed on the object by each force is the AREA between the curve and the x axis. The total work done is the total AREA between the curve and the x axis. For example, in this case, the work done by the three successive forces is shown. In this example, the total work accomplished is (1/2)(15)(3) + (15)(2) + (1/2)(15)(2) = ; work = 67.5 J. For a gradually changing force, the work is expressed in integral form, W = F · dx. Where the integral allows you to calculate the area.

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Kinetic energy Kinetic energy is the energy of an object in motion. The expression for kinetic energy can be derived from the definition for work and from kinematic relationships. Consider a force applied parallel to the surface that moves an object with constant acceleration. From the definition of work, from Newton's second law of motion, and from kinematics, W = Fx = max and v f 2 = v o ax, or a = ( v f 2 v o 2 )/2 x. Substitute the last expression for acceleration into the expression for work to obtain W = m ( v f 2 v o 2 ) or W = (1/2) mv f 2 (1/2) mv o 2. The right side of the last equation yields the definition for kinetic energy: K. E. = (1/2) mv 2 Kinetic energy is a scalar quantity with the same units as work, joules (J). For example, a 2 kg mass moving with a speed of 3 m/s has a kinetic energy of 9 J. The above derivation shows that the net work is equal to the change in kinetic energy. This relationship is called the work-energy theorem: W net = K. E. f K.E. o, where K. E. f is the final kinetic energy and K. E. o is the original kinetic energy.

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Potential energy Potential energy, also referred to as stored energy, is the ability of a system to do work due to its position or internal structure. Examples are energy stored in a pile driver at the top of its path or energy stored in a coiled spring. Potential energy is measured in units of joules.

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Gravitational potential energy Gravitational potential energy is energy of position. First, consider gravitational potential energy near the surface of the earth where the acceleration due to gravity(g) is approximately constant. In this case, an object's gravitational potential energy with respect to some reference level is P.E. = mgh, where h is the vertical distance above the reference level. To lift an object slowly, a force equal to its weight (mg) is applied through a height (h). The work accomplished is equal to the change in potential energy: W = P. E. f P. E. o = mgh f mgh o, where the subscripts (f and o) refer to the final and original heights of the body. Launching a rocket into space requires work to separate the mass of the earth and the rocket to overcome the gravitational force. For large distances from the center of the earth, the above equation is inadequate because g is not constant. The general form of gravitational potential energy is P.E. = GMm/r, where M and m refer to the masses of the two bodies being separated and r is the distance between the centers of the masses. The negative sign is a result of selecting the zero reference at r equal to infinity, that is, at very large separation.

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Elastic potential energy Elastic potential energy is energy stored in a spring. The magnitude of the force required to stretch a spring is given by F = kx, where x is the distance of stretch (or compression) of a spring from the unstressed position, and k is the spring constant. The spring constant is a measure of the stiffness of the spring, with stiffer springs having larger k values. The potential energy stored in a spring is given by P. E. = (1/2) kx 2. Change in potential energy is equal to work. The gravitational force and the force to stretch a spring are varying forces; therefore, the potential energy equations given above for these two cases can also be derived from the integral form of work, Δ P.E. = W = F · dx.

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Power Power is the rate of doing work, average P = W/t, where t is the time interval during which work (W) is accomplished. Another form of power is found from W = FΔ x and substitution of average velocity of the object during time t for Δ x/ t: average P = F Δ x/Δ t = F(average v).

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The conservation of energy The principle of conservation of energy is one of the most far-reaching general laws of physics. It states that energy is neither created nor destroyed but can only be transformed from one form to another in an isolated system. Because the total energy of the system always remains constant, the law of conservation of energy is a useful tool for analyzing a physical situation where energy is changing form. Imagine a swinging pendulum with negligible frictional forces. At the top of its rise, all the energy is gravitational potential energy due to height above the stationary position. At the bottom of the swing, all the energy has been transformed into kinetic energy of motion. The total energy is the sum of the kinetic and potential energies. It maintains the same value throughout the back and forth motion of a swing.

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Pendulum Energy At point C, the potential energy is dependent upon the height, and the rest of the total energy is kinetic energy.

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Work, Energy, Power… What is all the fuss?

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The Work-Energy Relationship Bar Chart Illustrations Objectives: the student will learn how to make an energy bar chart and will learn how to solve problems after making this transformation of information to such a problem solving tool.

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Bar Chart Illustrations KE i + PE i + W ext = KE f + PE f

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KE i + PE i-grav + PE i-spring + W ext = KE f + PE f-grav + PE f-spring KE i + PE i + W ext = KE f + PE f We will investigate the use and meaning of work- energy bar charts and make an effort to apply this understanding to a variety of motions involving energy changes or energy transformations.

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Procedure for Constructing Bar Charts What are the initial and final states of the object? Analyze the forces acting upon the object. Construct bars on the chart. Balance the energy in initial and final situations.

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Consider a ball falling from the top of a pillar to the ground below; ignore air resistance. The initial state is the ball at rest on top of the pillar and the final state is the ball just prior to striking the ground.

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Now Draw your Bar Chart! Observe that this work-energy bar chart reveals that there is no kinetic and elastic potential energy in the initial state. There is no gravitational and elastic potential energy in the final state. There is no work done by external forces. The sum of the heights on the right (5 units) equals The sum of the heights on the left (5 units) It is not important as to how high the two bars are in the above bar chart. If the bars were 4 units high instead of 5 units high, then it would be an equally acceptable bar chart. The decision about bar height is entirely arbitrary.

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The Bar Chart: Work with Energy

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Notice that in the initial situation the energy was gravitational potential energy. During the "interaction" negative work was done on the block by the frictional force. In the final situation, the remaining energy was "converted" in this particular problem to kinetic energy. Finally, we can represent the final energy as the sum of the initial energy and the work. Notice that the block will be moving at the "bottom" of the incline and that the distance above "ground" is essentially zero. Thus,

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Example: A Skidding Car Now that the analysis is complete, the bar chart can be constructed. The chart must be consistent with the above analysis. Observe that the bar for work is a downward bar. This is consistent with the fact that the work done by friction is negative work. Whenever negative work is done by external forces, the W ext bar will be a downward bar. Note also that the sum of the bar height on the left side (+5 plus -5) is the same as the sum of the bar heights on the right side of the chart. One final comment is in order: even though the height of all bars on the left equals the height of all bars on the right, energy is not conserved. The bar chart includes both energy and work on the left side of the chart. If work is done by external forces, then the only reason that the sum of the bar heights are equal on both sides is that the W ext makes up for the difference between the initial and final amounts of total mechanical energy.

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Example Energy of a Skier Examine the initial state of energy. Examine the final state of energy. Friction an air resistance have a negligible affect on this motion. Now that the analysis is complete, the bar charts can be constructed. The charts must be consistent with the above analysis.

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Check Your Understanding 1. A ball is dropped from rest from a tall bridge. As the ball falls through the air, it encounters a small amount of air resistance. The final state of the ball is the instant before it strikes the water.

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Check Your Understanding A volleyball player spikes the ball at just above net level and drives it over the net. The initial state is the ball just prior to the spike. The final state of the ball is the instant before it strikes the ground.

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Check Your Understanding A spring loaded projectile shooter is on a hill. The shooter is aiming across the valley at a tree house that is much higher than the hill. What would the energy bar chart application look like if you were to shoot the projectile into the tree house?

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Check Your Understanding In a physics lab, a Hot Wheels car starts at an elevated position, moves down an incline to the level ground, strikes a box and skids to a stop. Consider three states for the car: state A is the top of the incline; state B is the bottom of the incline before striking the box; state C is after the car has been brought to a stop. Use the diagram at the right and your understanding of the work-energy theorem to construct bar charts for the motion from A to B and from B to C.

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Are you taking a little break from work? Maybe you should just whistle? Here is a quick review do you understand these basic objectives yet? Define work and calculate the work done by a force. Use Hookes Law to determine the elastic force on an object. Calculate the power of a system. Calculate the kinetic energy of a moving object. Determine the gravitational potential energy of a system. Calculate a systems elastic potential energy. Apply conservation of energy to analyze energy transitions and transformations in a system. Analyze the relationship between work done on or by a system, and the energy gained or lost by that system.

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Work & Energy Additional Objectives Define the terms work and energy Determine the dot, or scalar, product of two vectors Define the terms kinetic energy and potential energy Identify the work-energy theorem and use it to solve problems

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Work & Energy: Work done by a constant Force

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Work is a Scalar Quantity

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Example Problem

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Use the net force rather than adding all values of work from each individual force

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Energy Energy Energy is the capacity to do work. There are several types of energy, however we will focus on mechanical energy Mechanical energy (E) comes in two forms: Kinetic Energy: due to motion of the object K = (½) mv 2 where: m = mass of the object v = velocity of the object

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Potential Energy: is mechanical energy that is stored in the object. It primarily is encountered in two forms: Gravitational Potential Energy: U g = mgy where: m = mass of the object y = height of the object over a certain reference level. (the choice of the reference level is arbitrary, so U is arbitrary)

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Spring Potential Energy: U spring = (½) kx 2 where: k = Spring constant (in N/m) x = Extension or compression in the spring.

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Power Power Power is the rate at which work is done. P = W/t The unit of power is Watt (W). 1 Watt = 1 Joule / 1 sec. A more commonly used unit of Power is horsepower: 1 horsepower (hp) = 746Watts

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Example Problem Power is Change in Velocity divided by Time Velocity initial = 10 m/s Velocity final = 20 m/s Position = 1000 Use kinematic equation Vf2-Vi2=2ax to solve for a Then Use Vf-Vi = at and solve for time.

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Climb the stairs as fast as you can! Calculations of Horsepower Lab Horsepower is a concept that is related to not only how much work you can do, but how fast? Hence Horsepower is the rate at which you can do work. Lets see if you can generate one horsepower of power output while climbing up a couple flights of stairs. What measurements you will need to get started. What is your weight in lbs?________________. What is the height of the stairs that you are running up?________________. What is the time it takes you to get to the top of the stairs?________________. Calculate your horsepower. mgh = work Work/Time = Power Use the Power in Watts to Horse Power Conversion to Complete Calculation What are your conclusions from your activities?

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Work - Energy Theorem Work - Energy Theorem When a system moves from an initial state to a final state, then the work done by all non-conservative forces (i.e. all forces other than springs & gravity) is equal to the change in Total Mechanical Energy (Kinetic + Potential) of the system: W non-conservative = E - E 0 where: W non-conservative = Non Conservative Work = F non- conservative d Cosq E = Total Final Energy = K + U (where U = U g + U sp, where applicable) E 0 = Total Initial Energy = K 0 + U 0 (where U 0 = U g0 + U sp0, where applicable)

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Energy Conservation Energy Conservation Where there is no non-conservative force acting on a system, the Total Mechanical Energy (Kinetic + Potential) of the system does not change. The Law of Conservation of Energy follows from the Work-Energy Theorem: If F non-conservative = 0, then W non-conservative = 0, So: W non-conservative = 0 = E - E 0 So: E = E 0 i.e. Energy does not change.

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Conservation of Mechanical Energy By the end of this part of the lesson you should be able to

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The Work-Energy Theorem An object which has energy has the ability to do work. The total amount of work that can be done is exactly equal to the energy available. This principle is called the work-energy theorem and applies to everything in the universe which we have been able to observe.

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Energy is conserved

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Work done by a conservative force

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Problem Solving Strategy

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Step 5: Calculate the Work done by non-conservative forces (eg. Friction), in going from one state to the next.

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Setting up a table of information State Vel. (v) Kinetic Energy (K) Vertical Position (y) Gravit. Pot. Energy (U g ) Spring Comp. (x) Spring Pot. Energy (U sp ) Total Energy (E) Work Done (W) 100 d Sin mg d Sin 00 - µ mg d Cos 2v2v2 ½ mv Steps 3, 4, 5 and 6 can be represented in the table as below:

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Step 7: Solve the Work-Energy equation set up in Step 6 to obtain the unknown quantity

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Try It On Your Own

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Block slides down a friction free hill and stops after reaching the bottom where friction exists.

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Atwood's Machine using Energy

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Interesting Web Site with additional references. In this image MIT Professor Dr. Walter Lewin Shows his beliefs in physics by standing in front of a 5 pound swinging pendulum. This is part of the MIT Open Courseware Project which included the Physics Course: Physics I: Classical Mechanics. mechanics-fall-1999/index.htm

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Work Done By a Variable Force, Conservative Forces And Potential Energy By the end of this part of the lesson you should be able to Determine the work done on an object by a variable force; Distinguish between a conservative force and a non-conservative force and be able to state the definition for each of these types of forces; Determine analytically whether a force is conservative or non-conservative; Determine the conservative force function that corresponds to a potential energy function.

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Graphical Expression of Work Calculate the Area under the curve. Notice the units will be N-m or Energy-Work Units

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How much work is done?

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Top View Sliding on Table

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The work done by a force is represented graphically by an area under the force vs. position curve. Can you calculate the work done by this force which moved the object some 14 meters? Just count the spaces below the curve. You can use the area of rectangles and triangles to do this. Every force in nature falls into one of two categories: Conservative and Non-conservative Conservative forces have potential energy functions associated with them. (Such as gravitational force)

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So now you have gone this far…Dont just set there. Use the force and continue on along your journey. A little work can get you far!

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