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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. Read Information on Slide

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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/s2. Read Information on Slide: If you could animate the gif so that the weights went up and down it would be pretty cool. At least you could show an arrow growing larger from the ground to the peak.

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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. Read Information on Slide

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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 2 + 2 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) mv2 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: Wnet = K. E. f − K.E. o , where K. E. f is the final kinetic energy and K. E. o is the original kinetic energy. SCHWINN: Read Information on Slide: Start a small object somewhere on the slide and have it move across. Maybe you could use the Bee and have him fly across the screen getting closer to you and then smaller.

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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. Read Information on Slide: In the figure you see a biker who has kinetic energy as he rides up to the top of the hill. When he reaches the top he is just barely moving, but he possesses a large amount of Potential Energy which can be converted back to Moving (Kinetic Energy) as he rolls back down the hill. At the bottom of the hill he will no longer have the potential energy but he will have converted it to Kinetic Energy…. Work will have to be done by something (Friction?) or Somebody ( the brakes on the bike being depressed) to cause the object to stop.

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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. Read Information on Slide: You could make this hot air balloon rise up along side of the text and have a meter which shows the Potential energy increasing as the balloon rises, or decreasing as the balloon descends. resources.yesican-science.ca/energy_flow/images/potential_energy2.png

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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) kx2. 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. Read Information on Slide: SNAP THE RUBBER BAND and send it sailing across the screen? This small car is powered by the energy of a wound up spring.

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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). SCHWINN: Read Information on Slide Can you have the power button glow if you push it?

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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. Read Information on Slide

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Pendulum Energy Read Information on Slide 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?**

This image shows you the fundamentals of power generation via a water flow spillway turbine system. This is an example of generating power from the potential energy stored behind a dam. Something, (Water) needs to be available with an energy potential to cause the rotation of the turbine blades. When the turbine blades are “forced” to spin then a rotor can also be made to turn and a generator can produce Electrical power from the transference of energy in an amount of time via this process.

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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. Read Information on Slide: You could use the animated gif you make from the first slide here as well.

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**Bar Chart Illustrations**

One tool which can be utilized to express an understanding of the work-energy theorem is a bar chart. A work-energy bar chart represents the amount of energy possessed by an object by means of a vertical bar. The length of the bar is representative of the amount of energy present, with a longer bar representing a greater amount of energy. In a work-energy bar chart, a bar is constructed for each form of energy. Consistent with the work-energy relationship discussed in your textbook. The sum of all forms of initial energy plus the work done on the object by external forces equals the sum of all forms of final energy. SCHWINN: What would really be amazing is if you would create a gif image that would actually work like a stereo equalizer where students could adjust each of these bar charts with a lever that would lift the bar up and down according to the type of energy that was associated in each of the problems. I added the 5 blue bars to give you a reference. KEi + PEi + Wext = KEf + PEf

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**KEi + PEi-grav + PEi-spring + Wext = KEf + PEf-grav + PEf-spring**

KEi + PEi + Wext = KEf + PEf 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. KEi + PEi-grav + PEi-spring + Wext = KEf + PEf-grav + PEf-spring In a work-energy bar chart, a bar is used to represent the amount of each term in the initially displayed equation. Consequently, the sum of the bar heights for the initial condition (initial energy + external work) must equal the sum of the bar heights for the final condition (final energy). Since the potential energy comes in two forms - the elastic potential energy stored in springs (PEspring) and the gravitational potential energy (PEgrav) - the previous equation is rewritten as

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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. The following procedure might be useful for constructing work-energy bar charts: 1. Analyze the initial and final states of the object in order to make decisions about the presence or absence of the different forms of energy 2. Analyze the forces acting upon the object during the motion to determine if external forces are doing work and whether the work (if present) is positive or negative. 3. Then construct bars on the chart to illustrate the presence and absence of the various forms of energy for the initial and final state of the object; 4. The exact height of the individual bars is not important; What is important is that the sum of the heights on the left of the chart is balanced by the sum of the heights on the right of the chart

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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. SCHWINN you can use the Key words in BlUE some how making them highlighted on the slide. Given this motion and the identification of the initial and final state of the ball, decisions can be made about the presence and absence of each form of energy. Since there is no motion at the top of the pillar, there is no initial kinetic energy. Since the ball is elevated above the ground while on top of the pillar, there is an initial gravitational potential energy (PEgrav). There are no springs involved; thus, there is neither initial nor final elastic potential energy (PEspring). In the final condition (just prior to striking the ground), the ball is moving. Thus, there is a final kinetic energy. And finally, the ball is no longer elevated above the ground so there is no final gravitational potential energy. The ball falls under the influence of gravity (an internal force) alone. Thus there are no external forces present nor doing work. The diagram at the right summarizes this analysis.

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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. To complete the bar chart, an arbitrarily decided height for each bar is decided upon and a bar is constructed for each form of energy. As mentioned before, it is not important exactly how high each bar is. It is only important that the sum of the bar heights on the left balance the sum of the bar heights on the right.

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**The Bar Chart: Work with Energy**

Consider a block sliding down an inclined surface in which the friction between the block and surface is very small, but cannot be ignored. Represent this statement with a work-energy bar chart. Then, we need to represent the initial energy on the bar chart. Since the block is not moving initially, it has no kinetic energy. But the block is above the "ground level" so it must have some initial gravitational potential energy, say about 4 units. NOTICE: There must be an interaction with an outside force (friction) how will we deal with that? It will be work from an outside source and it is work done against the system.

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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, 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. Now, we need to decide how to deal with the frictional (Outside force) interaction. When the block slides down the incline, the incline will hinder its motion due to the frictional force that it will exert on the block in the direction opposite to the motion of the block. This force will be applied during the distance that the block is actually moving. To give this interaction a "value", let's assume that it gives 1 unit of work to the block. Since the work due to the frictional force is in the - x-direction (negative), the work will be a negative 1 unit.

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**Example: A Skidding Car**

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 Wext 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 Wext makes up for the difference between the initial and final amounts of total mechanical energy. Read: Now we will repeat the process for a car which skids from a high speed to a stop across level ground with its brakes applied. The initial state is the car traveling at a high speed and the final state is the car at rest. Initially, the car has Kinetic Energy (since it is moving) but does not have gravitational potentioal energy (since the height is zero or elastic potential energy (since there are not any springs present). In the final state of the car, there is neither kinetic energy (since the car is at rest) nor potential energy (since there is no height nor springs). The force of friction between the tires of the skidding car and the road does work on the car. Friction is an external force. Friction does negative work since its direction is opposite the direction of the car's motion. Now that the analysis is complete, the bar chart can be constructed. The chart must be consistent with the above analysis.

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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. Read: Consider a skier that starts from rest on top of hill A and skis into the valley and back up onto hill B. The skier utilizes his poles to propel himself across the snow, thus doing work to change his total mechanical energy. The initial state is on top of hill A and the final state is on top of hill B. Suppose that friction and air resistance have a negligible affect on the motion. In the initial state, the skier has no kinetic energy (the skier is said to be at rest). There is no elastic potential energy in both the initial and the final states (since there are no springs). The skier has gravitational potential energy in both the initial and the final states (since the skier is at an elevated position). Finally, work is being done by external forces since the skier is said to be using "his poles to propel himself across the snow." This work is positive work since the force of the snow on his poles is in the same direction as his displacement. Observe that the bar for work is an upward bar. This is consistent with the fact that the work done by the poles is positive work. Whenever positive work is done by external forces, the Wext bar will be an upward bar. Note also that the sum of the bar height on the left side (+5 plus +2) is the same as the sum of the bar heights on the right side (+4 plus +3) of the chart. As mentioned earlier, the exact heights of the individual bars are not important. It is only important that the bars exist, that they are in the correct direction (upward) and that their sum on the left is the same as the sum on the right.

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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. Start with a blank bar chart and finally, Insert answer (THE BAR CHART is Correctly labeled).

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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. Start with a blank bar chart and finally, Insert answer: The bar chart is correctly labeled.

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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? Start with a blank bar chart and finally, Insert Answer: The bar chart is correctly labeled. You could design a shooter and a tree house if you want to.

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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. Insert Answer after placing two blank energy bar chart forms down first…

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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 Hooke’s 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 system’s 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. HERE IS A QUICK REVIEW: Work, energy and power are highly inter-related concepts that come up regularly in everyday life. You do work on an object when you move it. The rate at which you do the work is your power output. When you do work on an object, you transfer energy from one object to another. In this lesson you explore how energy is transferred and transformed, how doing work on an object changes its energy, and how quickly work can be done. SCHWINN: can you add a little Whistle here before you make a transition?

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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 We just reviewed what we have learned so far and now we are going to add some additional facts about Work and Energy. By the end of this half of the lesson, you should be able to: • 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**

SCHWINN: If you want to create a box and a hand that can pull the box across the bottom of the slide this would be great Just make sure that you have the red force arrow and the green d vector arrow showing as the box is being pulled across. Read Scripts: We have discussed that terms have specific meanings when they are used in science and engineering and that these meanings can be quite different than the everyday connotation we associate with the term. This is especially true of the term “Work”. In science and engineering, work is defined to be the application of a force through a distance. This definition of work has several subtleties. First, if a force is exerted on an object but that object does not have a component of displacement along the direction of that force, there is no work done on that object. Moreover, if a component of force is in the same direction as the displacement of the object, that component of force does positive work on the object; if a component of force is in the opposite direction as the displacement of the object, that component of force does negative work on the object. A Joule (J) is the unit of work. One Joule of work is performed when 1N of force is applied through 1meter. A more conventional unit of work, commonly used from a nutritional point of view is a Calorie. 1 Calorie = 4.18 Joules. This however should not be confused with the "Calories" indicated in nutritional labels. There: 1 Nutritional Calorie = 1 Kilo Calorie = 4180 Joules.

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**Work is a Scalar Quantity**

Work is a scalar quantity. Like the distance shown in the figure and the speed shown in the other graphic, these values only have a magnitude associated with them. NO DIRECTION IS REQUIRED! So, what does positive and negative work mean? If a force pushes in the same direction as an object is moving, that object will speed up. If a force pushes in the opposite direction as an object is moving, that object will slow down. Thus, when positive work is done on an object, it speeds up and when negative work is done on an object, it slows down. The amount of work, W, done by a force, F, in moving an object through a displacement, r is: W = F*r The unit of work is the unit of force multiplied by the unit of distance: The dot product is a way of multiplying two vectors to get a scalar result. Work is a scalar quantity. It has no direction associated with it!!

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Example Problem A 10.0 N force directed at 25° below the horizontal is used to move a 2.0 kg crate a distance of 4.0 m along a horizontal surface. If the coefficient of friction between the crate and the surface is 0.15, determine the work done by (i) the applied force, (ii) the gravitational force, (iii) the normal force, and (iv) the frictional force. What is the total amount of work done on this object? Schwinn: Draw a box and then a line with Rough Surface…. Add the arrow downwards at 25 below horizontal. Ask Students to draw the free body diagram for this picture…. Second Image is free body diagram Force Up = Normal Force Left is Friction Ff=Normal*Coefficient of Friction Force Right is Horizontal Component of Pulling force F= Fpull*Cos 25 Force Down is Weight Force down is also Vertical component of Pulling force F= Fpull*Sin 25 ANS Work done by (i) the applied force, DOWNWARD = 0 Forwards = 4 m x Force horizontal (ii) the gravitational force, IS DOWNWARD therefore Zero… Not moving in a y-direction (iii) the normal force, IS UPWARDS therefore Zero… Not Moving in a y-directon (iv) the frictional force. Work is Negative - What is the total amount of work done on this object?

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Example Problem A 3.0 kg box starts from rest at the top of a 3.5 m long incline that is elevated at 50°. The coefficient of friction between the box and the incline is How much work is done on the box by the gravitational force as the box slides from the top to the bottom of the incline? Schwinn: Draw a box and then a line with Rough Surface…. Add the arrow downwards along the horizontal surface of the incline plane. Ask Students to draw the free body diagram for this picture…. Suggest that they shift their x and y axis to be like the blue lines that are drawn to the right. Tell them to recall that work is related to a force that is parallel to the surface it is moving across. Therefore the weight is only contributing part of its value to “pull the object” a specific distance down the hill.

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

We can find the net amount of work that is done on an object by a group of forces, but what does this value tell us? Recall that Force times the distance where force and distance are parallel to each other gives the amount of work done by an individual force. Since we want to determine the net work done on the object, perhaps we should calculate that work using the net force acting on the object. So what is the net force on this object? It is the vector sum of the three forces. Note F1 shows 1 in the x direction and 5 in the Y. F2 shows -3 in the x direction and -2 in the Y. F3 shows 4 in the x direction and 0 in the Y. If we add these together we find the net force. The net force is 2 in the X direction and 3 in the Y direction. (See inserted arrow!) If you move the object 5 m to the along the path of the arrow you would need to multiply the 5 m times the total length of the force vector to get the work done by the net force. The vectors length is actually the square route of 13 because of the Pythagorean theorem.

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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) mv2 where: m = mass of the object v = velocity of the object SCHWINN…. Have a car move across the slide when you add the Kinetic energy statement. You could even include the sound of a race car squealing its wheels if you have one. 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) mv2 where: m = mass of the object v = velocity of the object Work is defined to be the application of a force through a distance energy is defined to be the ability to perform work. There are two types of energy that we will focus on initially: Kinetic Energy, Energy of Motion and Potential energy, Energy of position That is , by putting an object in motions (like a heavy truck) you give it the ability to apply a force though a distance (ouch) Likewise, if you were to pick that heavy truck straight up, you could give it the ability to apply a force through a distance by simply changing its location. We will discuss potential energy in greater detail in the next lesson. • Returning to our previous result, we see that a net amount of work done on an object results in a change in the speed of that object. Also, if we define the quantity (m v2)/2 as the Kinetic Energy 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: Ug = 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) SCHWINN…. Have a ball fall down across the slide when you add the Potential energy statement. You could even include the sound of the ball bouncing and it could bounce up and down a couple of times never returning to the same height but lower after each bounce. Potential Energy: is mechanical energy that is stored in the object. It primarily is encountered in two forms: Gravitational Potential Energy: Ug = 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: Uspring = (½) kx2 where: k = Spring constant (in N/m) x = Extension or compression in the spring. SCHWINN: Do you have a spring sound you could add when you squeeze down on this picture? You can make each of these pictures so that they follow the concept of a spring starting at equilibrium, being compressed by a force and finally being stretched by a force. You could make the image oscillate back and forth with a meter showing maximum potential energy when either compressed or expanded and minimum potential energy when at equilibrium.

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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 During each month you use energy to cook and light the place where you live. At the end of the month, you get a power bill. So, it should not be surprising that energy and power are related. Your text defines power as the rate at which work is done. This is a good definition for our needs now, but perhaps a better definition for power would be the rate at which energy is transferred from one system to another. Power Power is the rate at which work is done. If we use P to denote power: 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 The unit of power is then the unit of energy divided by the unit of time: • Unit of Power = (Joule)/s = kg m2/s3 = Watt • The symbol for Watt is W. The Horse power unit was historically derived from the amount of work that an average draft horse could do in a single day.

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Example Problem A 1000 kg car accelerates from 10 m/s to 20 m/s over a distance of 85 m. What is the minimum average power supplied by the engine that would accomplish this? 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. Finally use the Power = Velocity / Time equation and solve. 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? Just as energy is a scalar, so is power. A positive power means that the system is gaining energy; a negative power indicates that a system is losing energy. Let’s take a field trip to the stairwell- Try to generate a horse power. Can you do it? MAKE THE CALCULATION ON YOUR OWN. In the many years that I have been teaching most students are not capable of actually generating much more than 1 horse power. They either aren’t fast enough or large (mass) enough. Its not just about one variable its about how fast, how much mass, and how high you are moving the mass.

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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: Wnon-conservative = E - E0 where: Wnon-conservative = Non Conservative Work = Fnon-conservative d Cosq E = Total Final Energy = K + U (where U = Ug + Usp, where applicable) E0 = Total Initial Energy = K0 + U0 (where U0 = Ug0 + Usp0, where applicable) You NEED TO READ these letters as W work done by a non conservative force equals Energy minus Energy initial (subscripted with the “Not” sign) U is potential Energy and g is gravity, while sp is spring based. 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: Wnon-conservative = E - E0 where: Wnon-conservative = Non Conservative Work = Fnon-conservative d Cosq E = Total Final Energy = K + U (where U = Ug + Usp, where applicable) E0 = Total Initial Energy = K0 + U0 (where U0 = Ug0 + Usp0, 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 Fnon-conservative = 0, then Wnon-conservative = 0, So: Wnon-conservative = 0 = E - E0 So: E = E0 i.e. Energy does not change. 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 Fnon-conservative = 0, then Wnon-conservative = 0, So: Wnon-conservative = 0 = E - E0 So: E = E0 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 Define what is meant by the mechanical energy of an object. • State, in words, the principle of conservation of mechanical energy and the principle of conservation of total energy. • Solve problems involving energy conservation in a systematic way.

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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. Recall the Work-Energy Theorem: Wnet = change in KE (READ work net is equal to the change in kinetic energy) Since forces fall into one of two categories, the net, or total, work done on an object can be broken into two parts: Wnet = Wcons + Wnon-cons (READ work net is equal to the Work done by a conservative force and the Work done by a non conservative force) Where WCONS represents work done by conservative forces acting on the system and WNC represents the work done by non-conservative forces acting on the system. However, we can express the work done on the object by conservative forces in terms of the change in potential energy of the object: Wcons = - change in potential energy

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Energy is conserved If we define the sum of an object’s Potential and Kinetic energies to be the Mechanical Energy of the object: Total Energy = Kinetic Energy + Potential Energy We can rewrite our expression in terms of the mechanical energy of the object Wnc = Energy final – energy initial = change in energy That is, work done by non-conservative forces results in a change in the mechanical energy of a system. This is the generalized work-energy theorem. If WNC = 0 J, we get the mathematical form of the statement of conservation of mechanical energy: If Wnc=0, and change in energy = 0 then Energy final = Energy Initial Or: If all of the forces acting on a system are conservative, then the mechanical energy of the system is conserved (that is, remains constant).

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**Work done by a conservative force**

We now see what is meant by the term conservative force: A conservative force is one which does not change the mechanical energy of an object when the object is moved through a closed path. A conservative force is a force which conserves mechanical energy. It doesn’t matter what path you take to get up to a certain height, a force applied against gravity is conserved. However, if you have to slide an object along the floor you are doing work against a non conservative force such as the frictional force of the floor. This force is a path dependent force and is known as a non conservative force.

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**Problem Solving Strategy**

You should be able to solve Energy problems if you have a scenario like the one described below: Given: A system with all but one of the following: Initial velocity (v0), Initial vertical position (y0), Initial spring compression/extension (x0) Final velocity (v), Final vertical position (y), Final spring compression/extension (x), Frictional coefficient (µ), distance traveled (d), Any other external force (Fext). Find: The unknown variable from the list above. Example: A block of mass 'm', starting from rest, slides a distance 'd' down an incline plane of angle 'q', and compresses a spring of spring constant 'k'. Find.. (a) the speed of the block just before it strikes the spring, (b) the compression in the spring.

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**Problem Solving Strategy**

Problem Solving Procedure Step 1: On the diagram indicate various states that the system can be in, and label them 1, 2. Step 2: Pick a reference level for y. This is the vertical level at which y = 0. Typically, it is best to pick the lowest level in the system, if it is known. Step 3: In each state label the following: Velocity (v), Vertical Position (y) with respect to the reference, Compression (x), in the spring if any. [Note: x is NOT the horizontal position of the object]

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**Problem Solving Strategy**

Step 4: Calculate the various forms of mechanical energy in each state: Kinetic Energy (K = ½ mv2), Gravitational Potential Energy (Ug = mgy) Spring Potential Energy, if any (Usp = ½ kx2) [Note: x is NOT the horizontal position of the object]

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**Step 5: Calculate the Work done by non-conservative forces (eg**

Step 5: Calculate the Work done by non-conservative forces (eg. Friction), in going from one state to the next. Step 5: Calculate the Work done by non-conservative forces (eg. Friction), in going from one state to the next. SCHWINN: You can read the information on the slides and you can use these figures or make your own which ever works best for you.

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**Setting up a table of information**

Steps 3, 4, 5 and 6 can be represented in the table as below: State Vel. (v) Kinetic Energy (K) Vertical Position (y) Gravit. Pot. Energy (Ug) Spring Comp. (x) Spring Pot. Energy (Usp) Total Energy (E) Work Done (W) 1 d Sinq mg d Sinq -µ mg d Cosq 2 v2 ½ mv22 --- Step 6: Set up the Work - Energy Theorem between States 'a' and 'b' Wab = Eb - Ea Wab = (Kb + Ugb + Uspb) - (Ka + Uga + Uspa) Example: W12 = (K2 + Ug2 + Usp2) - (K1 + Ug1 + Usp1) So: -µ mg d Cosq = (½ mv22 + mgy2 + ½ kx22) - (½ mv12 + mgy1 + ½ kx12) -µ mg d Cosq = (½ mv22 )- (0 + mg d Sinq + 0)

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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 A 5.0 kg object is launched from the top of a 35 m tall building at 10 m/s directed at 30° above the horizontal. Assuming air resistance can be ignored, how fast is the object traveling when it strikes the level ground below? (Work this problem using energy methods.

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

SCHWINN: you can animate this as a picture similar to the one I have drawn and you can make the block slide down the hill and move across the bottom of the page. It would be really neat if the block started as a color blue and slowly the bottom of the block became red once it reached the ground because it was sliding down the hill without friction and then at the base it starts to interact with friction and gaining some temperature. A 2.0 kg block starts from the top of a 7.5 m long frictionless ramp which is elevated at 25°. At the bottom of the incline, the block slides onto a horizontal surface where the coefficient of friction between the block and the surface is How far from the bottom of the incline does the block slide before coming to rest?

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**Atwood's Machine using Energy**

A 4.0 kg block is connected to a 7.0 kg block by a massless string. The string is passed over a pulley so that both of the blocks are hanging down, as in an Atwood’s machine. If the 7.0 kg block is given an downward push so that it is initially moving at 2.0 m/s, how fast is each block moving when the 7.0 kg object reaches a point that is 0.85 m below where it started? Bar charts would be useful especially since you have to be aware of the kinetic and potential energy’s of each block in the problem. You have a series of examples similar to this one in your text book and you can also look this problem or a similar one up on the internet too.

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**Interesting Web Site with additional references.**

In this image MIT Professor Dr. Walter Lewin Shows his belief’s 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. The sum of the energy of position and energy of motion in a system is called the mechanical energy of the system. If all of the forces acting on a system are conservative, the mechanical energy of that system remains constant. WALTER LEWIN IMAGE

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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. 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**

Graphically, the work done by a force on an object as the object moves between two locations is the area under the curve between the two positions on a plot of the force versus position for the object. To see this, suppose that a constant force, Fo, acts on an object as it moves between x = xo and x = xf. If you plot the force acting on the object as a function of position: Remember that area below the x-axis is considered as negative area and area above the x-axis is considered positive area. Calculate the Area under the curve. Notice the units will be N-m or Energy-Work Units

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**A 2. 0 kg block sets on a horizontal, frictionless table**

A 2.0 kg block sets on a horizontal, frictionless table. It is acted upon by a single force of 3 Newtons. If the block moves only along the x-axis and has a speed of 7.0 m/s at x = 7.0 m, find the speed of the object at x = 12.0 m.

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How much work is done? A 5.0 kg object is lifted 3 m, carried horizontally from left to right a distance of 5 m, lowered 3 m, and returned to its original location by moving it 5 m from right to left. How much work is done by the gravitational force on the object as it is moved through this round trip path? Remember you can do work against gravity and gravity can do work on the object. One is positive and the other is negative work. How much work was done by gravity when you move horizontally perpendicularly to the pull of gravity? (ZERO is the ANSWER HERE!) SO make sure you answer the question that was asked…. How much work is done by the gravitational force on the object as it is moved through this round trip path? Mass times gravity times height… approximately 150 joules! The rest of the motion is either not parallel to the direction of the force or is work done by the lifting and not by gravity!

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**Top View Sliding on Table**

A 5.0 kg object sets on a horizontal table. The coefficient of friction between the object and the table is The object is moved from • (0 m, 0 m) to (0 m, 3 m) to (5 m, 3 m) to • (5 m, 0 m) and then returned to (0 m, 0 m). The x-y plane for this problem lies in the plane of the table. How much work is done by the frictional force on the object as it moves through this round trip path? THIS IS A PATH DEPENDENT PROBLEM: YOU WILL NEED TO KNOW THE FORCE AND THE ENTIRE DISTANCE TRAVELED. NOT THE DISPLACEMENT AND NOT JUST A MOTION DOWN LIKE THE PREVIOUS PROBLEM. DO YOU SEE THE SIMILARITIES AND THE DIFFERENCES. Friction is a Non Conservative Force while Gravity was a conservative force.

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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) The work done by a force is represented graphically by an area under the force vs. position curve. 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…Don’t just set there**

So now you have gone this far…Don’t just set there. Use the force and continue on along your journey. A little work can get you far!

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Work and Energy. October 2012.

Work and Energy. October 2012.

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