08 Work And Energy

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WORK ENERGY AND POWER WORK In physics, a force works if it is acting upon an object to cause a displacement. There are three key words in this definition - force, displacement, and cause. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. There are several good examples of work which can be observed in everyday life - a horse pulling a plough through the fields, a father pushing a grocery cart down the aisle of a grocery store, a weightlifter lifting a barbell above her head, etc. In each case described here there is a force exerted upon an object to cause that object to be displaced. But if a teacher applies a force to a wall and becomes exhausted there will be no work at all because there is no displacement. Mathematically, work can be expressed by the following equation. Work = F . d . cos θ where F = force, d = displacement, and the angle (theta) is defined as the angle between the force and the displacement vector The joule is the unit of work. 1 joule = 1 newton . 1 metre (cosines have no units) 1J = 1 N. m In fact, any unit of force times any unit of displacement is equivalent to a unit of work. Calculating the Amount of Work Done by Forces Work is calculated as force . displacement . cosine(theta) where theta is the angle between the force and the displacement vectors. Apply the work equation to determine the amount of work done by the applied force in each of the situations described below.

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Free-Body Diagram

Free-Body Diagram

Free-Body Diagram

A 10-N force pushes a block along a A 10-N force pushes a A 10-N frictional force slows a block frictional surface at constant speed for block along a frictionless to a stop after a displacement of 5.0 m a displacement of 5.0 m to the right surface for a displacement to the right. of 5.0 m to the right.

ENERGY Kinetic Energy Suppose a force (F) pushes an object along a displacement (d). For the sake of simplicity, let us suppose the object’s movement previous to the push and the push itself have the same direction. Then the angle θ is zero and cos θ = 1. So Work = F . d . 1. Because of the push the object will be accelerated: F (a characteristic of the interaction) according to Newton’s 2nd law can be replaced by m . a (both m and a are characteristics of the object being pushed, namely its mass and acceleration). But the displacement can also be expressed in terms of the object’s parameters and the time interval through which the body was pushed: F= m. a

d = ½ (v0 + vf).t

And as a = (vf – v0) / t

Work = F . d = m . a ½ (v0 + vf).t Work = m . (vf – v0) / t . ½ (v0 + vf).t

Cancelling t and working out the equation (use your maths), it will be simplified to Work = ½ m (vf2 – v02) = ½ m vf2 - ½ m v02 = ∆ ½ (mv2) The work that a force makes on an object is equal to the change in a property of the object before and after the interaction. That property (that can be calculated as half the mass times the square of the velocity) is called the Kinetic Energy KE of the object. If an object suffers no interactions its kinetic energy remains constant. But when a force acts on it, KE changes and this change is equal to the work made by the force to the object. The word ENERGY means “work inside”. We say that an object or a system has energy when it can work on other objects or systems. If an object stands at rest it can do nothing at

3 all on other objects. But if it is moving because we gave it kinetic energy now it can act on other objects. Kinetic energy is the energy of motion. An object which has motion, whether it is vertical or horizontal motion, has kinetic energy. Energy is a scalar quantity; it does not have a direction. Unlike velocity, acceleration, force, and momentum, the energy of an object is completely described by magnitude alone. Like work, the standard metric unit of measure for kinetic energy is the joule. As might be implied by the above equation, 1 joule is equivalent to 1 kg.(m/s) 2. Conservative and non conservative forces There are certain types of forces which, when present and when involved in doing work on objects, will change irreversibly the KE of an object. And there are other types of forces which can change the KE of objects but reversibly, that is, they take away the KE keeping it as if it were “stored” and eventually giving it back to the object. Think of a stone thrown upwards: it goes up but looses KE (goes more and more slowly) because there is a force working negatively on it. This force is the gravitational pull of the Earth (neglect friction). At a point the stone will have zero KE and will start falling down once again. Now the Earth’s pull accelerates the stone making a positive work on it. We have already seen in kinematics that the speed and hence the KE of the stone will be exactly the same as the beginning. We call this type of forces that take and give KE from objects conservative forces. On the other hand consider a book sliding on a table. It will stop after one or two seconds. This book had KE at the beginning and now it has lost it. Friction forces have worked on it taking away its KE. Will friction forces push the book back again and restore it its initial KE? Of course not! Kinetic energy has “disappeared” and no energy seems to be stored anywhere. Forces behaving like friction are called non conservative forces. When work is done upon an object by a non conservative force, the total kinetic energy of that object is changed irreversibly. If the work is "positive work", then the object will gain energy (a push or a kick). If the work is "negative work", then the object will lose energy (friction). Gravitational Potential Energy Now consider the inclined plane of the figure below and the block on it. There are three forces acting but we will just focus on the weight Fg. If the block moves (no matter why) from A to B then from B to C and from C back to A, the work done by Fg can be calculated for the three different stages: Work (A to B) = Fg x AB x cos 60º = Fg x CA Work (B to C) = 0 (why?) Work (C to A) = Fg x CA x cos 180º = Fg x CA x (-1) = -Fg x CA

4 If we add up the three partial works we will get ZERO! No matter the angle the result will be the same. This is the essential characteristic of conservative forces: their work along a closed path equals zero. Notice that no matter the angle of the incline or whether we go back and forth, as long as we go back to the starting point the work will be always zero. This can be shown with vector calculus but the necessary maths is beyond the secondary school syllabuses. As a corollary, the work that a conservative force does when moving from one point to another will depend just on this two points and not on the path followed (were this not true, then we would be able to choose different paths from, say A to B, and back from B to A so to make the total work different from zero.) In the case of Fg, the work when moving from C to A (and to all other points at the same level) can be easily calculated as Fg x AC. The general equation for calculating the work that Fg does on any object between any two points will be Work (gravitational) = m . g . h where m and g have the usual meaning and h is the difference in height (vertical distance) to any referential plane. This is the (negative) work that “eats up” the KE of an object projected upwards and the (positive) work that speeds it up again when it falls down. This is also the stored or Potential energy (gravitational) we have talked about. So Potential energy (gravitational) = (PEg) = m . g . h The other two forces (a contact force and friction) behave in a different way. The normal contact force will never work because it is always at right angles with the displacement and friction’s work will be always negative because it forms an angle of 180º with the direction of the motion at any point (remember that friction always opposes motion). In this case no Potential energy can be defined as there is no storing and restoring ability for this force. Potential energy refers to forces and positions; kinetic energy refers to masses and movement. So an object can store energy as the result of its position. For example, the heavy ram of a pile driver is storing energy when it is held at an elevated position. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not stretched), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy (PE) is the stored energy of position possessed by an object.

5 The two examples above illustrate two forms of Potential Energy - gravitational PE and elastic PE. Gravitational PE is the energy stored in an object as the result of its vertical position (i.e., height). The energy is stored as the result of the gravitational attraction of the Earth for the object. As the gravitational force (weight) is mass dependent so is gravitational PE. To determine the gravitational potential energy of an object, a zero height position must first be arbitrarily assigned. Typically, the ground is considered to be a position of zero height. But this is merely an arbitrarily assigned position which most people agree upon. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position; again this is merely arbitrary. If the tabletop is the zero position, then the potential energy of an object is based upon its height relative to the tabletop. For example, a pendulum bob swinging to and from above the table top has a potential energy which can be measured based on its height above the tabletop. By measuring the mass of the bob and the height of the bob above the tabletop, the potential energy of the bob can be determined. Elastic Potential Energy The second form of potential energy which we will discuss is elastic potential energy. Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device the more stretch, the more stored energy. Springs can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more the compression there is, the stronger force which is required to compress it further. The amount of force is directly proportional to the amount of stretch or compression (x). We have called them the extension of the spring (∆L) and the spring constant (k) when we studied Hooke’s law: F (load) = k . x (extension) If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. Elastic forces are restoring forces: they draw back the spring to its original position. If the

6 spring is stretched (pulled rightwards), the force points leftwards and if it is compressed (pushed leftwards) the force will point upwards. A stretched spring set free, will make a positive work “producing” kinetic energy, until it passes through the equilibrium position. Then it will make a negative work slowing down as it is compressed and changing KE into PE once again. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. We can calculate the stored energy in a spring is we know the work that has been made on it. The work as usual would be Work = force . displacement = F . x. But the force in this case is not always the same: it increases steadily from the initial position (x = 0, F= 0) to the final value (F = k . x). Hence the average force will be ½ k . x If we replace this value into the work equation and remember that this work is stored as elastic PE Elastic Potential Energy = (PEe) = ½ k.x2 To summarise, potential energy is the energy which an object has stored due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height position. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position. Total Mechanical Energy When work is done upon an object by a conservative force the total mechanical energy (KE + PE) of that object remains constant. In such cases, the object's energy changes form. For example, as an object is "forced" from a high elevation to a lower elevation by gravity, some of the potential energy of that object is transformed into kinetic energy. Yet, the sum of the kinetic and potential energies remains constant. This is referred to as energy conservation and will be discussed in detail. When the only forces doing work are conservative forces, energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical energy is conserved. Mechanical energy is the energy which is possessed by an object due to its motion or its stored energy of position. Mechanical energy can be either kinetic energy (energy of motion), potential energy (stored energy of position) or both. Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position (for example, a brick held at a vertical position above the ground or zero height position). A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its

7 high speed (kinetic energy) and its vertical position above the ground (gravitational potential energy). A book at rest on the top shelf of a locker possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A drawn bow possesses mechanical energy due to its stretched position (elastic potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy (abbreviated TME). TME = PE + KE As discussed earlier, there are two forms of potential energy discussed in our course gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten: TME = PEgrav + PEspring + KE Non conservative forces If only conservative forces are doing work, there is no change in total mechanical energy; the total mechanical energy is said to be "conserved." But when work is done by non conservative forces, the total mechanical energy of the object is altered. The work that is done can be "positive work" or "negative work" depending on whether the force doing the work is directed opposite the object's displacement or in the same direction as the object's displacement. If the force and the displacement are in the same direction, then "positive work" is done on the object; the object subsequently gains mechanical energy. If the force and the displacement are in the opposite direction, then "negative work" is done on the object; the object subsequently loses mechanical energy. The quantitative relationship between work and mechanical energy is expressed by the following equation: TMEi + Wext = TMEf The equation states that the initial amount of total mechanical energy (TME i) plus the work done by external forces (Wext) is equal to the final amount of total mechanical energy (TMEf). A few notes should be made about the above equation. First, the mechanical energy can be either potential energy (in which case it could be due to springs or gravity) kinetic energy or both. Given this fact, the above equation can be rewritten as KEi + PEi + Wext = KEf + PEf The previous equation is sometimes called the “work-energy theorem”.

8 ANALYSIS OF DIFFERENT SITUATIONS The quantitative relationship between work and the forms of mechanical energy is expressed by the following equation: KEi + PEig + PEis + Wext = KEf + PEfg + PEfs This equation will lead us through the analysis of some systems in which conservative andor non conservative forces act A Conservative System: The pendulum Consider a pendulum bob swinging to and fro on the end of a string. There are only two forces acting upon the pendulum bob. Gravity (an internal force) acts downward and the tensional force (non conservative force) pulls upwards towards the pivot point. This force does not do work since at all times it is directed at a 90degree angle to the motion. As the pendulum bob swings to and fro, its height above the table top (and in turn its speed) is constantly changing. As the height decreases, potential energy is lost; and simultaneously the kinetic energy is gained. Yet, at all times, the sum of the potential and kinetic energies of the bob remains constant. The total mechanical energy is 10 J. There is no loss or gain of mechanical energy; only a transformation from kinetic energy to potential energy (and vice versa). This is depicted in the diagram below.

As the 2.0-kg pendulum bob in the above diagram swings to and fro, its height and speed change. Use energy equations and the above data to determine the blanks in the above diagram. A Second Conservative System: The “Ideal” Roller Coaster A roller coaster operates on this same principle of energy transformation. Work is initially done on a roller coaster car to lift to its initial summit. Once lifted to the top of the summit, the roller coaster car has a large quantity of potential energy and virtually no kinetic energy (the car is almost at rest). If it can be assumed that no non conservative forces are doing work upon the car as it travels from the initial summit to the end of the track (where finally

9 an external braking system is employed), then the total mechanical energy of the roller coaster car is conserved. Conservation of energy on a roller coaster ride means that the total amount of mechanical energy is the same at every location along the track. As the car descends hills and loops, its potential energy is transformed into kinetic energy (as the car speeds up); as the car ascends hills and loops, its kinetic energy is transformed into potential energy (as the car slows down). The amount of kinetic energy and the amount of potential energy is constantly changing; yet the sum of the kinetic and potential energies is everywhere the same. This is illustrated below - the total mechanical energy of the roller coaster car is 40 000 Joules.

In this situation the air resistance and friction against the rails are assumed negligible (indeed, an idealized situation) and since the normal force acts at right angles to the motion at all times, it does not do work. The only force doing work on the roller coaster car is gravity; and since the force of gravity is conservative, the total mechanical energy is conserved. A Non Conservative System: Pulling up a car Now we will repeat the process for a car which skids from a high speed to a stop across a horizontal path with its brakes applied. The initial state is the car travelling 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 potential energy (since the height is zero) or elastic potential energy (since there are no springs). 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 or springs). The force of friction between the tires of the skidding car and the road does work on the car. Friction is a non conservative force and does negative work since its direction is opposite the direction of the car's motion.

Peeping Into Further Courses There is a relationship between work and mechanical energy change. If only conservative forces are doing work (no work done by non conservative forces), there is no change in total mechanical energy; the total mechanical energy is said to be "conserved." Whenever

10 work is done upon an object by a non conservative force, there will be a change in the total mechanical energy of the object. When a non conservative force is acting on an object (e.g. a push) the agent pushing the object experiences according to Newton’s 3rd law, an equal and opposite force. So if it is doing a positive work it will receive an equal amount of negative work! If it is giving energy it is loosing it at the same rate. Then one system is transferring energy to another one. Energy has not been created; it just changed from one system to another. One system acts as a source of energy to the second one. We could keep on analysing energy transfers between different systems. In the case of a car crashing against a wall, the wall sweeps out the energy of the car but this energy does not disappear: maybe mechanical energy does, but you will learn in further courses that mechanical energy has been either distributed randomly among the particles or molecules forming the interacting objects (this randomised mechanical energy is called thermal energy), or dissipated as a wave to the surroundings (sound energy) or used to change the shape of the crashing objects overcoming electric forces among their particles. POWER Work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity which has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber. Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation. Power = Work / time The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts. Most machines are designed and built to do work on objects. 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. Thus, the power of a machine is the work/time ratio for that

11 particular machine. A car engine is an example of a machine which is given a power rating. The power rating relates to how rapidly the car’s motor can accelerate the car. Suppose that Ben elevates his 80-kg body up the 2.0 meter stairwell in 1.8 seconds. If this were the case, then we could calculate Ben's power rating. It can be assumed that Ben must apply an 800-Newton downward force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben's body with just enough force to lift his body up the stairs. It can also be assumed that the angle between the force of the stairs on Ben and Ben's displacement is 0 degrees. With these two approximations, Ben's power rating could be determined as shown below. Ben's power rating is 889 Watts (about 1,5 HP) The expression for power is work/time. Now since the expression for work is force. displacement, the expression for power can be rewritten as (force . displacement)/time. Yet since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force. velocity. This is shown below.

This new expression for power reveals that a powerful machine is both strong (big force) and fast (big velocity). The powerful car engine is strong and fast. The powerful farm equipment is strong and fast.

PROBLEMS ON WORK ENERGY AND POWER A ball is thrown upwards with a speed of 20 m/s. Using work – energy concepts: calculate how high will it go. calculate its height when its speed is 3 m/s. 1-

A mass of 5 Kg changes its velocity from 4 m/s to 24 m/s in 10 s. Calculate: a- the force needed for this acceleration b- the kinetic energy at start c- the kinetic energy after 2 s d- the work made by friction forces to stop it after the driving force ceased (at 10 s) e- the power delivered by the driving force

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Two blocks at rest interact with each other. One of the blocks has a mass of 3 kg and at the end of the interaction it moves at 2 ms-1. If the mass of the other block is 2kg:

12 what will be its kinetic energy at the end of the interaction? Draw a diagram showing the situation. 3-

A trolley in a roller coaster falls down a hill and gets to the base travelling at 15 ms-1. If it wasted 30 % of its energy at the top of the hill because of friction, calculate the height of the hill.

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If the mass of the trolley was 500 kg calculate the work done by the motor that moved it up. The motor’s efficiency was just 50 %. If it took 60 s to drive the trolley up to the top, calculate the power used.

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Suppose the other 50 % energy used by the motor in point (3) was lost as heat to the surroundings. If 1cal = 4,18 J calculate the energy loss in calories.

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A car of mass 1.000 kg travels at 30 ms-1 and slows down to 10 ms-1 along an 80 m horizontal track a- calculate the energy change b- how much work have the brakes done on it? c- calculate the (average) stopping force

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When 100 g of petrol (gasoline) are burnt about 1200 kcal are produced. a- How many kJ are 1200 kcal? (1kJ = 0,24 kcal) b- If the efficiency of a motor in a car is just 25% calculate how much energy will be used to keep the car moving. c- The speed of the car is constant despite it is spending energy. Explain why. d- What happens to the rest of the energy in the fuel?

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A stone at rest (mass =2 Kg) is pulled along a horizontal path by a 10 N force along 0.3 m: what is the work done by the force? What is the change in the stone’s kinetic energy if the pull is a horizontal pull? What is its velocity? (Assume no friction force).

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If the stone of problem (9) were pulled vertically, its weight will be working too! Find this work and the change in kinetic energy in this case.

10- Calculate the work done by the gravitational pull on a 3 Kg stone thrown upwards with an initial velocity of 10 m/s as it moves from the floor up to its maximum height and as it falls back to the floor. Plot a graph work against distance. 11- Calculate the potential and kinetic energies [ Pe, Ke ]for the stone of problem (11) at the following heights: 0m, 1m, 2m, 2.5m, 4m and at maximum height. Calculate the mechanical energy [E(m)] at each point and plot the three of them E(k), E(p) and E(m) as a function of distance to the floor. Explain your results. 12- On the “FREE FALL” at Showcenter you fall freely from a 20m high tower down to a height of 7m. Considering there is no friction force, calculate your speed at that point.

13 How much energy do you loose while to a velocity of 1m/s? (suppose your mass is about 50 Kg). If the braking distance is about 2m find the force acting on you. 13- Calculate the average friction force for a sleigh sliding down a mountain if it starts at rest, falls from a 30m high slope and when it gets to the mountain’s base its speed is just 15 m / s. The length of the path is 200m. 14- Back again at Showcenter you go to the “ROLLER COASTER”. If you fall from 10m high find your speed at the base. If you suppose that friction sweeps away some 15% of the energy, re-calculate your speed at the base. If climbing up again from the base another 15% of the remaining energy is lost: will the trolley be able to climb up a slope 6.5m high?