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Mechanical Energy Give me a point to lean on and I’ll lift the world up ~ Archimedes (287-212 B.C.) CHAPTER 2 Mechanical energy, in the form of muscle power, was the first energy source that humans utilized. Hunters used muscle power to chase and wrestle animals or escape them when in danger. The use of tools for hunting and agriculture opened a new era in human development. The bow and arrow was used in hunting and in warfare, and wind was harnessed to navigate sailboats along rivers and on seas. As animals became domesticated, humans learned to exploit animal power - first oxen, then mules and horses - for pulling their plows and carts. The great civilizations of China, Egypt, Persia, Greece, and Rome relied on forced labor and animals to plow their farms; at the same time, their soldiers fought for the acquisition of more land and slaves. As the empires crumbled and slaves became scarce, the work of humans and animals was increasingly supplemented by power from watermills and windmills. Although the invention of windmills and watermills can be traced back to ancient Persians, Greeks and Romans, their use was limited primarily to grinding grains. They became widespread only at the end of the first millennium when Europeans used mills to operate such tools as saws and looms and to pump water from wells. The work from muscles, the work performed by windmills and watermills, and the work done by simple tools are all examples of mechanical energy. In this chapter we will define mechanical energy and how it manifests itself in the form of kinetic and potential energies. Hydro and wind energies will be discussed in Chapters 3 and 4, respectively. Work and Power In Chapter 1, we defined energy as the capacity to do work. This definition implies that work and energy are interrelated -- one cannot be defined without the other. Let us first introduce the more tangible concept - work - and then define energy in terms of it. Work Intuitively, work is the change in an object’s state of motion that results from the application of a force; no work is performed unless the object is displaced in the same direction as the applied force. Sideway displacements do not contribute, so no work can be done by a force that is perpendicular to the displacement. When force is at an angle, only the component of force in the direction of motion must be considered. To move an object twice the distance, with the same amount of force, twice the work must be done. Likewise, if it requires twice the effort to move an object the same distance, the work performed will still be two times as much. It is therefore understandable to define work as the product of the force (F) times the distance (d). W=F.d (2-1) W hen force is at an angle, only the component of force in the direction of motion must be considered. In SI units, force is given in newtons (N) and distance in meters (m). The work is therefore given in newton-meters (N.m) or joules ( J). Question: Which of the following represents the performance of work? a. An apple falls off an apple tree b. A horse pulls a carriage c. A balloon ruptures and air rushes out d. A woman carries a basket of fruits over her head e. A boy pushes against a wall until exhausted Answer: In each of the first three instances, a force has caused a movement in the direction of the force. In case a, the force of gravity causes the apple to fall. In case b, the horse applies a force in the direction of motion. In case c, the higher pressure in the balloon forces air out of the balloon. Instances d and e, however, do not constitute work. The force that the woman applies to the basket is upward and perpendicular to the direction of her motion. Similarly, when the boy pushes on a wall, there is no displacement and he performs no work. This might be quite confusing, as in both instances the person can become very tired. In fact, the body does continuous work to pump the blood to various organs of the body, to expand and contract various muscles in hands and feet, and force air in and out of our lungs; all of these actions serve to maintain the normal body functions. Although no mechanical work was performed in cases d and e, both the woman and boy use up a lot of chemical energy via metabolism. Question: A boy is moving a heavy box along a hallway. If the boy wants to move the box with the smallest amount of effort (putting in the least amount of work), which approach should he choose? a. Pull the box with a rope parallel to the floor. b. Push the box parallel to the floor. c. Pull the box with a rope at an angle. d. All of the above require the same amount of effort. Answer: The answer is d. Although the boy might be a bit more comfortable in one situation over another, the work done is exactly the same in all cases. Work is needed because of friction between the two surfaces; the amount of work is equal to the magnitude of the 28 Chapter 2 - Mechanical energy friction force multiplied by the distance traveled. The component of force perpendicular to the ground is not doing any work. The smart thing to do is probably to put the box on a wheeled cart or a dolly to reduce the friction. Question: You inflate a balloon by blowing into it. Do you perform work? By how much? Answer: Yes, you do work because you are applying a force (moving your diaphragm to push the air out) through a distance (expanding the balloon outward). The amount of work is equal to the stored elastic potential energy; that is, the amount of energy needed to stretch (deform) the balloon. Having defined work in terms of forces and displacements, we can now turn to defining mechanical energy as the energy acquired by an object upon which work is performed. Alternatively, it can be said that an object which possesses mechanical energy is capable of doing work. Mechanical energy can be described as either kinetic energy or potential energy. Kinetic Energy Kinetic energy is the energy of motion. Since any displacement can be viewed as a combination of a linear motion and a rotation, there are two forms of kinetic energy-- linear (the energy of motion from one location to another) and rotational (the energy due to rotational motion about the center of mass). The energy from wind and rotating flywheels are two examples of kinetic energy. In this book we concern ourselves mainly with the kinetic energy of a body in linear motion, which for an object of mass (m) and velocity (v) is: E = 1 mV2 2 (2-2) W hen mass is expressed in kilograms and velocity in meters per second, kinetic energy will be in joules. Example 2-1: Comets are time capsules that offer clues about the formation and evolution of our solar system, some 4.5 billion years ago. To better understand how the outer solar system was formed a team of NASA scientists designed a probe called “Deep Impact” which was to smash into a nearby comet (actually 83 million miles away!) at very high speeds and for the first time ever look into the comet. The 820-lb probe collided with the comet at 23,000 mph on July 4, 2005, making a hole the size of a football field. What was the kinetic energy of the probe at the time of impact? Solution: Converting the data into SI units, the probe’s mass is 820/2.2 = 372 kg; this mass collides with the comet at (23,000x1.609)/3600 = 10.28 km/s = 10,280 m/s. The kinetic energy of impact is: E= (372).(10,280)2 1 mV2 = = 1.97x1010J = 19.7 GJ 2 2 29 EX 2-1 Which is equivalent to 1.9x1010/4.2x106 = 4,680 kg = 4.68 tons of TNT. Potential Energy As its name implies, potential energy is energy at rest that is waiting, capable of changing in the shape or position of an object in a force field. Energy from a hydroelectric dam, from tides, and from a slingshot are all examples of potential energy. There are two types of potential energy: 1) Potential energy of form is the result of twisting, compressing, stretching, and bending matter out of its natural shape; 2) Potential energy of position is the energy of a mass caused by its higher elevation relative to its surroundings. The changes in potential energy due to deformation of bodies are not considered in this book. Potential energy of position (also called gravitational energy) can be calculated as the work needed to move an object from one location to a new location at a higher elevation. It is the product of weight (mg) and the height (h) to which the object was raised: E pot = mgh (2-3) In this equation, g is a constant signifying the gravitational acceleration between two objects (equal to 9.81 m/s2 on earth), m is mass in kilograms, h is height in meters, and Epot is potential energy in joules. Gravitational energy exists between any two objects separated by a distance. In the following example, the earth constitutes the other object. Example 2-2: Calculate the gravitational energy released as a result of the collapse of the 110-story World Trade Center in New York City on September 11, 2001. Each tower had a mass of about 0.5 million tons and a height of 415 m.1 Solution: If it is assumed that the mass of the tower was uniformly distributed between floors, then the average height of the fall was ½(415) = 207.5 m and the potential energy of the combined towers was E = 2x5.108x9.81x207.5 = 2x1012 J. Torque Figure 2-1 Torque Torque is a measure of work (energy) required to rotate an object. Intuitively, torque can be interpreted as “force with a twist,” since it results in rotation of the object upon which the force is applied. To tighten a screw or to pedal a bicycle we must apply torque. Torque can be calculated by multiplying the force and perpendicular distance between the axis of rotation (or pivot) and the “line of action” of the force. This distance is called “lever arm” or the “arm of the force.” (See Figure 2-1). (2-4) τ = F .d 1 See FEMA Evaluation Report at 30 Chapter 2 - Mechanical energy Work and Power G F YI ... iven enough fuel there is, in principle, no limit to how much work a machine can perform. However, the rate at which the work is performed is certainly limited. In comparing this with equation (2-1), we can see that torque has the same unit as work (N•m). Power A rock climber and a trail hiker of similar mass moving to the top of a cliff use the same amount of work. They each have to lift their own weight the entire height of the cliff. The hiker, however, can reach the top of the cliff much faster than the rock climber. The difference is the rate at which the rock climber and hiker are doing the work, or their rate of energy consumption or power. Power is defined as: (2-5) P=W t We know work is force times distance; therefore, power must be force times distance traveled per second. Since distance per second describes velocity, power equals force times velocity: P = F .V (2-6) The rock climber performs the same task at a slower rate, using less power. For the same reason, you cannot carry a heavy load as quickly as a light load because your power output is too low. Just as we showed that power for motion along a straight line is equal to force times velocity, we can show that rotational power is calculated as torque times angular velocity: (2-7) P = τ. ω In this equation w = 2pN is the angular velocity and N is the shaft speed in revolutions per second. From the definition, we see that the SI unit of power is J/s, commonly called a watt (W) in honor of James Watt, the inventor of the steam engine. Because a watt is a relatively small unit of power, a kilowatt (1 kW = 1,000 W) is often used instead. In the US, power is customarily expressed in horsepower, which is equivalent to 746 watts (1 h.p. = 746 W). Table 2-1 compares the power output of an average man and horse with that of several machines. Example 2-3: How much energy does it take an 80 kg man to run up a flight of stairs 6-m high? What is the power expenditure if it takes 10 seconds to climb the stairs? Assume g = 10 m/s2. Solution: The energy consumed is E = mgh = 80x10x6 = 4800 J. 31 Table 2-1. Power Outputs of Few Basic Machines Laptop Computer Man Horse Automobiles Water/Gas Turbines Power Plants Space Shuttle Main Engines 10 W 100 W 746 W (1 horsepower) 100 kW 1 MW 1 GW 30 GW F YI ... Watt and Horsepower I t is ironic that Watt coined the term “horsepower” to incite coal miners to buy his steam engine to replace their pit ponies. The power expenditure is P = E/t = 4,800/10 = 480 W (0.63 h.p.). Obviously, it takes the same person a bit longer than 20 seconds (maybe 25 s) to climb two flights of stairs. In this case, although the amount of energy spent is exactly twice as much (9600 J), the power reduces to 9,600/25 = 384 W (0.5 h.p.). Although humans can demonstrate impressive muscle power for a short time, even the best of athletes cannot sustain a power output of more than one-tenth of a horsepower. Example 2-4: To accelerate an automobile from 0 to 100 km/hr in 10 seconds, how much total energy must be provided by the fuel? W hat is the power rating? Would it make a difference if the car were accelerated to 100 km/hr in 20 seconds instead? Assume that the automobile has a mass of 1200 kg. Solution: The final velocity of the car is v = 100,000/3600 = 27.78 m/s. The kinetic energy is then equal to 0.5x1200x27.782 = 463,963 J. Whether this acceleration is carried out in 10 s or 20 s is irrelevant. The power rating, however, depends on the rate at which energy is consumed. If the car accelerates in 10s, power will be P = E/t = (463,963 J)/(10 s) = 46,396 J/s (46.4 kW or 62.2 h.p.). If the car’s acceleration takes place in 20 s, the power requirement would be reduced by half (23.2 kW or 31.1 h.p.). Example 2-5: It is expected that the world population will reach 10 billion by the year 2050. If, on the average, each person requires approximately 2 kW of electric power for his needs, how much total electric power must be produced in 2050? If 40% of all the electric power is produced by burning oil (as is done today), how much oil do we need annually to meet the demand? Each barrel of oil has an energy equivalent of 1,700 kWh. Solution: The total electric output must exceed (10x109 persons) x (2x103 W) = 2x1013 W = 20,000 GW of electricity. As we will see later, power plants produce electricity with efficiencies of around 33%; we need to use three units of petroleum energy to produce one unit of electricity. If 40% of the electric power is produced by oil, we will need (0.4)(60,000 GW) x (365x24 h) = 2.1x108 GWh = 2.1x1014 kWh of thermal energy. The amount of oil needed would be 2.1x1014/1,700 = 1.24x1011 barrels = 124 billion barrels of oil (bbo). The current world total petroleum reserves are estimated at 3021 bbo (See Table 6-5). Example 2-6: An electric motor runs at 1500 rpm and delivers 5.0 hp. How much torque does it deliver? 32 Chapter 2 - Mechanical energy Power of Words F YI ... S ound waves generally carry very little energy. While talking, a person normally emits a maximum of 10–4 watts of sound energy. This means that the average sound energy involved in giving a one-hour lecture is less than a joule. If such energy could be used to power a light 100 watt bulb, about a million people would be needed. Enlisting the services of the same crowd, about 10 minutes of casual conversation would be required in order to heat up an 8 ounce (200 gram) cup of coffee. So in terms of energy production, talk is really cheap! Solution: The angular velocity of an electric motor and the power delivered are: w = 2pN= (2p rad/rev)(1500 rev/min)(1 min/60s) = 157 rad/s P = (5.0 hp)(746 W/hp) = 3730 W Torque is calculated using equation 2-7, t = P/w = 3730 W /157 rad/s = 23.76 N.m Question: A cart filled with bricks is to be loaded onto a truck. Three ramps of different lengths are available (See figure). a. Which situation requires the least amount of force? b. Which situation requires the least amount of work? c. Which situation requires the least amount of power? Answer: Situation (a) requires the least amount of force, but the force must be applied over the longest distance. The product of force times distance, or work, is the same in all cases. The power expenditure depends on how long it takes to perform this work. If you pull faster, the time is shorter and more power is needed. Assuming speeds are the same in all cases, it takes longest for the cart to move up the longest ramp (a), and the least power is consumed. Can you verify these conclusions from equations 2-1 through 2-3? Question: Two identical cars are leaving Los Angeles at the same time. The first car arrives in San Francisco in 5 hours, while it takes 10 hours for the second car to reach the same destination. Which car consumes more energy? Which car puts out more power? Answer: Assuming that the cars’ efficiencies are equal at all speeds [in reality, cars are designed to give a better gas mileage at about 80-100 km/h (50-60 mph)], both cars use the same amount of fuel (energy), i.e. work the same amount. The first car, however, uses up fuel twice as fast, therefore putting out twice the power of the second car. Question: Suppose a small passenger sedan with a power rating of 80 hp can accelerate from 0-60 mph in 12 seconds. How long would it take a sports car with a power rating of 240 hp to accelerate by the same amount? Which car performs more work? Answer: Both cars accelerate to the same speed (and gain the same amount of kinetic energy), therefore they perform the same amount of work. The second car uses three times the power—which means it does the same work in one third of the time, or 4 seconds. 33 The Golden Rule of Mechanics The “golden rule” of mechanics essentially states that whatever you lose in power you gain in displacement. We commonly use pulleys and ramps to lift things which would otherwise be too heavy for muscle alone. In both cases, a smaller force achieves the necessary work, but over a greater distance. Question: Consider a bottle opener. In lifting the handle, you do work on the opener, which in turn does work on the cap it is lifting. W hich one is greater, your work or the work of the bottle opener? Answer: Because the opener cannot be a source of energy, the output work (work of opener) can never exceed the input work (your work). The opener simply aids in the transfer of energy from you to the bottle cap. In fact, nearly all of your work will go into deforming rather than into lifting the bottle cap. Of course, energy was not really lost; it was just not used in the way you intended. Although the output work is always less than the input work, output force does not have to be smaller than input force. This is exactly what a machine is supposed to do - transform part of the energy (work) into a form most convenient -- that requiring less force. Simple machines Machines are an important part of our daily existence; they vary in complexity from simple tools like screwdrivers and scissors, to more complex machines such as cranes and automobiles. Whether powered by engines or by people, machines make our tasks easier. A machine eases the load by changing either the magnitude or the direction of a force. Simple machines are used to reduce the amount of force required to do particular types of work, but the trade off is that you must apply this force over a greater distance. Others, such as bicycles, are considered complex (or compound) because they are made out of a number of simple machines. There are six types of simple machines (See Figures 2-2a-2-2f): 1. Inclined plane (any slanted surface used to raise a load from a lower level to a higher level). Work is carried out using less effort by moving a smaller force over greater distances. Examples include ramps for wheelchairs, ramps to load luggage onto a plane, and escalators. It is believed that Egyptian pyramids were built using similar ramps. Lever Fulcrum (a) (b) (c) (d) (e) (f) Figure 2-2 Simple Machines - (a) ramp, (b) wedge, (c) screw, (d) lever, (e) wheel and axle, (f) pulley 34 Chapter 2 - Mechanical energy 2. Wedge (essentially two back-to-back ramps or inclined planes turned on their sides). Wedges are used mainly to distribute force on both sides. It works by applying a smaller force over a larger area. Nearly all cutting machines (chisels, axes, scissors, and knives) use different variations of the wedges. 3. Screw (the circular version of the inclined plane). A screw is really an inclined surface wrapped around a cylinder or a cone. Some examples are ship propellers, corkscrews, meat grinders, ordinary screws, and jar lids. 4. Lever (a bar or a rigid object that rests and rotates about a point called a fulcrum). Levers make work easier by allowing a small force to be applied over a long distance, instead of a large force to act over a short load arm. A lever is characterized by a fulcrum or pivot, a force (effort) arm and a weight arm. Depending on the position where force and weight apply relative to the pivot, there are three kinds of levers. In a first class lever, the fulcrum is between the force and the weight (pliers, seesaw, and scissors). When the weight is between the fulcrum and the force, the lever is called a second class lever (nutcracker, and wheelbarrow). In third class levers the force is located between the fulcrum and the weight (baseball bat, shovel, tongs, and human arm). 5. Wheel and axle (a wheel or spoke attached to an axle or a lever). Here, a smaller force applied to the longer motion at the wheel results in a shorter more powerful motion at the axle. Examples are windmills, gears, doorknobs, and steering wheels. 6. Pulley (a chain, belt or rope wrapped around a wheel). A pulley is often used to lift a heavy weight with less force. A pulley makes work easier because it changes the direction of force. Instead of lifting up, you will pull down on the load using your body weight. Like other simple machines, the trade-off is the longer distance that the smaller force must travel at the end of the rope. Examples are the bicycle chain and the pulley used to move a curtain up and down. Question: W hich of the six simple machines reduce the amount of work needed to raise a load? Answer: None of them. Machines do not change the amount of work required. They merely make the work easier. Question: How can a small child lift a heavy load? Answer: In all the examples shown in Figure 2-3, the force required to carry out the tasks is reduced by making the lever arm bigger. In fact, we can show that the product of: force in x lever arm = force out x load arm 35 Effort Fulcrum Effort Fulcrum (a) Effort Load Fulcrum Fulcrum Load load arm Effort lever arm (b) Load Fulcrum Effort Fulcrum Effort Load lever arm load arm (c) Figure 2-3 Examples of levers; first class (a), second class (b), and third class (c). Mechanical Efficiency and Mechanical Advantage Two factors are important in selecting an appropriate machine for performing a given task: mechanical efficiency and mechanical advantage. Mechanical efficiency gives an indication of the losses that may be incurred while performing a certain job. It is the fraction of the input work Wi that is available for carrying out the desired output work Wo: W η m= W o i (2-8) In a real machine, some of the input work is inevitably lost as heat, so all real machines have efficiencies that are less than 1. Question: Suppose you hook two machines together, one with efficiency of 40% and the other with an efficiency of 60%. What is the efficiency of the two machines operating together? Answer: If you are tempted to say 100%, try to resist. The first machine will feed only 40% of the original input energy into the second machine, which then puts out 60% of the 40%. In other words, each machine loses some energy and the net efficiency is 40%x60% = 24%. Question: A transmission gearbox transmits power from the engine to the axle through a number of interlocking gears. To get more power we need to shift the transmission to a lower gear, which will cause the speed to decrease. Can we design transmissions that increase power and at the same time increase velocity? Answer: No. A machine can increase the magnitude of the effort (input) force or increase the velocity of the object to be moved, but not both. If a machine did both at the same time, the output work would exceed the input work, which is impossible. Mechanical advantage (MA) is the amount by which the machine multiplies the force. It is the ratio of the force exerted by a machine (or the resistance force Fr) to the force exerted up on a machine (or the effort force Fe). In an ideal machine no energy is wasted so the output work equals the input work (Wo = Wi), i.e; Fr. dr = Fe.. de. Therefore, MA = Fr de = Fe dr (2-9) where dr and de are corresponding displacements by Fr and Fe (lever arms). Forces are not, however, the only things that are magnified by levers. Since the longer arm of the lever has to travel a greater distance than the smaller arm of the lever in the same amount of time, it travels faster. Ve de (2-10) = Vr dr 36 Chapter 2 - Mechanical energy Lifting the Earth* F YI ... “Give me a point to stand and I will lift up the earth!” proclaimed the great Archimedes, the genius of antiquity who discovered the laws of leverage. He realized that a lever would allow one to lift a very heavy object while applying a very small force. In principle, his assertion about lifting the earth, though somewhat ambitious, does seem perfectly reasonable. By using a very long lever, for instance, you could attempt to balance one object weighing as much as the earth with an applied force on the other end. Let’s suppose, for argument’s sake, that Archimedes can apply a force roughly equal to his own weight. Clearly, the much lighter Archimedes would have to position himself much farther from the pivot than the heavy object. In fact, the ratio of the two lever arms would be precisely the inverse ratio of their respective weights (or masses). Since the mass of the earth is ME~6x1024 kg and the mass of Archimedes was probably around 100 kg (to keep the numbers nice and round), the ratio of the arms would be roughly 6x1022. A big number, but so what? Well, the difficulty arises when one considers the time involved in lifting the “earth” by any noticeable amount. For example, in order to lift the earth even by 1 millimeter, Archimedes would have to move his end through a giant arc 6x1019 m long. Just how long do you think that would take? Working against a force equal to his weight (1000 N), Archimedes would generate about 1.3 horsepowers(1 horsepower = 746 W) if he moved his end at the respectable rate of 1 m/s. At this rate, the task would take 6x1019 s or about 2 trillion years. Even if Archimedes were to move his end at the speed of light — nature’s fastest — the task would still take him over 6 thousand years. * Excerpts from a physics text currently under preparation jointly by this author and Professor Igor Glozman, Department of Physics, Highline Community College, Des Moines, Washington 98198. You can infer from equations 2-9 and 2-10 that the gain in velocity or displacement is inversely proportional to reduction in effort, and is therefore inversely proportional to mechanical advantage. Hence you may use a lever to gain either mechanical advantage or speed, but not both. Question: W hat is the mechanical advantage of a hammer driving a nail through a block of wood? Answer: The energy used to carry the hammer through its flight (from the time of raising your hand until the time that hammer hits the nail) is used to push the nail only a few millimeters into the wood. For example, if we lower our hand by 50 cm, but the nail is pushed down only by 0.5 cm, then we have reduced the displacement by100 times, and at the same time multiplied the force by a factor of 100. Question: Using the pulley arrangement shown in Figure 2-2, how much force is required to lift a 200-pound package? What is the mechanical advantage? Answer: The load is divided between two cables holding the pulley at the right. The force is one half (100 pounds) and the mechanical advantage is two. 37 W W/2 Unlike mechanical efficiency, mechanical advantage may or may not be greater than 1. Machines are designed to make a task simpler by increasing the force applied; therefore, for most machines mechanical advantage is greater than one. The muscles in our bodies also act as levers. Unlike machines, however, in many instances they do not make a particular task (such as lifting a weight) simpler. In fact, the human body is not designed for strength, but for agility, speed and wide ranges of motion. Question: Our muscles are much stronger than they seem. For instance, the bicep is attached to the forearm about 8 times closer to the fulcrum (the elbow) than one’s hand is when lifting a weight. W hat is the advantage of this seemingly inefficient arrangement? Answer: The advantage is speed (also a factor of 8), which is frequently more important in the animal world. Example 2-7: W hat is the mechanical advantage of the system of pulleys shown in figure Ex 2-7? Solution: The mechanical advantage of a pulley system is approximately equal to the number of supporting ropes or strands. A single pulley can only change the direction of the pull to lift a load. Therefore, pulley shown in (a) has a mechanical advantage of 1 (MA = 1). The pulley system in (b) has two supporting pulleys and therefore has a MA = 2. Example 2-8: In figure Ex 2-8, what is the force required for moving a 200 kilogram stone? What is the mechanical advantage? If we want to lift the stone at a rate of 5 cm a second, what is the speed at which we need to push down on the lever? Solution: From equation (2-9) we have (200x9.8)x0.2 = Fex 1.0, or Fe = 392 N. The mechanical advantage is: MA = Fr (200) (9.8) 1.0 = = = 5.0 392 0.2 Fe W W (a) EX 2-7 (b) EX 2-8 Biceps Triceps Bone Muscle Using equation 2-10, we have Ve = Vr . de 1.0 = 5. = 25 cm/s dr 0.2 Figure 2-4 A human elbow acts as a third class lever during lifting a weight. 30 cm 100¼ Human Body as a Machine 4 cm 40 cm The human body can be seen as a machine that consumes food and converts it to mechanical work and heat (See Chapter 5). Power is transmitted through the action of forces on the muscles, bones, and joints that make body’s lever system. As muscles contract, they pull a series of tendons and bones (levers) across joints (fulcrums, pivots, or hinges) to perform various tasks. Muscles always act as pairs, so when one contracts, its 38 Chapter 2 - Mechanical energy antagonist extends. Examples of the first, second and third class levers are the joint between skull and vertebrae (neck joints), the Achilles tendons, and the elbow joints, respectively. Most of the movements of the body are produced by third class levers, where the force is between fulcrum and weight. This design lends itself to speed of movement rather than force. For example, when we lift a weight, the biceps and triceps work as a pair; when the biceps flex the arm (effort) or lift a weight (load or resistance), the triceps relax to extend the arm (Figure 2-4). Question: A man is lifting a heavy weight by contracting the bicep muscle, and at the same time relaxing his triceps muscle. The elbow acts as the fulcrum. What is the force needed to lift a 10 pound weight? Answer: Force needed is higher by ratio of the arms (40 to 4); we need ten times the weight of the object to support the weight. In the discussions above we assumed that there are no losses involved. W hen losses are present, mechanical advantage will be reduced by a factor equal to the mechanical efficiency. Summary Mechanical energy is the total energy an object has excluding the thermal and internal energies associated with the structure of its atoms and molecules. In other words, it is the sum of its potential, kinetic, and rotational energies. Mechanical energy is what it takes to lift, move, turn, and twist an object. Examples of mechanical energy are energy possessed by a fast ball, energy stored in a spring or a rubber band, and the energy it takes to move a heavy object up a stairs. Simple machines are devices that are at our disposal to carry out the difficult tasks with less effort; they allow us to do the same amount of work by applying a smaller force (effort) but over longer distances. In performing any task there are always some frictional forces that we must overcome. For this reason, some mechanical energy is always converted to heat. In other words, in real systems, mechanical efficiencies are always smaller than “one”. 39 Exercises I. Essay Questions: 1. Which action requires more work: carrying a weight to the second floor from the ground using a ramp, using stairs or using an elevator? 2. A bicyclist in a road race like the Tour de France rides at about 20 mph (9 m/s) for over 6 hours a day. About 250 watts of power go into moving the bike against the resistance from air, gears, and tires. Another 750 watts are lost as heat. How many calories must the racer ingest per day in order to sustain such a rigorous regimen? (Use 1 calorie = 4.18 J) 3. How can levers and pulleys help in reducing the force needed to lift heavy objects? 4. Can a small child lift an adult with the help of a lever? How? 5. Suppose a tree root is under a section of cement sidewalk and over a ten-year period it lifts up the heavy slab. On the other hand, a strong worker with a crowbar could lift an identical section of sidewalk within a matter of seconds. Compare the work and power for both cases. 6. A car with a mass of 2,000 kg is traveling at 36 km/ hr. What is the car’s kinetic energy? 7. How much work do you do while climbing to the roof of a 30 meter tall building? Does it matter if you walk up the stairs slowly or quickly? If you skip every other step? 8. In the example above, what is the power you need to walk up the stairs if the vertical speed on the stairs is 0.5 m/s? 9. If the weight of an object being lifted is 200 kg and the number of supporting ropes the pulley system has is four, what would be the system’s MA? How much effort weight would you actually be lifting? 10. Determine whether each of the tools mentioned below are first class, second class, or third class 40 levers. a. Scissors b. Hammer used to insert a nail c. Hammer used to pull up a nail d. Fishing rod e. Nutcracker f. Screw g. Finger-nail clippers h. Wheelbarrow i. Achilles tendon, pulling or pushing across the heel of the foot j. Crowbar k. Elbow joint when lifting a load II. Multiple Choice Questions: 1. Which of the following statements are correct? a. Whenever forces are involved, work is performed. b. Work is performed only when forces are normal to the direction of displacement. c. Work is involved when an object is displaced in the direction of the force, but not in the opposite direction. d. Work is involved when an object is displaced in the direction of a force. e. None of the above. 2. Work is by definition a. The product of force multiplied by the time during which it is exerted b. The product of force multiplied by the distance through which it acts c. Equivalent to power/time d. Equivalent to energy multiplied by time e. None of the above 3. In the language of mathematics a. d = W/F b. F = W.d c. F = d/W d. W = F/d e. W = d/F 4. Which of the following objects possess mechanical energy? a. A car traveling on a flat surface b. A car going downhill Chapter 2 - Mechanical energy c. A drawn bow ready to be released d. A book sitting on the top of a bookshelf e. All of the above 5. After a skydiver has reached his terminal speed (constant speed) in the atmosphere, he a. Gains kinetic energy at the expense of potential energy b. Gains potential energy at the expense of kinetic energy c. Releases heat at the expense of potential energy d. Releases heat at the expense of kinetic energy e. None of the above 6. What happens to the kinetic energy of a moving car as it comes to a complete stop? a. It is lost. b. It is converted to heat. c. It is stored in the flywheel. d. It is stored in the battery. e. None of the above. 7. Which of these statements are true? a. Power is the rate of energy expenditure per unit time. b. Power can be calculated as the product of force times velocity. c. Power can be expressed in units of watts or horsepower. d. All of the above. e. None of the above. 8. There are 100 steps to get from the ground-floor of a building to its rooftop. Person A walks up to the roof in 100 steps, while person B climbs up skipping every other step. It takes person A two minutes to reach the top, while person B can do it in half the time. From the information provided one can deduce that a. Both A and B perform the same amount of work, but A has more power output than B b. Both A and B perform the same amount of work, but B has more power output than A c. A required more work because he has to walk 100 steps d. B requires less work because it takes him a shorter time to make the climb e. B requires less work, because he only has to work half the time 9. Which of the following statements are true? a. Pulleys and ramps allow for the lifting of heavy objects because less work is needed. b. Pulleys and ramps allow for the lifting of heavy objects because the same work can be performed over larger distances. c. Pulleys and ramps allow for the lifting of heavy objects because the same work can be performed over a longer period of time. d. Pulleys and ramps allow for the lifting of heavy objects because they perform some of the work for you. e. All of the above. 10. The “golden rule” of mechanics implies that a. Whatever you lose in power, you gain in displacement b. Whatever you lose in energy, you gain in power c. Whatever you lose in power, you gain in time d. Whatever you lose in energy, you gain in displacement e. All of the above 11. Which of the following is an example of kinetic energy? a. An automobile lifted by a hydraulic jack b. Water sitting on a reservoir on top of a mountain c. A rubber ball as it is hitting a solid wall d. A turbine being turned by a falling waterfall e. None of the above 12. The main function of simple machines is to multiply a. The force b. The work c. The power d. The energy e. All of the above 13. A fork is an example of a a. Pulley 41 b. c. d. e. Ramp Wedge Wheel and axle None of the above 14. A roller skate is a an example of a a. Lever b. Ramp c. Wedge d. Wheel and axle e. None of the above 15. A seesaw is an example of a a. Lever b. Ramp c. Wedge d. Wheel and axle e. None of the above 16. The lever arm is the a. Displacement from the pivot to the line of action of the force b. Displacement from the pivot to the point of application of the force c. Parallel displacement from the pivot to the line of action of the force d. Perpendicular displacement from pivot to line of action of the force e. Equal but opposite displacement from pivot to line of action of the force. 17. When a hammer is used to pull a nail out of a piece of wood, it acts as a a. Wedge b. Lever c. Ramp d. Pulley e. None of the above 18. Nearly all cutting machines use a. Ramps b. Levers c. Wedges d. Wheels and axles e. All of the above. 19. Which of the following can be represented by the 42 same units? a. Work, torque, and acceleration b. Work, torque, and kinetic energy c. Force, torque, and energy d. Force, power, and torque e. Work, torque, and angular velocity 20. The relationship of energy to power is like a. miles to miles per hour b. force to velocity c. distance to time d. kilowatt to killowatt-hour e. oil to electricity III. True or False? 1. Whenever a force undergoes a displacement work is performed. 2. Mechanical efficiency is the percentage of input work that can be converted to output work. 3. Unlike mechanical efficiency, mechanical advantage may or may not be greater than 1. 4. Torque is defined as the amount of force that is required to rotate an object. 5. When there is no friction, the sum of potential and kinetic energies remains constant. 6. In the metric system of units, torque measurements are usually given in newton-meters. 7. Torque is calculated by multiplying the force by the lever arm it is acting upon. 8. A wedge can be seen as a two-sided ramp with surfaces to distribute the load. 9. The sharper the wedge and the closer the threads on the screw, the greater the mechanical advantage. 10. A baseball bat is an example of first-class lever. 11. The human body can be seen as a machine that consumes food and converts it to mechanical work and heat. Chapter 2 - Mechanical energy 12. Mechanical efficiency is defined as the ratio of input to output works. 13. In an ideal machine the output work and the input work are equal. IV. Fill-in the Blanks 1. Work is the change in the state of motion that results from the application of _________. 2. No work can be done by a force that is _________to the displacement. 3. Torque has the same unit as _______ and is expressed in ________ in SI units. 4. Tweezers are an example of a class _______ lever. 5. A pulley system with two supporting ropes would have the mechanical advantage of __________. V. PROJECT I - The Mountain of Pharaoh The pyramids of Egypt were built to enshrine the pharaoh after their death. The ancient Egyptians believed that the pharaohs held the key to the afterlife for everyone, and the pyramids symbolize this strong belief. The most impressive of the Egyptian pyramids and one of the Seven Wonders of the World is the Great Pyramid of Giza. Its construction was commissioned around 2,550 B.C. by King Khufu of the Fourth Dynasty. The Pyramid was constructed using around two million blocks of stone each weighing an average of 2.5 tons. The heaviest blocks weighed 40 to 60 tons and were used in constructing the ceiling of the burial chamber. It has been estimated that it took 20,000 to 25,000 slaves to build the pyramid. Possibly the oldest structure in existence, the Great Pyramid took about 20 years to build. Its base covers 13.6 acres (55,000 meters squared); at 145 m tall, it was the tallest structure in the world until the Eiffel Tower was built in 1889. You are asked to research the probable methods that Egyptians used to build the Pyramids. Carry out a simple calculation to determine the amount of energy that was expended to build the Great Pyramid of Giza. Assuming that an average man could put out 100 watts of power continuously, estimate the number of slaves it took to work 12 hours a day over the entire year to finish the job in 20 years. Explain any discrepancy with data you may find in other literatures. Discuss different theories on achieving this task, and comment on their feasibilities. Hint: Energy can be estimated as the amount of mass which had to be lifted from the ground to its center of mass -- the weight of the pyramid times the height of the centroid above the base. The volume and the height of the center of mass of pyramid are: V = a2.h/3; and z = h/4. h a a Pyramid of Giza PROJECT II - Energy Consumption During a Climb In this project, you are asked to calculate the amount of work done and energy consumed during a climb to the top of a six-story building. To perform this assignment, you are asked to climb up from the first floor to the roof of a six-story building twice, once at a slow rate and the second time at a faster rate. Note the following data: 1. Your mass in kilograms 2. The number of stairs and the average height of each stair (in meters) 3. The time it took you to climb the stairs 4. The time it took you to descend to the first floor (take the average of the two runs) Now, answer the following questions: 1. What was the total height that you climbed? 2. What was your average speed? 3. What was the force used to perform the climb? 4. What is the work needed to overcome the gravitational potential energy? 5. How much energy did you burn during the climb? Note that the actual work is a lot more (by about a factor of 4) than the work of gravity. Additional 43 energy is necessary to power your muscles, nourish and repair cells, and to maintain your body temperature. 6. Calculate the average power consumption during each run by dividing work performed by time it took you to climb (or descend) the stairs. 7. Calculate the average power consumption during each run by multiplying your force by your average velocity. 8. In which instance did you perform more work, climbing up slowly or quickly? 9. In which instance did you put out more power, climbing up slowly or quickly? 10. Assuming the work performed in climbing up the stairs is used to generate electricity to turn on a 100-W fluorescent light, how long would the light stay on? Work Sheet for Project II Your name: ______________ Your mass: ______________ kg Number of stairs: ____ Your weight: _____________ N Total rise: __________ m Fast Climb Descent Average height of each stair: __ m Slow Climb Force (N) Time to climb, t (s) Average climb velocity, v (m/s) Total work (N.m) Energy consumed (kJ) Average Power output (W) P = W/t Average Power output (W) P = F.v Answer to question 7 Answer to question 8 Answer to question 9 Answer to question 10 44