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Work When you hold something, are you exerting a force on the object? Yes. When you hold something, are you doing work? If you set the object on a table, does the table exert a force on the object? Yes. Does the table do any work? No. When you hold something, then, you are not doing work either. Work What can you do that the table can’t do? You can lift the object up - which is work! We define the concept of WORK as the exertion of force through a distance. There is one more consideration, however. In tether ball, the pole exerts a force on the moving ball (via the rope). Does the pole do work? Work The pole does NOT do work! Does the player who hits the tether ball do work? YES. What is the difference between what the pole does and what the player does? Work The difference is in the direction. The pole was pulling on the tether ball perpendicular to the motion of the tether ball. The player was pushing on the tether ball in the same direction as the motion. In part two, however, we also had torque as being force across a distance. What is the difference? Work and Torque In applying torque, the direction of the force had to be perpendicular to the distance. This caused a turning force: t = r F sin(qrF) In doing work, the direction of the force has to be parallel (or anti-parallel) to the distance moved. We write this this way: Work = F s where the dot indicates the cosine of the angle between F and s: Work = F s cos(qFs) . Work Work = F s Although F and s are vectors (with magnitude and direction), Work is a scalar (magnitude only). Can we have positive and negative work? If the Force and distance are parallel, the amount of work is positive, but if the two vectors are anti-parallel, then the work is negative. Energy We can now define the concept of energy: Energy is the capacity to do work (in ideal circumstances). We all know that we can do work: exert a force through a distance. But to do that requires food. Thus we convert the energy in food into work. The same thing happens when we burn coal to generate heat which can be converted into electricity which can be converted into lots of useful work. Conservation of Energy A Natural Law Many such examples as we just saw lead us to propose a natural law. Remember that a natural law is a statement of how nature seems to work - it is not “derived” from anything more basic, it is observed to fit the results of observations (experiments). Energy can neither be created nor destroyed (that is, energy is conserved). However, it can be transformed from one form into another. Conservation of Energy The equation that comes from this law of conservation of energy is: S Energiesinitially = S Energiesfinally . Our job now is to find out how the amount of energy in different forms relates to the various parameters associated with that form. That is, we need to derive formulas for various kinds of energy. Units The units of energy (and work) are: Nt*m = Joule. A British unit of energy is the BTU (British Thermal Unit). 1 BTU = 1,054 Joules (This is the energy necessary to heat one pound of water 1 oF.) Another unit of energy is the calorie. 1 calorie = 4.186 Joules (This is the energy necessary to heat one gram of water 1 oC.) However, the calorie we refer to when we eat is really a kilocalorie = 4,186 Joules. Units The units of torque are: Nt*m = Nt*m. Note that even though torque and energy both have units of Nt*m, they are different quantities, and so they have different formal names. Energy units are in Joules, while torque units are simply specified as Nt*m. In the British system, the unit of torque is simply called the foot-pound (ft-lb). Positive and Negative Can we have energies that are negative? First, can we have negative money? Yes - it’s called debt. You need to earn money to pay off the debt and reach up to zero. In the same way, some energies can be negative - we need to gain some energy to reach what we define as zero energy. Forms of Energy Kinetic Energy Energy of motion, called Kinetic Energy: should depend on mass and speed of object. Your car has energy when it is moving. The wind has energy when it is moving, and we can convert this wind energy into electric energy via windmills. Potential Energies Energy of position, called Potential Energy: should depend on why that position has energy. The water stored behind a dam has energy due to it’s height above the base of the dam. We can use this to run a hydroelectric station. The energy in food is due to the molecular binding of the atoms in the food. The same is true for coal, oil and gas. There is also energy stored in the nucleus of atoms - nuclear energy. Some other forms • Heat: should depend on temperature, type and amount of material. We burn coal to get heat to turn water into steam and use the steam pressure to get work (or electricity). • Light: should depend on type and intensity of the light. • Sound: should depend on type and intensity of the sound. Kinetic Energy - derivation If we let an object fall, it gains speed. It also gains what we call kinetic energy. By the Conservation of Energy law, the amount of work going into the object (from gravity) will equal the amount of energy the object has (kinetic): F s cos(q) = mg h (1). But if an object falls a distance h with an acceleration of g, how fast is it going? Kinetic Energy - derivation KE(m,v) = mgh (The amount of kinetic energy, which depends on the quantities mass and speed in this case equals the amount of work done by gravity, mgh). From our motion equations, v = vo + gt and h = ho + vot + (1/2)gt2 or in this case (ho=0, vo=0): h = (1/2)gt2, or t = (2h/g)1/2 so v = gt = g(2h/g)1/2, or v = (2hg)1/2, or h = (1/2)v2/g; thus mgh = mg(1/2)v2/g = (1/2)mv2 = KE . Kinetic Energy - formula KE = (1/2)mv2 . Note that the kinetic energy depends on m (the more mass the more kinetic energy) and on v2 (if you double the speed, you quadruple the kinetic energy). Also note that the kinetic energy must always be either zero or positive - it can’t be negative. (This is like cash in the money analogy.) Kinetic Energy - considerations You’ve probably heard the expression: “speed kills”. This comes from the fact that KE depends on the square of the speed. If you double your speed, you quadruple the amount of energy of the object. And remember that energy is the capacity to do work - for either good or bad. Uncontrolled energy can exert large forces through significant distances - which can be very dangerous! Kinetic Energy - considerations Note that the difference between (1 m/s)2 and (2 m/s)2 is 3 m2/s2, whereas the difference between (99 m/s)2 and (100 m/s)2 is 199 m2/s2. What this indicates is that it takes more and more energy to move faster and faster. This explains why there is so little difference between first and tenth in a speed race between trained athletes! Gravitational Potential Energy If we let the force of gravity act on an object as it moves, gravity is exerting a force through a distance and may add or subtract energy from the object. We can work with this near the earth (where gravity is constant) with PEgravity = mgh . Farther from the earth’s surface, gravity changes, and we need a different formula (from calculus): PEgravity = -Gm1m2/r12 . PEgravity Considerations In the simpler formula near the earth’s surface, PEgravity = mgh both m and g are positive numbers, but h is a height measured from some point that you determine. It can be the ground, but doesn’t have to be. Note that h can be either positive or negative! h and Dh The formula for PEgravity really should be DPEgr = mgDh instead of PEgr = mgh. We often use h when we mean Dh, as in the case of “what is your height?”. What we mean is, “what is the difference in height between the top of your head and the bottom of your feet?”. We often assume that we are measuring from the floor, ground, or some other standard position. If you use (or see) PEgr = mgh, be sure to interpret it as DPEgr = mgDh, and know where the “standard” position of h=0 is. PEgravity Considerations In the more general form, PEgravity = -Gm1m2/r12 , the PE is always negative, with the highest (least negative) value being when PE=0 or r12 goes to infinity. Note in particular, that while the force of gravity goes as r122, the PE of gravity goes only as r12. PE versus DPE We have a similar case here as we did with the near-earth PE: when should we use PE and when should we use DPE ? If we use PE = mgh, we are assuming that there is some standard position where h=0 (usually the ground, the floor, or table top). If we use PE = -GM1m2/r12 , we are assuming that the “standard r12” is a position that makes PE = 0; to have this, the “standard r12” would have to be infinity, since 1/infinity = 0. Problems Having KE and PEgravity, we can start solving some problems using the Conservation of Energy. Problem: How high will a ball go if it is thrown with a speed of 25 m/s? We could solve this problem using Newton’s Second Law and the equations for constant acceleration, or we could use Conservation of Energy. Tossing a ball up Let’s try this problem using Conservation of Energy: We recognize that we have kinetic energy (since we have motion), and we recognize that we have gravitational potential energy (since we have gravity); also vi=25 m/s; hi=0 m (start from the ground); vf=0 (highest point). These are all related by the Conservation of Energy: Tossing a Ball Up S Energiesinitially = S Energiesfinally . KEi + PEi = KEf + PEf (1/2)mvi2 + mghi = (1/2)mvf2 + mghf (note that this is equivalent to saying DKE = DPE, or ½mvf2 – ½mvi2 = mg(hf-hi) (1/2)*m*(25 m/s)2 + m*(9.8 m/s2)*(0 m) = (1/2)*m*(0 m/s)2 + m*(9.8 m/s2)*hf . Here we see that the mass cancels out, and we have one equation in one unknown (hf): hf = (1/2)*(25 m/s)2 / (9.8 m/s2) = 31.89 m. Observations We should note two things from this example: Conservation of Energy is a scalar equation, and so has no information about directions. This makes it easier to solve, but gives less information in the answer. Conservation of Energy makes no mention of time (only initial and final). This removes t from the problem - making it easier but also giving us less information in the answer. Escape Speed In the previous example, we threw something up that went about 32 meters high. How fast would we have to throw something to make it escape from the earth altogether? We can use Conservation of Energy again, but we need the more general form for potential energy due to gravity. To escape the earth, rf = infinity! We start with ri = Rearth = 6.4 x 106m. Escape Speed S Energiesinitially = S Energiesfinally . KEi + PEi = KEf + PEf (1/2)mvi2 + -Gmearthm/ri = (1/2)mvf2 + -Gmearthm/rf (note that this is equivalent to saying DKE = DPE, or ½mvf2 – ½mvi2 = -Gmem/rf - -Gmem/ri) We see that m is in each term, so we cancel it. (1/2)*(vi)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(6.4x106m) = (1/2)*(0 m/s)2 - (6.67x10-11 Nt*m2/kg2) *(6.0 x 1024 kg)/(infinity) We again have one equation in one unknown (vi). Escape Speed We have used vf = 0 m/s since this is the minimum speed we need at the end. We could have more speed when we escape, but we’re looking for the lowest speed for the object to still escape; this means that both the terms on the right side = 0. Solving for vi = vescape = [2*G*Mearth /Rearth ]1/2 = 11,180 m/s = 25,000 mph. Friction and Energy Loss Can we use Conservation of Energy if we have friction? What happens with friction? We convert kinetic energy into heat! What “formula” do we use for how much energy is “lost” to friction, that is, how much energy is converted from kinetic to heat? Friction We start from the basic definition of energy: the capacity to do work, where work = Force thru a distance: Elost = Ffriction * s. We still have Ffriction = mFc. Where does this Elost go in the equation for Conservation of Energy: on the initial or final side? Is it a positive or negative amount of energy? Friction Since some of the initial kinetic energy will go (transform) into some heat, the Elost should be a positive term if it is on the final side, or a negative term if it is on the initial side. S Energiesinitially = S Energiesfinally . KEi + PEi = KEf + PEf + Elost where Elost = + Ffriction*s = + mFcs . Friction - example Problem: If the coefficient of friction between a block of wood and the concrete floor is 0.50, how far will a block of wood slide on the floor before coming to rest if it starts with a speed of 10 m/s ? We recognize this as a Conservation of Energy problem with kinetic energy and with friction (Elost). Friction - an example We are given: vi = 10 m/s; vf = 0 m/s; m = .65 We are looking for s (the distance of slide). S Energiesinitially = S Energiesfinally . KEi = KEf + Elost where Elost = + Ffriction*s = + mFcs . From S Fy = 0, we have Fc = mg. Therefore: (1/2)*m*vi2 = (1/2)*m*vf2 + m*m*g*s . Friction - an example ½*m*vi2 = ½*m*vf2 + m*m*g*s We notice that there is an m in each term so it cancels out! ½*(10 m/s)2 = 0 + (.65)*(9.8 m/s2)*s This is one equation in one unknown (s): s = ½*(10 m/s)2 / (.65)*(9.8 m/s2) = 7.85 m. Power We now know what Force and Energy are, but what is Power? Power We now know what Force and Energy are, but what is Power? The definition of Power is that it is the rate of change of Energy from one form into another: Power = DEnergy / Dt . The units of power are: Joule/sec = Watt. Another common unit is the horsepower, hp. The conversion factor is: 1 hp = 746 Watts. Example: Power What is your power output when you climb stairs? In this case, you are changing your potential energy (mgh) in time, so … P = Dmgh / Dt if your mass = 70 kg, gravity is 9.8 m/s2, and you climb steps of height 10 meters in a time of 20 seconds: P = 70 kg * 9.8 m/s2 * 10 m / 20 sec = 343 W or 343 W * (1 hp / 746 W) = .46 hp . Example #2: Power What is your average power output per day? If you eat 2,000 Calories per day, (and assuming you do not gain or lose weight), that energy must be converted into energy you use throughout the day. P = 2,000 Calories / day = (2,000 Cal)*(4,186 joule/Cal) / [(24 hours)*(60 min/hr)*(60 sec/min)] = 97 Watts. Example #3: Power How powerful must a car engine be (on average) if it is to accelerate a 2,000 kg car from zero to 65 mph in 20 seconds? This is a power question. The change in energy is in the form of kinetic energy. We should convert 65 mph into metric form: vf = 65 mph * (1 m/s / 2.24 mph) = 29 m/s. vi = 0 (starts at rest); Dt = 20 sec. Example #3: Power Power = DEnergy / Dt = [(1/2)*m*vf2 - (1/2)*m*vi2] / t = [(1/2)*(2,000 kg)*(29 m/s)2 - 0 ] / 20 sec = 42,000 Watts * (1 hp / 746 Watts) = 56 hp. Note that this is the average power. Force and Power We know how force is related to energy, and how energy is related to power. Can we relate power to force? Work = F s , Power = DWork /Dtime = F Ds /Dt (but Ds/Dt = v), so Power = F v . Power, like work, is a scalar. Note that if F is constant, Power must go up as speed (v) goes up! If Power is constant, F must go down as v goes up. Force and Power At very low speeds, even a small power will give a rather large force! On cars with manual transmissions, you normally don’t rev up the engine (high power), and then pop the clutch! This causes tremendous forces that can break the car! What do you pay for: Force, Energy, or Power? What do you pay MLG&W (for example) for: force, energy or power? What do you pay the gas station for: force, energy or power? What do you pay for: Force, Energy, or Power? What do you pay MLG&W (for example) for: force, energy or power? What do you pay the gas station for: force, energy or power? In both cases you pay for ENERGY! Cost of Energy What is the cost of energy? We saw before that we could do work at the rate of a couple 100 Watts, but that was hard work!. If we worked for 40 hours a week, how much useful work would we perform? Work = Energy = Power * time. The MKS unit of energy is the Joule, but this is a very small unit. Another common unit of energy is the Kilowatt*hour. Cost of Energy In terms of kilowatt-hours, if you worked at the rate of 200 Watts for 40 hours, you would do 8 KW-Hrs of work. How much does the power company charge for a KW-hr of energy? How much would you make for your 8 KWHrs of work (assuming MLG&W paid you exactly what it charges us for that same amount of energy)? Cost of Energy In Memphis, MLG&W charges about 9 cents per KW-hr. Thus, if you were to work for the power company providing power, you would earn about 8 KW-hr/week * $.09 = $.72/week (72 cents per week)! As we see, energy is quite cheap! The reason our utility bills are so high is that we use so much energy - especially when we heat (or cool) things (like air and water)! Computer Homework The computer homework program, Energy and Power, is the first program on Volume 2. This program has questions about the concepts we have covered in this set of slides. The labs on Atwood Machine and Hooke’s Law also deal with these concepts.