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by: Graciela Homenick


Graciela Homenick
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This 23 page Class Notes was uploaded by Graciela Homenick on Saturday October 3, 2015. The Class Notes belongs to PHYS 1111 at Armstrong Atlantic State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/217875/phys-1111-armstrong-atlantic-state-university in Physics 2 at Armstrong Atlantic State University.




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Date Created: 10/03/15
Notes follow and parts taken from College Physics Wilson amp Buftal and Physics 16 Edition Cutnell amp Johnson Vibrations amp Waves When a mass executes the same pattern of motion repeatedly like a mass bobbing on a spring or a pendulum swinging back and forth in a clock we call it simple harmonic motion SHM A force that produces simple harmonic motion has a form simpler than anything except a constant force like what gravity is approximately near the Earth s surface where the acceleration is just g and one example ofthis is called Hooke s Law This is the force exerted by an ideal spring 7 it s proportional to the extension or compression of the spring but it s in the opposite direction to re ect the fact that the spring opposes being stretched or compressed We can write it as F3 kx where k is the spring constant which we ve seen before and x is the amount by which the string is stretched or compressed relative to its rest length which we can write as x0 Sometimes you ll see this stated explicitly when the formula is written as FS k xxO The reason this force causes oscillations is that when the spring is extended and released the force above acts on the mass to accelerate it back towards the equilibrium position The force gets weaker but is still in the same direction as the mass approaches its original position When it reaches its initial position the force is zero but the mass now has some velocity and therefore shoots past the position x0 Once it gets past the original point now on the other side of it the force acts to drag it backwards The force will eventually slow it down until it stops and bring it back towards the equilibrium position again If there were no friction this would continue forever It will be convenient to describe the motion of the mass in terms of a few parameters The position of the mass at any time is called its displacement this is one of the first vectors we ever saw and the maximum of its displacement is called its amplitude A It s a scalar and it just represents the distance between the equilibrium position and the furthest extension of the spring As the mass bobs back amp forth the time for it to make one complete cycle maximum positive displacement to equilibrium to maximum negative displacement to equilibrium and finally back to maximum positive displacement is called the period T Amplitude is measured in meters and period is measured in seconds as you would expect Sometimes the inverse of the period is also a useful thing to know This is called the frequency f or sometimes v and it is the number of cycles completed in one second You ve probably figured out by now that the relationship between frequency and period is just l 1 We know from long ago that a stretched or compressed spring has a potential energy which is equal to Ulkx2 2 where x is still the extensioncompression from the rest length The total energy will be the potential energy plus the kinetic energy remember that the mass is ying back and forth which we can write as Ezlmv2 lkx2 2 2 This is of course conserved in the absence of friction The energy just moves back amp forth between kinetic and potential terms never changing its total value We can use this to find the total value of the energy 7 we know that when the displacement is equal to the amplitude the mass is as far out as it s going to go by the definition of amplitude and is going to turn around Its velocity is momentarily zero so at that point the energy can be written as l 2 E 7k A 2 Knowing that this is the total energy we can substitute it in the previous equation and find the velocity of the mass at any pointx along its path We can write lkA2 imvzlkx2 gt vzi 2 2 2 We expect that v should depend on all of these things 7 the mass will certainly go faster if you extend the spring more initially or if it is small or if the spring is very stiff so that k is large The thing to notice here is that v is also a function of x which is changing constantly That also makes sense 7 the mass is stopped at the endpoints where xA and it is moving the fastest at equilibrium because it s been accelerated in the same direction for a long time At equilibrium where x 0 the velocity is k Vmax i A 7 m Equations of Motion A helpful fact about simple harmonic motion can be found by looking at circular motion If an object is moving in a circle and it casts a shadow in the same plane its shadow moves back and forth along a line in simple harmonic motion The shadow will move back and forth along a line in simple harmonic motion If we call the position along this line x and the circle s radius is A the two are connected by xACOs6 If the angular velocity is held constant we know from our earlier work that 9 n twhere n is the angular velocity and tis the time Where we called this angular velocity earlier we can also call it the angular frequency since it represents the number of radians per second the object moves through Since there are 2 11 radians in a circle the number of cycles per second f the previous definition of frequency is related to the angular frequency by oo275f We can also write this in terms of the period T rather than the frequency If we do that we get 2 xACOs it T You can see that this motion will repeat every time t is a wholenumber multiple of T that is every period We can find the period for simple harmonic motion by taking advantage of this circular reference system For the object moving in a circle the period is just the distance traveled divided by the velocity in this case the velocity of the object in the circle is at all times equal to the maximum velocity of the shadow which is in simple ha1monic motion The reason the speed changes for the shadow but not the real object is that the component of velocity in the x direction changes as the ball moves around the circle The magnitude of its velocity is the same but the shadow only reveals the x component which grows and shrinks We know the maximum velocity for something in SHM and the distance around the circle is obviously just 2 nA so we get T 275A 27 m A km I for the period An important result here is that the period depends only on the stiffness of the spring and the object s mass It does not depend on the amplitude so pulling the mass very far from equilibrium and letting it go will produce oscillations with the same period as a small displacement from equilibrium this is approximately true This is why a pendulum clock works so well 7 it does not depend on the pendulum moving exactly x centimeters from left to right every time A pendulum works in a way very similar to the mass on a spring When it is displaced from equilibrium there is a force which tries to move it back to its resting position When the pendulum returns to that position it has a velocity which carries it past rest and to the other side where the force acts on it again to drag it back down See the diagram below for the explanation of this Resting position Restoring force is mg Sin 0 In SHM the restoring force is proportional to the displacement Here the restoring force is proportional to the Sine of the angular displacement This means the equation of motion what is the angular displacement at a given time t after releasing the pendulum is in general very complicated and not solvable analytically we can t put it into a simple form Ifthe size of the angular displacement is small in radians 7 that means it s less than about 1520 the Sine of the angle is very close to the angle itself this only works in radians For example if we look at a displacement of 57 which is 01 radians we see that the Sine of 01 radians is 00998 or approximately 0 1 If we make the approximation known as the small angle approximation that the Sine of the angle is equal to the angle we get a restoring force approximately equal to m g 0 and we re back to a restoring force proportional to the displacement The formula which gives the position of a mass in SHM as a function of time uses the Cosine function to connect time and position In fact we could just as easily use Sine instead The initial conditions of the problem determine which one we should use For example if we use the Cosine function that says that the x position at time F0 is A since the Cosine of zero is 1 If the mass is really at x A at time F0 we need to add an angle to the Cosine argument to show that We would write instead of the formula above 2 I xACOs E180 j The 180 angle here is called the phase constant or phase shift This is actually very much like the constants we saw in the kinematic equations For example one of those equations was x x0 Vt In this equation x0 is a constant which depends on where you start measuring distances If we start time when the mass is passing through the point x 0 we need for the formula to re ect that We could either use Sine since Sine of zero is zero or we could do the equivalent thing which is to make the phase shift 90 You can check for yourselfthat Sin 9 is equal to Cos9 90 We ll always get oscillation at the same frequency so the period is also the same but the position at time t depends on where the mass is when we start the clock The phase shift sometimes we use a Greek delta 5 to represent phase shift stands for the choice we made We can find the velocity and acceleration by looking at the formulas we already have for v and x If the phase shift in our problem is zero we get 2 x2gt xA2 A2Coszool m 4 l we can simplify this more and get vzf A1 C0s203t fASinoat gtoa ASinoat m m taking advantage of the fact that D 4 k m Of course if we have a phase shift that makes us write x as a Sine instead of Cosine we ll get a Cosine term for the velocity This all boils down to the relationship between Sine and Cosine If we look at the slope of a Sine curve different everywhere because it s curved and plot the slope as it changes it will form a picture of a Cosine curve If we plot the slope of a Cosine curve everywhere we ll get the negative of a Sine curve The slope of a negative Sine curve is a negative Cosine curve and the slope of a negative Cosine curve is a Sine curve again Remember that the slope of a position vs time curve is the velocity at that time The acceleration is easier to find since it s just the force divided by the mass Remember though that since the position is changing in time so is the force and therefore the acceleration so we ll see a t in our formula for acceleration We know that the force has the form F kx so we get FkxkAC0soo l 032AC0soo I m m m This shows us that the acceleration is exactly out of phase with the displacement This is what we expect 7 if the displacement is positive it will take a negative acceleration to bring the mass back to equilibrium This oscillation back and forth could theoretically continue forever In reality though there is always some energy loss with each cycle This energy loss is known as damping and it gives us damped harmonic motion instead of pure SHM Frequently the damping is small and we can ignore it for our purposes Including it makes the equations significantly more complicated We can correct for this effect by adding energy from an external source This is why a grandfather clock has hanging weights As they drop they return lost energy to the pendulum to keep it sw1ng1ng For some applications damping is desirable In your car you have springs to soften the ride when your car hits a pothole If you didn t have big dampers called shock absorbers your car would keep bouncing for quite some time and you wouldn t like the feel of it at all One of the tests for bad shock absorbers is to push each corner of your car up amp down a few times so that you get a large amplitude of motion and then stop The car should quit bouncing very quickly 7 usually within one cycle Solids amp Fluids Solids amp Moduli When a force is applied to a body it can be expected to deform by some amount We can talk about the forces involved and the body s response to these forces by defining stress and strain Stress is just the force per unit area on a body Force divided by area is also called pressure The SI units of pressure are Nmz or pascals abbreviated as Pa For reference atmospheric pressure is about 100000 pascals F stress 2 7 A If an object is compressed compressional stress or extended tensile stress it will change its length in that direction by some amount It s reasonable to expect that this amount will depend on the unstressed length of the object Imagine a bungee cord which has a rest length of 50 m Under the application of a particular stress it stretches to 60 m We would expect this stretching to be distributed evenly along the cord so that the top half stretched from 25 m to 30 m and the bottom half did the same For this reason we talk about the fractional change in length or strain AL strain 2 7 0 In our example above the strain would be 10 m 50 m or 02 Notice that there are no units for strain The amount of strain we get in a material depends on the amount of stress we apply not a big surprise 7 if you re stretching something it will stretch more if you pull harder In general the relation between stress and strain is not simple but if we limit ourselves to small stresses we ll find that they are approximately linearly proportional The constant of proportionality is called the elastic modulus and we can write the formula as stress Elastic Modulus 7 stram Since strain is unitless the units of elastic modulus are the same as the units of stress The three kinds of modulus we ll look at are those describing how length changes Young s modulus shape changes shear modulus or volume changes bulk modulus Young s modulus symbol Y can be found from stress FA F L0 7 gt 7 Modulus strain AL L0 AL A Basically a large value of Y represents a material that can be pulledpushed by a large stress without changing its length very much We expect metals to have large values of Young s modulus The shear modulus symbol S is a measure of how an object s shape changes under the application of oppositelydirected forces which don t act along the same line and therefore produce a torque A shear will distort a cube so that the face becomes a parallelogram instead of a square The cube only front and top faces drawn is distorted by the application of the forces and the amount of the distortion can be described by the ratio of x to h which approximates the angle I by which the cube s sides have been distorted Really we want where tan I 9W1 but because tan I I for small angles we canjust use 9W1 Our formula is then SFAFAFh p xh xA Finally an object may be compressed which will change its volume The relevant quantity here is called the bulk modulus symbol B which relates to the other quantities like this FA Ap V0Ap AVV0 AVV0 AV This quantity is the only modulus we ve seen that applies to liquids and gases which don t support shear tension or compression along a line Solids liquids and gases all resist compression to some degree We can also talk about compressibility instead of the bulk modulus but they are just reciprocals of one another If B Fluids amp Pressure For a uid we can only apply a force over some area it s hard to push water around with a needle We re then back to talking about force per unit area which is pressure just like stress Only the component of the force which is normal to the surface will increase the pressure 7 any component parallel to the area will do nothing to squeeze the uid We could then write iFcosG A A We re able to sense relatively small differences in pressure 7 when we drive up even a small mountain our ears tell us something s happening Similarly if you dive to the bottom of a pool you ll notice a pressure difference in your ears When we re talking about air the dependence of air pressure on altitude is more complicated than in the case of liquids For the atmosphere it s an exponential dependence which basically means that most of the Earth s air is close to the surface We will assume liquids are incompressible This isn t a bad approximation for most liquids but it s horrible for gases and we can t make it there Gases are very compressible 7 that s part ofthe definition of a gas In reality liquids can be compressed by a very small amount under high pressures but it s not generally significant for us This means that we ll consider the density of the liquids to constant at all pressures The source of atmospheric pressure or the pressure at the bottom of the ocean is just gravity pulling on the material Since pressure is forcearea and force is massgravity we can solve for the pressure at a given depth just by knowing the density of the material F mg pVg pAhg 277 h p A A A A pg where we ve used the fact that the volume of the liquid sitting on our areaA is just equal to that area A times the height h of the column of uid If we re on Earth s surface and if we re not that g doesn t belong there we also have to add the atmospheric pressure to this because it s all around us In other words the pressure at the surface of the liquid isn t zero it s just normal atmospheric pressure We can rewrite our formula as ppopgh where p0 is atmospheric pressure at the Earth s surface about 101325 Pa or 147 lbsin2 One of the interesting things about a uid is that pressure applied to part of it is transmitted throughout the uid This m n0t mean that pressure is a constant throughout a liquid We still have the changes with depth shown above It does mean though that if we magically doubled Earth s atmospheric pressure that change by p0 would be felt at the top bottom sides and all other points of the liquid This idea is behind hydraulic assist devices of all kinds like your power brakes garage car lifts the jaws of life etc Basically we can use the fact thatFA is constant except for height variations to turn a small force on a small area into a large force on a large area We can take a sealed tube of uid with a small piston on one end and a large piston on the other We push lightly on the small piston and the large piston pushes forcefully as shown below m Small force on small area gives pressure p Pressure p acting on much larger area produces much larger force Since the pressures are equal we see that the force on the big piston is equal to the ratio of the areas multiplied by the input force Ermall Fbig Abig TZPZT Fbig A Ermall small big small Are we really getting something for free here No We re multiplying the force but there s no way that a tube of water can do any work on its own The work done by the large end will be the same as the work done on the small end The reason is that Work Force distance Let s see what a worked example looks like Assume the area of the small piston is 01 m2 and the area of the large piston is 1 m2 Ifthe force on the small piston is 50 N the formula above shows us that the force the large piston can exert is just 1 m2 01 m2 50 N 500N The force does get multiplied How about the work done at the small end If the small piston moves forward by 02 meters the work done on it will be 50 N02 m 10 J That movement will push a volume of uid out of the small piston s cylinder into the larger part of the tube The volume moved is just V A d and we knowA and d already so we get V 01 m2 02m 002 m3 That same volume being pushed into the big tube is what pushes the larger cylinder out of the way again assuming the uid is incompressible How far does the big piston move We use the same volume formula to see what distance it has to travel to make room for this 002 m3 of water Now rearrange V A dto get d VA 002 m3 1 m2 giving a displacement of 002 m for the large piston How much work will it do Just use WFd and get 500 N002 m 10 J As we expected we re multiplying force but we re not multiplying work which would require an external source of energy It s the same idea as a lever 7 you re not doing any more work you re just trading off between force and distance If you re willing to move the long end of the lever a great deal a small applied force becomes a huge force but only over a small distance When we talk about pressure we need to specify whether we re measuring absolute pressure or gauge pressure Absolute pressure would be the pressure as compared to a vacuum In the formula above p p0 pgh p is the absolute pressure and p0 is the atmospheric pressure Gauge pressure would be the p g h part of the pressure 7 it s the absolute pressure minus the atmospheric pressure pabsolute Zpatmospheric pgauge At sea level on the Earth absolute pressure should be about 101000 Pa above gauge pressure Archimedes Principle amp Buoyancy Why do some things oat in water while other things sink We know there must be more to it than weight 7 an iceberg can weigh millions of tons or more amp still oat but a penny will sink There s also more to it than just material 7 large ships are made of steel in some cases over a foot thick and they oat Obviously there must be an upward force that counteracts the downward force of gravity That force is known as the buoyant force We can find it from what we know so far Assume we have a cube with faces of areaA submerged in water The force trying to push it to the surface must be an imbalance in the force on the top face and the bottom face we know this because it moves vertically 7 we don t drop blocks in the water and see them move horizontally Also if we look at the symmetry of the problem where would any difference in the forces on the faces come from How could a larger or smaller force know what side it was supposed to be associated with The forces on the sides must balance each other Let s look at what s left ppgh pipth The pressure on the top face is p p g h while the pressure on the bottom face is p2 p g 112 These pressures are different and pressure is just force divided by area The areas are the same so the forces acting on the faces are different The net force is F 2 7F 1 notice that F 2 is directed upward and F1 is directed downward 37F2 E7pgh2A pgaApgh hA Notice though that 112 7 h1A is just the volume of the block Also we re multiplying that volume by the density of the water We could then restate this in terms of the mass of the water that would fill the same volume as the block known as the displaced volume Fnet p g de39splaced m uid displaced g W uid displaced The net upward force is then just equal to the weight of the uid that s displaced If the cube has a side length of 10 cm its volume is then just 10 cm 10 cm 10 cm 1000 cm3 1 liter or 0001 m3 Water has a density of 1 g cm3 so that means we re displacing 1000 g or 1 kg of water 1 kg of water has a weight of 98 N so the upward force on the cube is 98 N If the weight of the cube is less than 98 N it will bob to the surface If it s more than that it will sink to the bottom if released Assume the cube has a mass of 025 kg so its weight is 245 N How far out ofthe water will the cube oat It can t go all the way to the top and come completely out of the water because the buoyant force there is essentially zero not exactly zero because air is a uid also We don t usually see things oat in air because it s around 1000 times less dense than water There s a buoyant force due to the air on everything including you but it s generally so small as to not be noticeable Gravity would pull it back down into the water The balancing point occurs when the buoyant force is equal to the object s weight The volume of water which would have a mass of 245 N is 245 N 98 N 1000 cm3 This would be 250 cm3 Ifthe cube s face area is 100 cmz this means the lower 25 cm of the cube is in the water and the rest of it is out of the water We see that average density is the important thing here Steel ships oat because their mass divided by the volume of the whole ship is less than the density of water Incidentally we can talk about densities by using something called speci c gravity This is essentially just the density of something usually a liquid compared to water If the speci c gravity of something is 135 my high 7 this is actually mercury it s 135 times as dense as water so one cm3 of it would have a mass of 135 g Mercury would definitely sink in water Incidentally this is the idea behind antifreeze testers which are basically fat eyedroppers with lots of little spheres in them Each sphere has a different density and as the specific gravity of your coolant increases more of them will oat The specific gravity of the stuff in your radiator depends on the percentages of water and coolant in the radiator If it s all water the antifreeze tester will show that your coolant will freeze around 0 C A 5050 miX is ideal and should give you protection down to 410 C Fluid Dynamics amp Bemoulli s Equation We can use the principles of conservation of energy and conservation of mass to discover the way uids behave in motion This is in general a horribly complicated subject but if we make some assumptions we can greatly simplify it We will therefore consider the dynamics of an ideal uid sometimes jokingly called dry water We assume that the ow of the uid is steady which means all particles of the uid have the same velocity as they pass any given point This approximation is OK for low velocities Think of the ow of a slow stream If the velocity is large like rapids the ow becomes turbulent The ow should also be irrotational meaning there are no whirlpools in the stream Also if we dropped a small paddlewheel in the stream it wouldn t rotate as it went downstream We assume the uid is nonviscous meaning there s no internal friction Syrup and peanut butter are very viscous and water is not very viscous The realworld effect of viscosity is to slow the uid when it gets near the walls of the pipe since it s stopped Finally we assume the uid is incompressible meaning its density doesn t change We can use the conservation of mass to determine how the uid speeds up or slows down in response to changes in the size of the pipe carrying it We call the input end of the pipe 1 and the output end 2 The mass going into end 1 in a short time At depends on the pipe s size and the uid s velocity amp density We can write this mass entering the tube in that time as Ami 2 p1 AVI 2 p1 A1V1 At At end 2 the mass leaving in time At is Amz 92 AVz 2 92 A2 V2 At The mass added at end 1 in time At has to be the same as the mass leaving end 2 in time At Since the masses are equal we can set these two equations equal to one another We can also get rid of At which is the same on both sides Since we ve assumed there can be no change in density p1 and p2 are the same and can be cancelled We re left with the ow rate equation which says A1V12A2 V2 For example ifthe area of the pipe at point 1 is 025 m2 and the uid velocity is 10 ms there and the pipe shrinks at point 2 down to an area of 005 ml the uid s velocity at that point is 025 m2 005 ml 10 ms 50 ms This effect is the reason why you use your thumb on the end ofa garden hose to get a faster stream of water The water leaves at a higher velocity but the volume delivered per minute doesn t change You won t ll a bucket any faster with your thumb over the end of the hose We can now use the conservation of energy to derive Bernoulli s equation which describes the ow of uids in a gravitational potential If we look at the work done onby the uid at each end of the pipe the difference will give us the net work WZFI x1F2 x2 p1A1V1tp2A2V2t By the ow equation A M A 2V2 The time is of course the same in both terms These three terms multiplied together are just the total volume which we can also write as the mass divided by the density We get A W p1p2 where Am is just the mass involved This net work is the same as the total of the changes in the kinetic and potential energies The kinetic energy change is as we ve seen before AKziAm v22 V12 while the change in potential energy of the gravitational eld is AUAmgy2 y1 Equating W with the sum of these two terms we get Am 1 p1p2EAmV22 V12Amgy2 y1 We get rid of the Am and move p over to get Bemoulli s equation 1 1 p1Epvfpgy1p2Epv pgy2 We haven t said points 1 and 2 have to be anywhere in particular so this must be a constant everywhere in the pipe 1 p5pv2pgyC0nsz The units on each term would be energy except we have p instead of In so we really have energyvolume or Jm3 Under our original four assumptions if we know the value of this constant somewhere we know it everywhere in the uid A common example of this is to nd the speed of water exiting a hole at the bottom of a bucket For this purpose we ll assume that the atmospheric pressure is the same at the top of the bucket and the hole a very good approximation for a real bucket so p1 p2 The surface ofthe water will drop with a velocity we ll call v2 and we ll call the unknown velocity of the water leaving the hole V For this part we can use the ow rate equation to prove thatA 1v A 2V2 where A 1 is the area of the hole and A 2 is the area of the bucket s open top For a small hole A 2 is much greater than A 1 The ow rate equation tells us that this means v2 must be much smaller than V We can therefore make the approximation that v2 0 which basically states that the lowering of the water level happens so slowly that the waterm the bucket doesn t have any signi cant kinetic energy of course it s a different story outside the hole Taking all of this into account and representing the height of the water s surface by y2 and the height ofthe hole as y we then cancel p and get V1222gy2y1 3 V1 2gy2y1 The picture below illustrates this tn Temperature We sometimes think of heat and temperature as interchangeable but they re different things Heat is energy which is transferred from an object with a higher temperature to an object with a lower temperature The microscopic de nition of temperature is the average random translational kinetic energy of the molecules making up the substance This is distinct from the total internal energy of an object because there are other internal motions the molecules can have which require energy but don t contribute to temperature Other terms we ll need to know include thermal contact two systems exchanging heat energy are in thermal contact and thermal equilibrium two systems with the same temperature will no longer have a net heat exchange this is thermal equilibrium Celsius Fahrenheit ampKelvin The two temperature scales we re most familiar with are Celsius C and Fahrenheit F Zero on the Fahrenheit scale was set to be the lowest temperature Daniel Fahrenheit could create with an ice and salt mixture He also called body temperature 96 F not 986 and was the first to use a glass tube filled with mercury as a thermometer The most common and oldest thermometer design relies on the expansion of materials as their temperature increases and contraction as they cool When the mercury in a thermometer heats up amp expands it climbs the narrow glass tube holding it Thermostat switches are frequently made of a bimetallic strip This works because different materials have different coef cients of thermal expansion A strip which is straight at room temperature will bend if it is heated or cooled This bending is frequently used to make or break electrical contact Two reference points are needed for calibration of the thermometer this assumes a linear expansion with temperature which is a good approximation for most of our needs The Celsius scale was developed by dividing the liquid range of water into 100 increments 0 C is ice 100 C is boiling water both at regular atmospheric pressure Since we know the location of these same points on the Fahrenheit scale we can find the conversion from one to the other FC32 or CF 32 Fahrenheit amp Celsius are familiar to us but a more useful temperature scale for us is the Kelvin scale which is absolute It s called absolute because it stops at zero You may have heard weather people say something like It s going to be 80 F today twice as hot as it was last night when it was 40 F 80 F is n0ttwice as hot as 40 F One way you can verify that is to imagine lower temperatures If it was twice as hot it would be 80 times hotter than l F but how much hotter than 0 F or l F The Kelvin scale provides an absolute way to talk about temperatures because 80 K really is twice as hot as 40 K There s no problem with 0 K or negative temperatures because 0 K is set at absolute zero We can get very close to absolute zero but we can never actually reach it Kelvin degrees are the same size as Celsius degrees 7 the zero points arejust offset The conversion between the two scales is KC27315 gt CK 27315 Thermal Expansion We ve seen that gases expand when heated but the same is true of liquids and solids as well We nd that the amount of expansion is approximately proportional to the change in temperature This means that if an object gets 1 longer after being heated from 300 K to 400 K temperature difference of 100 K it should get 2 longer if it s heated from 300 K to 500 K twice the temperature difference Notice that we give these numbers as percentages The actual size increase in meters for example has to depend on the initial length of the material as well as its composition We can write this formula for expansion in one direction as OLAT 0 where L0 is the rest length AL is the change in length a is the thermal coef cient of linear expansion which depends on the material and AT is the change in temperature since it s the difference between two temperatures and Celsius degrees have the same size as Kelvin degrees it can be in either C or K and we ll get the same answer Typical values of 0c are around 10396 7 10395 C The units have to be inverse degrees so that we get a dimensionless number when we multiply by the change in temperature We can also examine the change in the area or volume of a solid Notice that we can write the formula above as ALzLoa AT L LozLoa AT gtLL0 1a AT Since this change will be happening in both directions for an area or all three for a volume we could just square or cube the right hand side of our final equation Because at is so small though squaring it will give us something that s almost equal to zero What is commonly done in cases like this which are frequent in physics is to take the term linear in ac and drop higherorder terms like at 2 or at 3 etc This is known as keeping terms to order at Verify for yourself that if we square the right hand side of the last equation and drop higherorder terms we will get L2 2L3 12a AT gt AA0 12a AT and using the same arguments for volume we get L24303QAT where V L3 For example if an aluminum 0c 24 X 10396 C cube with sides of length 2 m is heated from 100 K to 500 K what happens to it Well V0 2 m3 8 m3 AT 500 K 7 100 K 400 K so V 8 m3 1 3400 K24 X 10396 C 8 m3 10288 823 m3 Heat Heat which is de ned as energy being transferred from one place to another can be measured using the unit of energy we re used to the Joule Another common unit of heat is the kilocalorie kcal This is the amount of heat needed to raise the temperature of 1 kg of water by 1 C from 145 C to 155 C The calorie is just 1 1000 11 of this amount In the US when talking about the calories in food we re really talking about kilocalories so that a 900 calorie slice of pie really has 900 kilocalories of energy When talking about heating and cooling capacity we in the US still sometimes use the British Thermal Unit or btu This is 252 kilocalories and is de ned as the heat needed to raise the temperature of 1 pound of water by 1 F To connect these ideas to Joules we need to know that 1 calorie not a food calorie is 4186 J a food calorie which 1 kcal is 4186 J This relationship was found by examining the mechanical work which was needed to produce a given amount of heat in water Specific Heat Notice that the definition of these measures of heat has included a material water If every substance behaved the same way we wouldn t bother to specify water All substances don t behave the same way Adding the same heat energy to the same masses of different materials will cause different increases in temperature The temperature increase in some material will depend on 1 the heat added 2 the mass of the substance and 3 the nature of the material a constant which is different for every substance The relation is szcAT where m is the mass of the material AT is the change in temperature Tfmgl 7 Tmnml Q is the heat energy added to the material and c is called the speci c heat of the substance The specific heat tells us something about how energy is distributed in a given material We know from the definitions above that the specific heat of water is 4186 Jkg C For lead on the other hand it s 130 Jkg C This means that while adding 4186 J of heat to 1 kg of water will change its temperature by 1 C it will change the lead s temperature by 322 C This means that water is putting less of the energy into the random translational energies of its particles It therefore has more places to put internal energy more modes than the lead does This means that it is much easier to heat up a lump of lead than water Also it s much harder for a refrigerator to cool 1 kg of 50 C water than it is to cool 1 kg of 50 C lead Phase Changes amp Latent Heat There are four phases of matter three of which you re familiar with Solids retain their shape and volume under a wide range of conditions Liquids retain their volume but not their shape you can pour water from a cylinder into a piggy bank amp it ll take the shape of a pig but it will keep the same volume Gases retain neither shape nor volume they can be put into a container of any shape and can also be compressed to take up a smaller volume Plasma is the fourth state of matter and one we won t study but it s basically a substance in which some or all of the electrons are not attached to particular atoms They roam around inside the material staying within it but not locked down to particular points If enough energy is added to a solid to break the bonds between molecules it will melt and become a liquid If we remove energy from the liquid it will return to the solid state Solids are generally crystalline orderly arrangements of atomsmolecules which repeat themselves Amorphous solids don t have this kind of orderly structure and have a range over which they get progressively more uid If energy is added to a liquid the molecules will be able to increase their distances from one another If the molecules can separate far from each other the substance is now a gas This happens gradually through evaporation the molecules near the surface have enough energy and are traveling in the right direction to escape the surface or rapidly through boiling Dry ice which is frozen C02 doesn t become a liquid on the way to being a gas It undergoes sublimation and moves directly from the solid to the gaseous state It takes a certain amount of energy to make the phase transition from one state to another even if the temperature doesn t change Going from ice at 0 C to water at 0 C actually takes a reasonable amount of energy This is called the latent heat of fusion Going from water at 100 C to steam at 100 C takes an input of energy called the latent heat of vaporization This same heat must be removed from a gas to get it to condense to a liquid or from a liquid back to a solid The expression for the amount of energy necessary is slightly simpler in this case since there is no temperature change exactly at the transition point The heat needed or released still depends on the particular substance involved and the amount of it present We write this as szL where Q is the heat requiredreleased m is the mass of material and L is its latent heat Therefore if we want to know the energy required to turn 1 kg of ice at 710 C into water at 5 C we really have to add three different heat energies QTotal kgcice 00 kgLwaterice cwater 0 Using numbers from the book this would give us 1 kg2100 Jkg C10 C l kg333 X 105 Jkg l kg4186 Jkg C5 C 374930 J Notice that in this case almost 90 ofthe energy was used to melt the ice There s usually a tremendous amount of energy involved in a phase change Latent heats of vaporization are also typically about this size Air conditioners are typically designed to take advantage of this by using a coolant which is a gas on the hot side and a liquid on the cold side thereby allowing a small amount of coolant to transfer a large amount of heat energy The pump can change the pressure of the coolant which also changes its boiling point This is also why a steam burn is so serious 7 when the steam condenses on your skin it releases its latent heat of vaporization which is large into your skin The exact temperatures at which substances boil or freeze depends on the atmospheric pressure At high altitude where the pressure is low water boils at a lower temperature This is why some products have highaltitude cooking instructions generally involving more time cooking to make up for the lower temperature if it s in boiling water The change in transition temperatures with pressure is part of what makes ice skating possible Water is unusual in that higher pressure gives it a lower freezing point Evaporation is a method of cooling which takes advantage of the latent heat of something usually water The water doesn t have to be boiling for this to work The fastest molecules near the surface and therefore the hottest ones can leave and become vapor When they do this they take a great deal of heat with them This is how perspiration works to cool you off Of course if the air around you is already saturated with water molecules the air is less likely to pick more of them up from your body That s why humid days in the summer seem so much hotter than dry ones Heat Transfer There are three ways to transport heat from place to place conduction which involves direct contact between materials of different temperatures to exchange energy without any largescale motion of matter convection which moves heat by moving the hot matter from one place to another like boiling water bringing the hot water from the bottom of the pan to the top and radiation which involves photons carrying the heat away Conduction works like a fight or disturbance in a crowded room 7 the jostling of people gets passed through the crowd but individual people can t really move much The hot and therefore fast molecules in one substance slam into their neighbors and give up energy allowing heat to move from the highenergy hot end of something to the lowenergy cold end Different materials have different abilities to move heat this way 7 we distinguish between thermal conductors which are good at moving heat this way copper pans etc and thermal insulators which are not good at moving heat pot holders Generally things with more free electrons like metals are good conductors while things with few free electrons cloth wood glass are good insulators We can reason that the ow of heat through something should depend on its makeup metal wood etc its area a large window makes a room much less efficient at staying warm in winter or cool in summer its thickness Kleenex may be a thermal insulator but you wouldn t use one to grab a pan from a hot oven and the temperature difference across the material we expect heat to ow more quickly through a fork with one end in a fire than if that end is just in wa1m water We can combine all of these ideas to get kAAT At d Here the At is the time over which an amount of heat AQ is transferred A is the surface area of the material AT is the temperature difference d is the thickness of the material and k is a constant dependent upon the material and called the thermal conductivity A little dimensional analysis will quickly show that the units for k must be Jm s C For reference copper has a k of 390 Jm s C while Styrofoam has a k of about 0042 Jm s C That factor of almost 10000 in the thermal conductivities is the reason why cookware is frequently made of copper so as to transfer the heat from the stove to the food more quickly and coffee cups are made of Styrofoam you d rather not have that heat transferred to your fingers Convection is restricted to liquids and gases because it involves movement of the molecules making up a material Basically a hot liquid or gas will expand to fill a greater volume than a cool one will If the same mass now has a greater volume that means its density will be lower than the surrounding material which means its buoyant force will overpower its gravitational weight and it will rise As it rises to the surface it will lose energy to the air Once that energy has been lost the temperature will drop the material will contract and the density will increase causing it to fall back to the bottom This process sets up a convection cycle the overall effect of which is to move heat from the bottom of a material to the top This cycle is found in weather patterns boiling water the surface of the sun and the interior of the Earth In fact the cycle acts to power the conveyorbelt like motion of the Earth s continental crust which causes earthquakes and mountain building If you ve ever heard someone say hot air rises now you know why Radiation is the third method of heat transfer and it does not require an intervening medium All of the heat we get from the sun comes to us via radiation Essentially photons are produced by the hot object and then travel away from it through a vacuum if necessary The lamps that keep food warm in a restaurant or cafeteria work on this principle They emit photons whose wavelengths are larger than red photons and for this reason this end of the spectrum is called the infrared Your remote control puts out this kind of radiation and the army s nightvision goggles receive it We ll see later that hot where hot means above absolute zero radiate a range of wavelengths As the temperature increases the peak or most common wavelength decreases the exact relation is called Wien s Law It s very simple and it just says that the relationship between temperature and this peak wavelength is km T00029mK The 00029 m K is known as Wien s constant This tells us that something at 310 K you radiates more photons with a wavelength of 00029 m K310 K 94 x 10396 m than any other kind If the temperature goes up to 5500 K however the peak wavelength drops to 530 x 10399 m green light That s about where the Sun s surface is Notice that we didn t say anything about the composition of the material 7 we used the same formula for a human body as for a large collection of hydrogen and helium To a very good approximation it doesn t matter A 400 K lump of gold radiates the same kind and number of photons as a 400 K pork chop We call this approximate behavior blackbody radiation If a body was perfectly black perfectly absorbing this is the way it would act Here s the shape of a few blackbody curves Marsz 1 gtlt In x gtlt 1n Wavelengvhru 25gtlt1 7 5gtlt1 7 15mmquot lxl quot 125gtlt1 quot15gtlt1 175gtlt1 quot hm H d Vtufan law b1aekbody as A P Q 6 A e T4 AT 615 ea11ed the Stefmanltlmann eonstant and1s equal to 5 67 n 103 Wrn2 K A 15 the area of one2 1s ea11ed the emissivity of the rnatena1 ranges from 0 to 1 and desenbes how e1ose to a blackbody rt rea11y 1s wrth 21mean1ng atrue blackbodyandT1s the abso1ute ternperature of the object Notree that ternperature enters to the fourth power 7 gomg from 300 K to 600 K doublmg the M 716 urnes1 In that ease 300 K p 100 K eryogenre eoo1erbutrtwr11 absorb rnueh more energy than rtradates 1fwe put rt rnto a 500 K oven netpower 15 then ea11 a n dependent arnount of energy regard1ess of1ts sunoundmgs However 1f1t39s gomg to be m raddatmg For that reason a good absorber rnust be a good ernrtter Gas Laws amp Absolute Temperature Gas thermometers are potentially easier to make than liquid ones because all gases behave about the same way at low densities Two of the earlier observations about the properties of gases are Boyle s Law p1 V1 p2 V2 gt pV Const af xed temp where p is the pressure of the gas and Vis its volume Of course we have to keep the amount of gas constant for this to hold we can measure this by mass or by number of atomsmolecules Essentially this just says that if we have this gas in an expandable container like a bottle with a sliding piston at the top we ll see the pressure drop if we increase the volume of the container If we decrease the volume the pressure rises There s also a relationship between the volume occupied by a gas and its absolute temperature measured on the Kelvin scale if the pressure amp amount of the gas are held xed This is called Charles law and it is written V V V 71 72 gt 7 C0nst at xed pressure T1 T T For lowdensity gases we can combine the fact that pVis xed with the fact that VT is xed and get the ideal gas law actually pretty good for any lowdensity gas which is pTVz C onst or speci cally pTV N k B The constant here N kB is a measure of how much gas is actually there The N tells you how many molecules of gas are present If you remember from chemistry you can go from the mass of a gas to the number of atomsmolecules of that gas if you know its atomicmolecular weight When the number of grams of a gas present is the same as its numerical atomicmolecular weight you have one mole of that gas A mole is just a shorthand way of talking about a number like a dozen Instead of 12 of something though a mole of something is 602 x 1023 of it You probably remember hearing about this 7 it s called Avogadro s number As an example let s say we re talking about carbon dioxide C02 The molecular weight of C02 is just the atomic weight of carbon 12 plus 2atomic weight of oxygen 16 44 If you have 44 grams of C02 that s one mole of it or 602 x 1023 molecules of it The k3 is just a constant which lets us use metric units for everything It s called Boltzmann s constant and its value is 138 x 103923 JK remember that K is for Kelvin You may have noticed that Boltzmann s constant is about as small as Avogadro s number is large that is if we multiplied them together we d get a number around one If we actually multiply them together we get something called the universal gas constant written as R which has a value of 831 Jmol K We can then rewrite the ideal gas law in what is sometimes called its Macroscopic form pVnRT The new thing here is the n which now represents the number of moles of the gas ie 11 g of C02 is 1144 1A of a mole so 71 025 We can solve this equation for the volume occupied by one mole of a gas any gas at 0 C 27315 K 7 we have to convert and normal atmospheric pressure known as STP for standard temperature amp pressure and we get 224 liters Under STP conditions if you have a 224 liter container of any gas you also have one mole of that gas Avogadro s number of molecules of that gas Notice that if we put conditions on the ideal gas law we can recover Charles law or Boyle s law If we keep temperature and the amount of a gas xed we get since R is of course xed ini p2V2 If we instead keep the pressure fixed and let the temperature change we get by putting the changing things on the same side lJTJ V275 We can use these laws to see why things like overheated radiators or malfunctioning pressure cookers are dangerous As you raise the temperature in a closed container xed volume V you will also raise the pressure Something will eventually have to give or you ve literally got a bomb If you open the cap on the radiator abruptly the pressure will drop and the volume of the gas superheated steam in this case will increase dramatically If you re in the way you ll get covered by the steam and severely burned The ideal gas law provides a way to make a very accurate thermometer If we put a gas in a xed volume sealed container and measure its pressure it should be proportional to the absolute temperature This is called a constant volume gas thermometer Kinetic Theog of Gases If we want to use the equations of mechanics we ve found so far to study the molecules in a gas we need to use statistical methods There are too many molecules in any reasonable sample of gas to draw the freebody diagrams To simplify things we will for now ignore the fact that the gas molecules could have internal energy associated with a Vibration or rotation We can do this by concentrating on monatomic gases These are gases made of single atoms 7 helium would be a good example because it doesn t form molecules with anything under any reasonable conditions even itself We will use the kinetic theory of gases to gure out what s going on This says that collisions between gas molecules or atoms in this case or between molecules and the walls of the container are perfectly elastic Also we ll assume the molecules don t interact with one another except during collisions Through a mildly complicated derivation we won t worry about here we nd that we can connect the pressure in and volume of the container with the mass amount and average speed of the gas The connection is pVivafm In this formula p and Vare the pressure and volume N is the number of gas molecules m is the mass of a molecule and vm is the root mean square speed of the atoms This is basically an average speed 7 we square all speeds to get rid of negative signs average them and then take the square root to get back to something with dimensions of speed If we go back to the ideal gas law microscopic version we have You may recognize the left side of the second equation above as being the kinetic energy average Since the only nonconstant thing on the right side is T this tells us that the temperature is a measure of the average random kinetic energy of the molecules in the gas Next semester you ll see that this formula falls apart at absolute zero where it would predict motionless atoms Quantum mechanics won t allow that but that s not generally something we need to worry about Because this is a monatomic gas there is no storehouse of internal energy other than kinetic energy If the temperature absolute temperature goes up by 50 the internal energy of the gas goes up by 50 The formula above connecting temperature and kinetic energy gives an interesting result if we compare two different kinds of gases which are at the same temperature According to our formula they have the same kinetic energy at the same temperature but if the masses are different the speeds will be different We can find the average speed of the atoms in a helium balloon at room temperature about 300 K All we need to know is the mass of a helium atom and since its atomic weight is 4 that means that Avogadro s number of helium atoms has a mass of 4 grams The mass of an individual atom therefore is about 7 x 103927 kg Plugging in the numbers gives us vms of about 1300 ms around 4 times the speed of sound Keep in mind that this is an average only and that an individual atom just bounces around from collision to collision very very quickly Anyway compare this to the speed of a Radon atom at 300 K The atomic weight of Radon is 222 so an atom ofthat has a mass of37 x 1039 kg We ll get a speed of about 180 ms This speed difference affects diffusion or the rate at which something spreads out If you ve ever dropped food coloring or milk in water you ve seen diffusion The time between opening a container of something like perfume and smelling it is the time spent waiting for diffusion to carry it to your nose During World War II diffusion was used to separate Uranium 235 from a mixture of U235 and U238 U235 is lt l of all uranium The uranium in the form of uranium hexa uoride was heated until it became a gas Some of the molecules had 6 F atoms surrounding a U atom but most 99 were 6 F atoms around a P38 atom If you look at the differences in molecular weight it s 2356 l9 349 compared to 2386 l9 352 This means you can verify this the molecules with the U235 move less than 12 faster than the molecules made with If This process had to be repeated many many times before enough P35 was collected to make a bomb This was and still is one of the biggest challenges in building the atomic bomb Diatomic Gases For diatomic gases the molecules can have internal energy other than kinetic energy The two atoms are connected by a force that acts like a spring so they can oscillate back amp forth vibration This provides a place for the gas to store energy which does not affect its temperature This dumbbelllike structure can also rotate around any of three perpendicular axes Along an axis


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