PRINCIPLES OF PHYSICS I
PRINCIPLES OF PHYSICS I PHYS 2211
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Date Created: 10/03/15
PHYS 2211 Notes follow and parts taken from Fundamentals at Physics 17 Edition Halliday ResnickI amp Walker College Physics Wilson amp Bn al and Physics 16 EditionI Cntnell amp Johnson Units of Measurement One of the hardest things for new physics students to get used to is the fact that most physics professors seem to be obsessed with units of measurement The reason for this is simple if you don t mention the units you re using your statements will usually have no content For example if a friend is traveling around the world and tells you that his hotel room was 4000 that means nothing If it was 4000 US dollars it must have been a nice room If it s 4000 lira or 4000 yen it might have been a phone booth There are no physically important quantities which can t be measured somehow except for things like 7 and other constants from mathematics rather than physics itself and if they can be measured they have units of measure The basic metric units are meters length kilograms mass and seconds time Various pre xes attached to these units provide a shorthand way to create new units which are more appropriate to a given situation When we discuss the size of light waves we ll use nanometers nano one billionth If we re talking about the distance from the Earth to the Sun gigameters giga one billion would be more appropriate Here are some of the most common prefixes in the metric system and the values they correspond to These prefixes are among the very few constants you ll be expected to memorize Dimensional Analysis Units of measurement are also involved in dimensional analysis Dimensional analysis is basically a way to look at your answer and see if it makes sense and it can sometimes tell you exactly what piece you ve left out For example if we re talking about a velocity we know that it must be expressed as Length Time We re used to seeing things like miles per hour for this but the US is essentially alone in sticking to the English system of units In the metric system velocity would be meters second Areas would be measured in square meters area of a square side lengthz area of a circle furl etc If you ve ever bought carpet you probably bought it in square yards instead Volumes are measured in cubic meters very large 7 dumpsters may be measured this way or liters much smaller 7 a cube with a side length of 10 cm PHYS 2211 As an example of how this can help you you may do a lab experiment which involves hanging weights from a string and then shaking it back amp forth to see how quickly waves travel down it You ll see that the velocity ms is directly related to the square root of the tension in the string We ll worry about what tension is later but for now all you need to know is that the units of tension can be written as kilograms meters secondsz Just by looking at this you know something must be missing as you can see below k km m gmxivgim ss2 s s We can look at this and see that we have a factor of kg that we don t need and we still need one more factor of x m to get meters per second What we ll see later in the semester is that the missing piece is the mass per unit length of the string ie is this dental oss or heavy rope Can you guess where that would have to be in the formula to make the units work out Unit Conversions Frequently we need to go back and forth between different units when doing a problem There are a few things to keep in mind First remember that we can only convert between units measuring the same thing 7 there s no way to convert from meters to seconds for example Also remember that the only thing we can ever do to an isolated quantity 220 miles to Atlanta for example is multiply by one That sounds pointless but in fact the whole idea of unit conversion lies in the number of ways you can write one For example 1 mile is equal to 1609 kilometers Below we find the distance to Atlanta in km 1609km 220 miles X 1 354km Notice how we wrote one as 1609 km over 1 mile That quantity is in fact equal to one just like 12 inches is equal to one foot How did we know not to write it as 1 mile over 1609 km That would still be one right Look below at the difference Z M354km but 220milesXMl37m mile 1609km km 220 miles gtlt Written the first way the miles which we want to get rid of go away and leave only km behind The second way we get miles squared as well as km on the bottom That s why we always write our one in such a way as to get rid of what we want gone and replace it with what we want Significant Figures It s important to keep an eye on the number of figures to record after a calculation If you re trying to decide how to divide a kilogram of sand into 3 piles you can use your calculator to find PHYS 2211 out how much to put in each pile hopefully you don t need to but we ll do it anyway If you put 1 3 into your calculator you ll get back an answer of 03333333 depending on how many digits your calculator reports Ifyou look at this for a minute you can see that the output has many more digits than the input Ifyou measured the sand carefully maybe you know that it was 1000 grams Maybe you went to a chemistry lab to borrow a nice scale and you ve found that it s 1000000 grams Writing it this way indicates that you re sure of the mass down to the last figure shown the final 0 in 000 This says you know the answer to within one milligram because if you measured your sand and got 1000001 grams that would be one milligram of difference Ifyou only know the mass of the sand to within a milligram you can t then write down that each pile will contain 1000000 03333333 33333333 grams because the final 3 here would represent 30 micrograms and your scale is not that accurate In other words the number of significant digits in your final answer is limited by the number of significant digits in your input If you know the total mass to within a milligram you ll only know the mass of each pile to within a milligram at best You may have noticed referees in football ignoring this fact for the first three downs the referees make an estimate with their eyes about where the ball should be placed On the fourth down the chains come out and the position of the ball is measured to within a fraction of an inch It makes no sense to attach a very precise measurement to several other rough estimates As another example police frequently make damage estimates to the cars involved in a wreck If the estimate for a car s repairs is 4000 does it really matter if the officer didn t notice that your air freshener was damaged in the crash Should he change his estimate to 400098 There are some relatively simple rules for deciding about the number of digits to keep during a calculation If you are multiplying or dividing the answer can only have the same number of significant figures as the leastprecise number involved For example if you re finding the area of this room and you measure it to be 121 meters in one direction and 15255 meters in the other direction your calculator will claim that the area is 1845855 m2 However you can only keep 3 digits because your first measurement only had that many The fourth digit tells you how you ll record the other three In this case the 4 11 digit is a 5 which means we ll round the 3 d digit up Your answer should be 185 m2 Remember that this happens because of your uncertainty in the 121 m measurement It could really be 1205 m or 121499 m or anything in between The thing to remember is that the calculator is not smart 7 it has no ability to evaluate and make conclusions 7 it s just there to speed number crunching When you re talking about addition and subtraction the number should have the same number of decimal places not necessarily the same as number of figures as the leastprecise number For example let s say I want to know how far the top of a 40story building is from the center of the Earth The book says it s 6378 km from the center of the Earth to the surface A 40story building is approximately 120 m tall If I try to add these together Ineed to put them in the same units 6378 km 0120 km 6378 has the fewest number of decimal places zero so we have to limit the answer to zero decimal places whole number That means we round 0120 km to 0 km In other words if the only data we have is the back of the book we don t know the distance to the center of the Earth well enough to make the distinction between the ground oor amp penthouse of this building We PHYS 2211 can put the numbers in the calculator and it will happily produce 637812 for the answer but the answer can t be trusted Chapter 2 Kinematics Distance and Speed vs Displacement and Velocim Most people feel comfortable with the ideas of distance and speed Distance just talks about the separation between two things and speed just describes how fast we can get from one thing to another Both of these quantities are known as scalars Scalars are objects without direction your car s odometer measures distance miles since it came from the factory and its speedometer measures speed how fast are you currently going Displacement includes directional information For example if you were to drive your car back to the factory its total displacement since it was built would be 0 km You might have gone through 3 sets of tires and an engine rebuild but its displacement is zero Quantities with both magnitude and direction are called vectors and they re drawn as arrows The pointy part of the arrow indicates the direction the quantity is going To find displacement put the tail of the arrow at the initial point and draw a line to the final point ending with an arrowhead In the example of the car returning to the factory the line has zero length Pay careful attention to the distinction between units and direction 75 C is a scalar because it has no direction attached 35 m East or up or down etc is a vector because a direction is attached Velocity is also a vector There s a big difference between driving 100 kmhr East and 100 kmhour West very noticeable if you happen to be on the Eastgoing side of the interstate In three dimensions vectors have 3 parts called components We usually only think of two of these if we re driving 7 how far East and how far North negative numbers would then put us to the West or South Ifwe re looking at an airplane its velocity vector has 3 obvious parts East West North South and updown The airplane may be moving 200 kmhr to the North 300 kmhr to the East and 10 kmhr up If you continue in physics you ll see stranger cousins of vectors called tensors but we won t be using them We will soon look at how to deal with vectors in more detail but first we should think about what we really mean by velocity We can talk about average velocity or instantaneous velocity Your speedometer if it gave directional information gives instantaneous velocity Average velocity is what you might calculate after a long day of driving if you ve gone 600 km West in 6 hours your average velocity is 100 kmhr West Each of these quantities can be useful but you should always be aware of which one you need and why If a friend tells you she averaged 100 kmhr on her drive that is a summary of information Maybe she drove the full 600 km in 6 hours with no stops and the cruise control set or maybe she spent 10 minutes stopped for food and later had a highspeed chase with the police If you only know the average you ll miss out on all the interesting stuff like that If however your friend s acceleration was zero on her whole trip except for her initial start and final stop the average velocity by itself tells the whole story The formula for instantaneous velocity is just dx v 7 air The average velocity formula uses finite instead of infinitesimal displacements and times and is usually written with a bar over the v like this PHYS 2211 xf xi AX v 7 7 If ti At In this formula dx is the change in position from beginning to end also known as the displacement dt is the change in time from beginning to end As the size of the time interval dt gets smaller and smaller the distance traveled also gets smaller and the velocity gets closer to its instantaneous value Acceleration When an object s velocity changes in magnitude and01 direction we say that it has undergone an acceleration Acceleration has both magnitude and direction and is therefore a vector The de nition of acceleration is the change in velocity per unit time As with velocity we can de ne both an average acceleration and an instantaneous acceleration If acceleration is de ned as change in velocity divided by change in time we can write that as dv a 7 air The units of acceleration are therefore Length Time Time or LengthTime2 In the metric system we would use msz Incidentally another common unit for acceleration is the g or the same acceleration experienced by things falling in Earth s gravitational eld One g is equal to an acceleration of 98 msz In other words for every second an object falls neglecting air resistance it gains 98 ms of speed about 22 mph added on every second For reference a car in a panic stop with good tires on a dry road can decelerate at about one g This means you can stop from 60 mph in about 120 ft A very fast sports car can accelerate at about 23 of one g 060 mph in about 4 seconds This by the way is a good example of a nonconstant acceleration in general a car will have the highest acceleration in the lowest gear If we want to be more correct we should take note of the fact that displacement velocity and acceleration are vectors Ifwe stick to onedimensional motion only back amp forth along a line like a train on straight tracks we don t have to worry about it Later when we look at two and three dimensional motions we ll be more care ll and write vector quantities in boldface type a for acceleration When you re writing them by hand you can just draw a small arrow over the letter to make it clear that you re using a vector a for acceleration Because acceleration amp velocity are both vectors the acceleration can be nonzero even when the speed of an object doesn t change For example if you re spinning a yoyo around your head at a constant 30 revolutions per minute the velocity is not constant The direction of the yoyo s motion is changing all around the circle so the yoyo is accelerating while maintaining a constant speed You ll feel this acceleration as a tug on the string PHYS 2211 Kinematic Equations There are 4 basic equations we ll need to talk about the way things change position and change velocity As with any formula you learn in here you have to understand when it applies These four equations are valid only when the acceleration is constant Using the wrong formula for a given situation will give you the wrong answer almost every time The formulas we ll need are xx0 vt v v0at 1 2 xx0v0tEat Each of the three equations above can be found by integrating the earlier differential expressions for v and a We can combine them to find the fourth kinematic equation v v0 2ax x0 The first of these four equations just describes how distance is covered as an object moves with constant velocity This is a very important thing to remember 7 if an object is accelerating this formula will not accurately tell you what s going on In this formula the x is the final position of the object after it has moved with velocity v for time t The final position is therefore vt plus whatever the initial position was represented by x0 This is the formula you would use if you were on aroad trip and fell asleep for an hour not while driving hopefully You d wake up look at your watch and realize you were probably 65 miles past wherever you fell asleep The second equation gives the final velocity v of an object after it has moved with a constant acceleration a for a time I As before this velocity change must be added to the initial velocity v0 This is one way to find the speed of a falling object 7 you just need to know how long it fell how fast it was going in the first place and how quickly it accelerates when falling Notice the similarity between the first and third equations In fact we don t really need the first equation as long as we have the third one because the first one assumes constant velocity or zero acceleration If we put in a value of zero for a in equation 3 equation 1 pops right out The advantage of equation 3 is that it is still valid even when a is not zero This is the formula we use to find out how much distance is covered by an accelerating object For example if you ve ever dropped a penny off of a tall bridge amp timed it you can use that time to find the bridge height If it takes 4 seconds for the penny to hit the ground and we neglect air resistance we can fill in the numbers to nd the height First we ll choose to set your hand as the zero displacement point That means x0 will be equal to zero and disappear Since we re not throwing it but just dropping it its initial velocity Va is also zero so the v0t term also disappears We know a l g 98 msz so we re set xx0 v0t at2 0 04sec98msz4sec2 784m PHYS 2211 So you re over 250 feet in the air The fourth equation is useful when you want to relate initial and nal velocities to acceleration and the distance over which that acceleration occurred If we look at another car example the fastest dragsters in the world can complete a run usually 1A of a mile at a speed of around 300 mph We can find the average acceleration with equation four Keep in mind that we can only find the average because our data is limited to the initial and nal velocities and the distance covered The instantaneous acceleration is certainly changing throughout the whole trip down the track This will tell us that the driver experiences an acceleration at least equal to the average but probably greater at some points The distances and velocities become in metric units imilexlf09m 24023m 300mzles gtlt1609Mgtlt lhour 7134 ms mzle 1 hour lmz39le 3600 sec Our equation is now remembering that the car starts from a dead stop v2 v2 2axf x0 l34ms2 0ms2 2a4023m 0m 0 Solving this equation gives us a value of 223 ms2 for the acceleration or about 23 times the acceleration due to gravity This equation is also very useful when you want to know how high something can be shot or thrown straight up You know that the final velocity which in this case means the velocity at its maximum height has to be zero at the top of its ight and you know that a g You can either measure the height to find the speed of the throw or if the speed is known you can accurately predict the maximum height Notice that the object s speed when it returns to your hand will be exactly the same as the speed when it left This ignores air resistance but it s why shooting guns into the air is generally not a smart idea One of the biggest parts of understanding the kinematic equations is learning when to use which one The first for example is not valid when acceleration is occurring With practice you ll learn how to apply these to solve various problems While we can t use the kinematic equations in their present forms if a is not a constant we can still find a or v working in the other direction For example if you are told that the formula for an object s position as a function of time is xt pt4 qt2 rrs you can still find v by taking the derivative with respect to time You ll get 3 vl4pl 2qlr The acceleration involves one more time derivative so we get PHYS 2211 at212pl2 2q We now know that at time t 3 seconds we should nd that x 81p 9q 3r s v 108p 6q r and a 108p 2q You could turn these into numbers if you were given the values of p q r and 3 By the way what are the units for these variables Is there any way to gure them out Free Fall The kinematic equations above can be used to analyze the behavior of objects in free fall falling under the in uence of gravity alone near Earth Basically all we do is replace the a for acceleration with the known value of g 98 msz except that we make the 98 ms2 negative since the object falling is accelerating downwards towards the ground Even if you red a bullet straight up it would be accelerating downward the entire time This is sometimes a hard concept for students to keep in mind 7 acceleration and velocity don t have to be in the same direction When we say all things accelerate downward at 7g we re ignoring air resistance For things which are reasonably dense and not falling very far this is a good approximation For things like feathers paper and soap bubbles it s a bad approximation We ll explain why this is a problem when we discuss the relationship between force and acceleration The only other change we ll make to the kinematic equations is to replace x with a y This is just a convention 7 it s something people do but it has absolutely no physical signi cance The formula works the same way no matter what variable you decide to use but when you re working in more than one dimension as we soon will be it s convenient to call one direction y and the other one x To compare a couple of situations let s go back to the very tall bridge we used as an example not long ago We know that if we drop something and ignore the air it will take 4 seconds to hit the water from our 784 m bridge What if we give the rock an initial velocity Ifit s up it should take more than 4 seconds to hit the water but if we throw it down it should take less Assume you can throw the stone at 30 ms about 67 mph First throw it up in the air Up is the positive direction so Va is 30 ms We have another choice 7 we can either let x 784 m and x0 0 m or we can letx 0 m and x0 784 m Does it make a difference Not really Ifyou look at the third kinematic equation we could take the x0 over to the left hand side and have x xg over there If we do that either choice above gives us 784 m on the left side it ll be negative because we already chose down as the negative direction and the change in position or x xo is de nitely down Now we have 784m30mst 98msz t2 We now have a quadratic equation which you ve probably seen before amp solved before If we have an equation of this form all btc0 PHYS 2211 then the solutions for t are tz bi b2 4ac 2a In our case our two choices for solutions are t 198 sec or 81 sec We clearly want the positive root here although the negative root has some physical signi cance which we ll see in minute How fast is it rock going when it hits Well the rock will climb up until its velocity is exhausted v 0 The height the rock reaches before stopping can be found by using equation 4 Omsf 30ms2 2 98ms2y784m y is found to be 1243 In As a check on our math the time it will take to reach this new altitude is just v v0at 0ms 30ms 98ms2t This tells us that the rock will have reached its 1243 m height after 306 seconds The rock is momentarily stopped at that altitude and begins to fall The time it takes for this part of the fall is simpler we use equation 3 and set the initial velocity v0 0 ms We get 1243m 98ms2 t2 so I 504 seconds You ll notice that the time it takes to reach its maximum height plus the time to hit the water from that point is 81 seconds exactly what we calculated with the quadratic formula Finally how fast is the rock going when it hits Back to equation 4 v2 vg 2 98msZ 1243m Remembering that our initial velocity v0 was 0 this gives 494 ms for the speed just before striking the water Now we can say what the negative root of the quadratic equation actually meant 7 ifthe rock is supposed to be moving upwards at 30 ms at time t 0 and a height of 784 m above the water it could have been fired from the water s surface at 494 ms at time t 198 seconds Check for yourself that if we had thrown the rock down with the same velocity it would have reached the water in 198 seconds does that time sound familiar PHYS 2211 Components of Motion and Vector Addition This is a summary of the information you can nd in the page about trigonometry amp triangles located at httpwwwchemistryarmstrongedubairdtrigtripdf So far we ve looked at obj ects moving in one dimension 7 either up and down usually called y in 2D or leftright x These cases are a little more specialized than what generally happens when something is moving and has motion in both the x and y directions When both of these motions are happening at once it s time to start using vectors In some cases we ll use the individual components for example we may say something is moving 4 ms E and 3 ms N and at other times we ll talk about the overall magnitude and angular direction of the vector See below for both methods together V y 0 gt 4ms E The vector here is V and the components of V are called v and vy Notice that the components are not in boldface ithe components are not vectors The magnitude of the velocity vector is found by treating the components like legs of a right triangle and using the Pythagorean theorem The magnitude of it is then 17vv For the example above that would give us 5 ms for the total velocity The angle 9 can be found by using any of the trig functions on the three legs of the triangle made from the two components and the final vector We could use Sine or Cosine since we know all three sides of the triangle now but it s more common to use the Tangent Tangent is the only trig function that would connect the two components and the angle directly We find the angle by V V Tan6iy 0r OzTarf1 i VX VX In our case we get 369O Something important to notice is that we chose to measure the angle from the x axis We could just as easily measure it from the y axis or the negative x axis or wherever If we do that though our formula above for the angle will be altered PHYS 2211 If we re combining vectors we can do it one of two ways The most mathematically accurate way to do it is to add the components If we take our velocity vector above and add another vector to it let s say it has a ycomponent of 10 ms and an x component of 72 ms we need to add the components individually and then recombine to form the nal vector The sum of the x components would be 4 ms 2 ms 2 ms They components add to give 3 ms 10 ms 13 ms What s the length of this new vector Square each nal component and add them taking the square root oftheir sum WZ ms2 13 ms2 Wl73 mZsz 1315 ms We can nd the angle by the method above and we get 81250 This makes sense because we re measuring the angle from the x aXis and we have a vector that is very much larger in the y direction than the x direction so it shouldn t be a surprise to see an angle close to 90quot The other way we can add vectors is called the graphical method or polygon method where we put the vectors together head to tail and see where they go In the example below we start with our two vectors mentioned above and then move the tail of one to the head of the other keeping it in the same orientation 7 we can t rotate it at all or we ll get garbage The nal vector also called the resultant is the arrow from the rst tail to the nal head We can add more than two vectors the same way 7 we just keep putting tails at the heads of other vectors and when we re done we connect the rst tail to the last head Subtracting vectors is no harder 7 for the component method we just subtract the components For the graphical method we just add the negative of the vector which is the same as subtracting it A concept which is also useful is that of a unit vector This is just a vector with magnitude equal to one in whatever units are being used We can take advantage of this to give a shorthand way of talking about vectors Instead of saying a vector has an x component of 7 meters and a y component of 73 meters we could also write this vector call it r as r 7 x 73 y These are sometimes written with hats over the letters instead of arrows 77 m fc 3 m Other Vector Operations There are two other common things we can do with pairs of vectors We can look at their similarities the amount of overlap between them by taking the dot product of the two vectors To do this we combine the x y and 2 components of each to form a scalar like this PHYS 2211 abzax bx ay by 612 2 We can also nd the dot product by multiplying the lengths of the two vectors together and then multiplying by the cosine of the angle between them ab HaHHbHCOSG As this form makes clear if two vectors are perpendicular so 9 90quot their dot product will be zero This means there is no overlap between the two the dot product of a Northpointing unit vector and an Eastpointing unit vector is zero since North has no Easterly component and vice versa If the two vectors are parallel 9 0quot the overlap is total and the dot product is just the product of their lengths We can also combine two vectors to form a third vector This is known as a cross product or vector product Your book has an appendix with the component form of the cross product in it but there is another much simpler form 52a X b HaH HbHSmG The direction of the new vector 0 is chosen so that it is perpendicular to the other two You can use the right hand rule to clear this up point the ngers of your right hand in the direction of a and then curl them into the direction of b Your thumb will be pointing in the direction of c In contrast to the dot product you get the largest value of the cross product when the two vectors are perpendicular and you get zero if they are parallel Projectile Motion When an object is moving under the in uence of gravity we call the resulting motion projectile motion because it describes anything thrown or shot as long as it s not powered like a rock baseball or cannonball At rst this sounds like something that would be very complicated 7 a baseball is thrown into the air at some angle measured relative to the horizontal usually and some speed and we need to predict things like where it will land how fast it will be moving when it hits and how long it will take to get there The pa1t that saves us lots of work is the fact that we can separate this twodimensional motion into two onedimensional motions Gravity acts only on the ycomponent of velocity and position 7 the velocity in the x direction is the same from the moment the pitcher releases it until it hits the ground This is making the approximation that the air does not affect the path of the ball at all This is obviously a very poor approximation for a pingpong ball but it s a pretty good one for most other projectiles Stop for just a minute amp reread that It s hugely important and makes our calculations much easier Since we can break this otherwisecomplicated motion into two separate onedimensional problems we can use the 4 3 really kinematic equations we ve been using to solve each piece PHYS 22 ll separately One painful problem becomes 2 simple problems This also illustrates why we ve spent some time talking about vectors and components Breaking the 2D motion into two problems means we have to get the components of initial velocity We do that with trig Once we ve solved our two problems we have to join the results back together which means using the Pythagorean theorem and more trig The simplest case is when a projectile is red horizontally so that the angle involved is 0quot Assume that the velocity of the object is V 25 ms in the x direction We also have to specify height from the ground 7 it s what will determine when the projectile hits the ground we ve already said the speed in the x direction has nothing to do with time in the air Let s say that the height from the ground is l m of course we ll also say that the ground is level How long will it be in the air We go back to our 4 kinematic equations and we need the third one We ll replace x s with y s since we re looking at the effects of an acceleration in the 7y direction Also since the initial velocity had no y component we can drop the term involving v0 t We get this 0mlm 98msztz t0452s If the ball is in the air for 0452 seconds and it travels 25 ms in the x direction we get a range of 113 In What will the new velocity be just before it hits We still have the same 25 ms in the xdirection but in the ydirection we ll have using the second kinematic equation v98 msz 0452 s 443 ms The total velocity is then W443 ms2 25 ms2 254 ms What direction Using 9 tan Yvvx we get an angle oflOquot This makes sense because we can see from the quot J of the r that the y r of velocity is still much larger than the ycomponent so we expect an angle much closer to 00 than 90quot Also the angle should be negative since the ycomponent of velocity is negative Things aren t much more difficult when the projectile has components of velocity in both the x and y directions The main change is that our time of fall equation 3 d kinematic equation now has the v0 t term that we left out in our first example The effect of this is that we now have to solve the quadratic equation just like we did in Chapter 2 Once the time is known range is found by multiplying it by the x component of the velocity Sometimes we can use shortcuts to find these answers The curve made by a projectile is called a parabola E a projectile lands at the same elevation from which it was fired the parabola will be symmetric We can use that symmetry to find the total time the projectile spends in the air The time to go from the initial elevation to the maximum height is the same as the time needed for gravity to change the y component of velocity to zero That time from kinematic equation 2 is 0 v0 Sine 98 msz t The total time in the air is then 2 v0 Sin 9 g Since the range is just the x component of velocity multiplied by the time in the air we get I 2 R vtv0 C039 v0Sm29 g g As with every formula you shouldn t use it if you don t understand where it comes from Knowing the origin of this one tells us that if the firing and landing points are at different PHYS 2211 elevations firing from a building to the street below this range formula will not give us the correct answer Incidentally you ll notice that this formula says the maximum range occurs when the projection angle is 45quot This is because of the tradeoff between small angles which have large x components of velocity but which don t spend much time in the air and large angles which spend longer in the air but don t travel too far because they have only small x components of velocity The plot below shows this dependence on angle Projectile Motion 1 075 a 3 o 5 a a 025 0 0 15 30 45 60 75 90 Angle Relative Velocity When we talk about velocities the important thing is really relative velocity In fact there is no such thing as an absolute velocity 7 all velocities are measured relative to something else For example if you and a friend are playing catch on an airplane which is moving at 600 kmhr relative to the ground that doesn t mean that you have to be able to catch a ball thrown at 650 kmhr The important thing is that the velocity of the ball relative to you is probably something like 50 kmhr If someone on the ground could see it they would say that its velocity is 550 kmhr when you re throwing towards the rear of the plane and 650 kmhr when you re throwing towards the front If the plane had an open window and you pitched the ball to someone on the ground the speed relative to the ground is suddenly my important to that person In general relative velocities are found by adding or subtracting the two velocities involved In the case of vectors subtraction is just the addition of the negative of the vector As mentioned above a plane s speed is typically measured against both the air and the ground The airspeed determines whether the plane will y or drop out of the air The ground speed determines how quickly you get to another place on the Earth s surface This is why planes typically take off into the wind 7 if they re pointed into the wind they already have wind owing over the wings at for PHYS 2211 example 30 kmhr so they only need to be moving at 100 kmhr wild guess relative to the ground to take off if we assume that you need air moving over the wings at 130 kmhr to take off Trying to take off with the wind at your back would mean you would need a ground speed of 160 kmhr before the wind over your wings was moving at 130 kmhr Force Net Force and Newton s Laws of Motion You probably already feel familiar with the idea of force Basically a force is something that acts to change the motion of a body Force is a vector 7 it has both magnitude and direction The important part is generally the net force The net force is the vector sum of all the forces acting on a body For example in a tugofwar if the two sides are evenly balanced the rope doesn t move If one side is slightly stronger than the other the net force may be small so the rope will slowly move towards the stronger side The first of Newton s three laws of motion talks about this I n the absence of external forces a body at rest tends to stay at rest and a body in motion tends to stay in motion This sounds somewhat obvious to us now but the part about a body in motion staying in motion would have seemed strange to people several centuries ago Watching things on Earth suggests that the natural state of all objects is to be at rest In fact it s just as natural for something to continue moving at a constant velocity but we usually see the effects of frictional forces on moving objects Those external forces gradually slow objects until they stop In space where there is no air resistance or friction objects keep moving at the same speed forever unless they fall under the gravitational in uence of some other body Friction is an example of a contact force because the two bodies involved are actually in contact If you push a cart across the oor that s also a contact force The other type of force is called an action at a distance force Examples of this are gravity and electromagnetism These forces can in uence the motion of bodies when they are far apart these two actually have no limit on their range 7 the force gets weaker with increasing distance but the gravitational force between two things is not zero for any finite distance The resistance to being accelerated or decelerated is called inertia or mass This is also part of Newton s second law of motion This law boils down to F m a In this formula F is the applied external force m is the mass of the body on which the force acts and a is the acceleration of the body When we look at rockets we ll see that this is a slightly simplified version of what Newton s 2quotd law really says but this is good enough for now This tells us that mass is really what resists motion Larger masses are harder to accelerate which seems obvious 7 pushing a wheelbarrow will make it accelerate much more quickly than exerting the same force on a car The SI unit of force is called the Newton N We can look at the dimensions of acceleration times mass to see that a Newton is 1 kg l msz l kgquot ms2 l N For comparison a pound is the English unit of force which is equal to 445 N This brings up the distinction between weight and mass Mass is fundamental and unchanging and it essentially measures the amount of matter inside something You can think of it as a rough count of the number of protons neutrons and electrons pieces of atoms in an object A lkg mass for example is the same on the Earth the Moon the Sun in space or anywhere else It may be weightless in space but it s not massless For example if the astronauts in the space shuttle take a baseball bat up with them it will be weightless once they re in orbit If it were also massless they could hit each other with it and it wouldn t hurt PHYS 2211 Weight is the pull of gravity acting on a mass we re usually talking about the pull of Earth s gravity on a mass We can then say that your weight would be only 16 11 as much on the Moon as it is on Earth even though your mass would be unchanged For that reason we can t really convert from pounds to kilograms 7 pounds are force and kilograms are mass Of course we know that in almost every case we re talking about objects which are on the Earth and therefore their masses and weights are related by the formula below which stems directly from Newton s 2nd law Fma or Wmg From this we can see that one kilogram multiplied by one g will give 1 kg98 msz 98 N We already know that one pound is 445 N so we can see that one kilogram on Earth s surface will be pulled down with a force of 22 pounds 98 N This is an incredibly important result 7 when we want to know how anything moves what we will always do is sum up all of the forces If their sum is not zero the object will accelerate This is another point to go back amp read again It doesn t seem like there is any content in this so far but try this example Let s say a weightlifter can exert an upward force of 1000 N If a mass of 125 kg is dropped on him slowly of course what will happen Will he be able to lift it or will it hit the ground How long will it take to go up or down We can find out using Newton s 2nd law The weight of 125 kg on the Earth s surface is W m g or W 125 kg98 msz 1225 N This is too heavy for the weightlifter so it will hit the ground or his chest How quickly Will it accelerate downward at g We now look at the net force on the barbell There s a downward force of 1225 N but an upward force of 1000 N is also acting on it The net force if we make downward negative is 1225 N 1000 N 225 N negative so it s downward We use Newton s 2nd law to nd the acceleration of this barbell The acceleration is the net force divided by the mass at F m 225 N 125 kg l8 msz How long will it take for this to hit the weightlifter s chest Assume it will drop 06 m from the point of handoff until contact with the chest We know distance initial velocity 0 ms if it was slowly handed to the person and acceleration How long it takes is then found using our 3 d kinematic equation again 0 06 1218 ms2t2 This gives 082 s Quick but not nearly as quick as a pure fall which would have had an acceleration of g Newton s Third Law Newton s 3Yd law says that for each action there is a reaction In other words when one body the Earth for example exerts a force on another body you the second body exerts an equal and oppositely directed force on the first body You pull on the Earth with the same force that it pulls on you Newton s 2nd law explains why that force alters your motion so much more than it alters Earth s This concept confuses people because they think that if the forces are equal and opposite there should be no motion The crucial part that is usually missed is that these two forces act on different objects Go back to the idea of atugofwar This time tie one end of the rope to a car that s stuck There may be no motion initially but the force of the people on the rope is the same magnitude as the force of the rope on the people Once the motion starts the force of the people on the rope and the car is still balanced by an opposing force PHYS 2211 Even when we re talking about things just sitting around on the Earth there is still a force which balances gravity we know there is because if the only force was gravity Newton s 2quotd law would force things to start accelerating at g This force is called the normal force normal meaning that it is directed normal perpendicular to the surface it s resting on Objects exert a force due to their weight on the surface they rest on and the surface responds with a normal force It s important to keep in mind that Newton s laws apply in inertial non accelerating and non rotating frames Free Body Diagrams When analyzing a problem in mechanics the first step is always to make a free body diagram which shows all forces acting on all bodies The resultant of all of the forces on a body together with the mass of the body determines the motion of that body An example of this found in the book is called the Atwood machine This consists of two masses suspended from either side of a pulley The only function of the pulley is to change the direction of the forces 7 there s no way it can change the magnitude of the forces We ll usually ignore complications like any forces besides gravity and tension in the rope or the mass of the rope itself when we look at this machine although we could include things like friction in the pulley In this approximation we have three possible cases which depend on the masses involved Ifmi gt n12 mi will move down and M2 will move up If M2 is larger the reverse will happen Finally the two masses could be equal and we would then expect no motion at all See the diagram below for one possibility Let s see how to find the value of the acceleration the same for each mass since they re connected and the tension also the same throughout the rope We can use these two facts to figure out what s happening From the drawing we see that there are exactly 2 forces on each mass the weight of the mass and the tension in the rope which we ve said is the same for both masses The sum of the two forces on m will be equal to m times its acceleration and we can write a similar but not quite identical formula for m2 Again the accelerations will also be equal so let s see what we get PHYS 2211 mlg Tm1al Tngmzaz Notice from the drawing that a and a2 must be oppositely directed We can then call one of the a and the other 7a Because m1 is larger we know that it will accelerate towards the ground which is the negative direction For that reason we ll set a a2 and 1 all Now we get T mlg m1a T ngzmza Let s get the tension term by itself in each case When we do that whatever is on the other side of each equation will equal T and since the T s are equal the other side of the equation for m will equal the other side of the equation for m2 szlg mla Tm2gm2a m1gam2 g39l39a Solve this for a again it s the same for each mass and we get m1 m2 g a m1 m2 It s important to notice that in this case we looked at each mass separately we found the forces acting on m to be its weight and tension We applied a similar procedure to m2 One way to take a shortcut here is to look at the net forces involved There s a downward force of m g on the rst mass but there s also an upward force of ng on the rst mass This upward force is really the ordinary downward force of gravity on m2 which has had its direction changed by the pulley The net force on the masses is therefore m g 7 mg but this force acts on the total mass of the system which is m m2 By Newton s 2 d law the acceleration is the net force divided by the total mass on which it acts so we the result above In this case the system is the combination of the blocks and the rope so tension is an internal force and therefore doesn t appear in our equations Notice that we got the same answer either way we just have to be clear about what constitutes our system and therefore what forces are external to it You can also find the tension by realizing it would be equal to the smaller of the masses multiplied by gravity if the machine were motionless Since it s actually accelerating we have to multiply the smaller mass by a g to get the tension We should also be able to multiply the larger mass by g a and get the same number for tension The general idea in solving these problems is to sketch the setup and then draw in the force vectors on each moving point Something else to consider is the fact that you can always choose your own orientation for your x amp y axes For example if we re looking at a mass on an inclined 18 PHYS 2211 plane we are free to keep the same orientation for our axes that we used in projectile problems ie updown is y and leftright is x We can do that but it s not smart We should take advantage of the problem s physical characteristics to orient things so that we have to do less work For the inclined plane we usually do this by choosing the y axis as normal to the plane and the x axis as up and down the plane Equilibrium If all of the forces on an object are balanced so that there is no net force we say that the object is in translational equilibrium This is distinct from rotational equilibrium which we ll discuss later Equilibrium can be static or dynamic Static equilibrium is what we expect from things like a hanging sign We want the net force and therefore the net acceleration to be zero but we also want the velocity to be zero relative to us Dynamic equilibrium is illustrated in the example below where two boats triangles are cooperating to tow a water skier ST ill If each boat pulling the water skier is pulling with 600 N of force and each is pulling at a 450 angle to the xaxis what is the magnitude of the retarding force exerted on the skier by the water if the boats amp skier are traveling at a constant velocity First we can draw the forces on the skier In component form F R the retarding force is entirely in the 7x direction and if there is no net force the x components of the other forces balance this Those components are F1 cos45quot and F 2 cos45quot The only y components involved are from the boats These must also balance so we get the equations below 2F F1 cos45 F2 cos 45quot FR 0 2Fy F1sin45 F2 sin 45 0 Keep in mind that the two angles involved are not both 45quot The angles must be measured in the same direction for things to make sense This means we can call the second angle 3150 or 450 check for yourself to see that it makes no difference in the math Anyway the formulas above show us that F1 F 2 in magnitude but we knew that anyway since the problem told us that each 19 PHYS 2211 boat was applying a force of 600 N When we plug those numbers into the formula for the x components of the force we get that FR 600 N0707 600 N0707 849 N in the 7x direction Friction In realworld situations frictional forces are frequently important in the analysis of motion Friction behaves very differently from most of the other forces we ve looked at so far A frictional force only acts in response to an applied force it always resists motion and it varies in size from zero if nothing is trying to push the object to a maximum level we ll discuss later The cause of friction between solids in contact is generally accepted to be due to bonding between the microscopically small parts of the surfaces which are actually touching each other This is a kind of microwelding between the surfaces and it takes a force to break those tiny welds We won t look much more closely at the causes of friction because it s a complicated subject The types of friction are more important for us When an object is stationary and you re trying to move it you re working to overcome static friction If you re able to get it moving you still have to exert a force on it to keep it moving 7 the force you exert now is opposing dynamic friction Finally when you re trying to push a car on a at road you re working against rolling friction In general static friction is greater than dynamic which is greater than rolling In other words when you re moving a refrigerator it s easier to keep moving than to get moving Also if you put it on a wheeled cart it s even easier to move it The general form for frictional force in the static case is fSSMSN Notice that this formula has the less than or equal sign In other words the frictional force will be somewhere between 0 and the quantity on the right It s easy to see why it varies 7 assume the quantity on the right is equal to 10 Newtons for some situation If you are only pushing with a force of 5 N we know that the object should sit still If it resisted with a force of 10 N in the opposite direction it would actually start moving towards your hand if you pushed it lightly We know that it won t do that 7 we may not be able to apply enough force to move it but we know things don t come after us if we re too weak to move them What do the quantities on the right represent The N in the formula is the normal force of the object we re talking about The It is called the coef cient of static friction This has to be a dimensionless quantity we know that because N is a force Newtons and the frictional force is also in Newtons Once the object is in motion the formula for the dynamic frictional force is szi39LkN The k is for kinetic friction another name for dynamic friction If you apply a force less than the coefficient of static friction multiplied by the normal force the object you re pushing will not move If the force is greater it will start moving and the equation for kinetic friction will determine the force needed to keep it going Applying aforce greater than to the object will cause it to accelerate 20 PHYS 2211 Inclined plane problems are favorites for illustrating friction We know that as the gravitational force tries to drag the block or whatever down the plane the frictional force tries to prevent the motion We expect the gravitational force to increase as the angle of the plane increases a steeper slope will allow gravity to more directly pull on the block The thing that you might not have expected is that the frictional force will decrease as the tilt angle increases This is because the normal force is no longer antiparallel to gravity N is equal to mg cos 9 on the plane and that force drops as the angle increases You re already aware of this if you ve ever noticed that things don t tend to sit on walls as well as they do on floors One of the ways to nd the coefficient of static friction involves the plane 7 since frictional force will be equal to the force trying to drag the block down the plane until it actually moves we know that mg sin 9 We also know that when it starts to move it s equal to MJN My mg cos 9 Using the equation above we see that it boils down to My tan 9 This tells us that an object which only starts sliding at an angle of 450 has a coefficient of static friction of exactly 1 The table in your book shows that in general these coefficients are less than 1 When solving problems relating to friction the fact that the equation for static friction has a less than sign in it is a slight but very important complication The procedure is this first ignore friction and add up the forces involved as if friction did not exist When you get to the end of this preliminary problem you need to compare the net force with the maximum force of static friction which is HSN If the net force is less than this maximum the object does not move and the frictional force is exactly equal and opposite to the force calculated in the absence of friction If the frictionfree net force is greater than the maximum frictional force take the difference between the two and that is the true net force on the object and it w move One of the most common mistakes students make when solving problems involving friction is to put in the frictional force at its maximum just like any other force and solve away This mistake will be obvious if at the end of your problem you find your mass moving u an inclined plane with friction present Now take a look at the situation below and see if you know just by looking which way the block should move with and then without friction O Should block 1 go up the plane or down There s no way to know without knowing m m 2 the angle of the inclined plane and the coefficient of static friction MJN Air resistance is a different kind of frictional force and is a velocitydependent force The shock absorbers in your car and on selfclosing doors also provide velocitydependent forces 7 if 21 PHYS 2211 you try to open the door very quickly or if you hit a bump at high speed the shock absorber will strongly resist the motion unless the force is large enough to break it You can demonstrate this in a pool Waving your hand quickly underwater takes much more effort than slowly moving it The exact form of the velocity dependence is itself velocity dependent Air resistance is important because it s what makes us think that heavy things fall faster We know that all falling objects near Earth s surface accelerate at g initially but we also know that sheets of paper fall more slowly than lead weights It s because the object only accelerates until the force of air resistance matches the force of gravity When that happens the object stops accelerating and falls at a constant velocity known as its terminal velocity The sheet of paper has a large area and a shape that tends to catch air very efficiently These factors combine to give it a very small terminal velocity which it reaches very quickly as it utters to the ground A sphere made of lead has an aerodynamic shape and will have a small surface area compared to its weight that s why we say lead is dense This lead weight will have a very high terminal velocity and it will have to fall for quite some distance before it reaches that terminal velocity so it will make it to the ground much more quickly unless both things are falling in a vacuum To calculate the drag force more exactly we need to look at what factors in uence it the area of the object moving through the air or uid is clearly important since carrying a piece of plywood in the wind is much easier than carrying a stamp in the wind Also the force must depend on velocity as you can confirm by placing your hand outside of a car window as it accelerates The density of the uid you re moving through will also matter 7 it s much easier to walk through the air than through chestdeep water Finally the shape of the object will matter The next time you see a car from the 70 s notice how square it looks compared to today s heavily rounded cars That s not just a styling choice All of these combine in the formula below 1 D7C A112 2 P If D is a force we can see that C must be dimensionless Additionally the area here is the area perpendicular to the direction of motion which is in the direction of v We can use this expression for D to find terminal velocity v by noticing that v must be the point where D is balanced the object s weight We can write Dzwsovzz CpA For two objects with the same shape surface texture so C is the same and the same cross sectional area moving in the same medium we find that the heavier of two objects does fall faster Of course in a vacuum all things will fall with the same acceleration 22 PHYS 2211 Uniform Circular Motion amp Centripetal Acceleration A relatively simple kind of circular motion is known as uniform circular motion The motion is uniform in the sense that the object s speed is constant The object can t have a constant velocity though because only things moving in straight lines at constant speeds have constant velocities When an object moves in a circle its velocity vector is tangential to that circle We know that objects tend to continue in straight lines unless acted on by a force so something has to be forcing that object back towards the center of the circle See below The velocity vector would like to carry the rotating blue ball towards the bottom of the page If the ball is going to remain in a circle some force must be bending it back towards the person spinning it smiley face That force is called the centripetal force meaning centerseeking and produces a centripetal acceleration when acting on the moving mass Your book shows that the change in the direction of the velocity vector divided by the velocity is the same as the change in position along an arc of the path divided by the radius We can use these geometrical arguments to prove that the direction of the centripetal acceleration is radially inwards towards the center and that its magnitude is V2 can t a The acceleration am is the acceleration necessary to keep the object moving in a circle If this is not applied the string on the yo yo you re swinging above your head breaks the object will y off tangentially If we multiply this acceleration by the mass it acts on we ll get the centripetal force acting on the object Very much like what we have above we can write mv2 1 FC ent ma cent Something worth keeping in mind is the idea of a centrifugal force This is commonly described as the force that throws you to the side of your car as you go around a curve That s actually not quite right What throws you to the outside of the curve is just Newton s lSt law 23 PHYS 2211 When you approach a curve the car and its passengers want to keep going straight It takes the application of a force to prevent that The applied force is called the centripetal force and the friction between the tires and the road provides it Once you slide across the seat amp hit the door the door will provide the force needed to change your body s direction For this reason centrifugal force is called a fictitious force It really represents the absence of centripetal force rather than the presence of some other force Ifyou aren t belted in you ll go this way Car and passengers want to obey Newton s lst law and go this way Tires are providing a force in this direction The introduction of the centrifugal force provides us with a way to preserve Newton s laws in the noninertial frame of the car For example let s say the driver isn t too reckless and that there s enough friction between your clothes and the car seat to keep you from sliding You ll say that your acceleration along the seat is zero We could then write 2 2 mv mv ma0 0r f5 f 7 Notice that the fictitious centrifugal force is directed oppositely to the frictional force so it comes in with a negative sign but it s on the left side of the equals sign since it s a force The other way to do this without the fictitious force is to say that the only applied force is friction but that since you re moving in a circle you must be feeling an acceleration of mvzr instead of zero The applied force goes on the left and the ma goes on the right We get 24
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