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Lab 1: Reflection and Refraction Introduction ▪ What were the goals of the lab? ▪ To demonstrate the law of reflection ▪ Law of Reflection: the angle of incidence (incoming light) equals the angle of reflection ▪ To demonstrate the law of refraction ▪ Snell's Law (Law of Refraction): ▪ This says that when light is traveling through some medium (air) and it hits the surface of another medium (say an acrylic block), the angle between the ray of light and the normal will change (usually become smaller) ▪ To use Snell's Law to determine the indices of refraction of solid and liquid optical media ▪ To study the deviation of a light beam on passing through a transparent plate ▪ What is the index of refraction? What did we assume about it for this experiment? ▪ Its formula is: ▪ Basically, it is defined as the ratio of the velocity of light in free space VS. velocity of light in the material ▪ We assumed that the index of refraction for all the different wavelengths of light in the visible spectrum was the same ▪ This allowed us to use an incandescent white light, which contains all the wavelengths, and assume a single index of refraction for the white light ▪ What else did we assume about the light source in this experiment? ▪ We assumed that the light coming from the incandescent light bulb was a series of straight lines, even though this may not be fully true ▪ However, since the dimensions of the experiment were large, diffraction and interference effects could be ignored ▪ Before the experiment started, how did we ensure that the light source was centered? ▪ We position the light source at the left end of the bench, and put the angular translator on the right ▪ The angular translator is just a thing that is on the bench which we can use to measure angles ▪ It has 3 main components: ▪ Angle plate: this is a fixed plate that sits on the optical bench and has angles on it (obviously the line through 0o and 180o runs parallel to the bench) ▪ Rotating table: the rotating table sits on top of the angle plate and is able to swivel freely on a vertical axis in the middle ▪ It has marks which allow you to tell how many degrees it is away from 0o, when you compare it with the angle plate ▪ Translator arm: this is also attached to the angle plate, and is able to swivel independently of the rotating table ▪ It also has a mark which tells you how many degrees it is off-center ▪ Then we placed and centered the viewing screen on the translator arm and then positioned the arm at the 180o mark ▪ The translator arm has "holders" so we can attach things to it -- so in this case we are going to attach a viewing screen on it so we can use it to look at stuff ▪ The 180o mark refers to its position relative to the angle plate: in this case, the translator should be on a line going parallel to the length of the bench, and on the far side of the angle plate from the light source ▪ Then we put a slit (allows a slice of light through) on a component carrier, then put the carrier on the rotating table at the 90o-270o mark. Ensure that the slit is set up such that when you look at the light which it lets through using the viewing screen on the translator arm, the image is 1 cm off- center ▪ If you think about how the rotating table is set up, the 90o-270o mark is going to be perpendicular to the length of the table ▪ Rotate the table 180o and note the new position of the slit image. If the positions are equally spaced about the center line of the viewing screen, the bulb must be central ▪ So we just rotate the rotating table half a revolution, which means that the slit should now be off-center to the other side The Law of Reflection ▪ How did we study the law of reflection? ▪ OK, so what we do is we add a slit to the optical bench so that the light coming from the light source is in a single plane ▪ Note that before the slit was on the rotating table, but that was just for alignment purposes -- now the slit is on the bench, between the rotating table and the light source ▪ Also note that the distance ("d") between the slit and the center of the rotating table is the SAME as the distance between the first analyzer holder on the translator arm and the center of the rotating table -- this is because the slit is where the light "begins", and the first analyzer holder is where we will be measuring the "end" of the light ▪ Then instead of a slit on the rotating table, we put a mirror ▪ It is now very simple to tell what the angle of incidence is: remember there are parallel and perpendicular lines on the rotating table, and so if we look at the one which (originally) is at 0o and then look at where it goes as we rotate the table, we know that light is hitting the flat surface of the mirror at whatever angle ▪ Then we swivel the arm around the angle plate until the light which reflects off the mirror hits the viewing plate on the arm -- and when this happens, we can ALSO see how many degrees "off center" the arm is, and so we can figure out what the angle of the REFLECTED light ray is (look at the listing on the angle plate, then subtract from that the listing of the rotating table) ▪ So what we do is we look at several different angles for the rotating table, and then see what the reflected angles are using the translator arm: they should be the SAME! ▪ So we plotted the angles of incidence vs. the angles of reflection, and ensured that there was a straight line for them ▪ How did we ensure that the incident ray, normal to the reflecting plane (mirror), and reflected ray were all on the same plane (coplanar)? ▪ We made the slit horizontal and we used the viewing screen to look at how "high" it was, vertically ▪ And then we looked at the reflected ray of light to see if it was at the same height ▪ It should have been all the same… The Law of Refraction (Snell's Law) ▪ How did we study the law of refraction? ▪ Well instead of the mirror on the rotating table, we take a block shaped like a semi-circle ▪ We position the block so that (at least initially) the light is hitting the flat side dead-on, like a 0o angle from the normal ▪ If you think about it, the light will come straight out the other side and so if we put the viewing screen right behind the block, it will see the light ▪ But then we start rotating the block and now, putting the viewing screen directly in the straight path of the light will NOT allow you to catch the light ▪ Instead, you have to realize that the path of light will bend as it goes through the block, and move the viewing screen accordingly ▪ The cool thing is that since the "back" surface of the block is curved, the light will hit the normal of the BACK surface at a 0o angle: this means that however much the angle of light changed, it was all due to refraction at the FLAT or "plane" surface ▪ As with last time, use the angle plate to see what the angles are coming in and going out ▪ We recorded incident angle, refracted angle, sin(incident angle), sin(refracted angle) ▪ Since we can rearrange Snell's Law like so ( ), the slope of sin(incident) vs. sin(refracted) is the same as n'/n, and since n = 1, the slope is equivalent to the index of refraction of the acrylic block ▪ Explain how we looked at index of refraction for two different media. What was interesting about the second one? ▪ Well we used the acrylic block, but then also we used distilled water Refraction of Light Through a Plate ▪ How did we measure the refraction of light through a plate? ▪ OK, so we have the slit in front of the angular translator so the light is once again coming through in a single plane ▪ Then, we place a glass plate of known thickness on the rotating table ▪ The ray of light will be refracted a) when it moves from air into the glass, and then b) when it leaves the glass and moves out into the air again ▪ Because the light moves through the glass at an angle, the place where it enters will NOT be lined up with the place where it exits ▪ The lateral distance between the entry and exit points is what we want to measure, because then we can use the provided formulas (should probably memorize for the final exam) to figure out what the angles were ▪ Once we know what the angles are, we can figure out the index of refraction for the glass ▪ So what we do is we put a "scale" on the side of the glass plate which is AWAY from the light source -- it is a millimeter scale which we can use to see where the light is coming OUT of the glass plate ▪ At first, we set it so the light is coming straight through (0o angle to the normal) the glass plate and so when it leaves, it should be at the 0 mm "baseline" mark ▪ However, as we rotate the glass plate, the light rays will start to be refracted and so they will not leave the glass plate at the same lateral mark as where they came in -- we can use the millimeter scale to see how much they deviated, and from there we can figure out what the refracted angle was when the light passed from air into the glass FORMULAS: Part A: Lab 2 Saturday, January 20, 2007 1:20 PM Lab 2: Simple Harmonic Motion on a Linear Air Track SHM with Mass and Spring ▪ Describe the equipment set-up for this part of the experiment. ▪ The idea here was that we had a "linear air track", which is basically a track which a mass and spring can "slide" on ▪ We propped the air track up on one side to form a ramp, and the mass and spring system were at the top of the ramp ▪ Thus the equilibrium position for the mass was affected by gravity, as the mass slid down the ramp and was counteracted by the tension in the spring ▪ Briefly describe the equations which were used in this section of the experiment, how they were derived, and how they were used. ▪ Explain each part of this section of the experiment. ▪ Firstly, we only used one spring between the mass and the "base" ▪ We displaced the mass/"glider" 2 cm and ALSO 3 cm from the equilibrium position and then observed the time required for 50 oscillations ▪ We used this to find the period, and then furthermore the spring constant (using the equations above) ▪ It (should have been)/was noted that for the 2 cm trial and the 3 cm trial, the periods (and spring constants, etc.) were the same ▪ Secondly, we attached 2 springs together in "series" ▪ We only did one trial, with a 2 cm displacement from equilibrium ▪ The idea with this one was to determine whether the following equation for springs in series was true: ▪ Lastly, 2 springs were connected in "parallel" ▪ Again only one trial was done (2 cm displacement) ▪ Again we wanted to confirm the equation for parallel springs: SHM with Pendulum ▪ Draw a diagram showing how the pendulum was set up. Give equations which demonstrate the relevant relationships in this experiment. ▪ What is a moment of inertia? Explain the TWO moments of inertia which were present in this experiment, and how they were related. ▪ A moment of inertia is a property of some (rotatable) object which affects the amount of angular acceleration resulting from a given amount of torque. Here is the formula: ▪ On a given object, there can be multiple moments of inertia, depending on WHAT PART of the object is in question ▪ For the pendulum, there is a moment of inertia about its axis of rotation, and also one about its center of mass ▪ There is a theorem called the "parallel axis theorem", and it states that the moment of inertia about any axis that is PARALLEL to and a distance "b" away from the axis that passes through the center of mass is given by: ▪ This theorem applies to the pendulum because we can consider the moments of inertia about the actual axis, and also the center of mass ▪ Since we can now relate the moment of inertia for the center of mass (which we KNOW -- see formula above) to the moment of inertia for the rotation point of the pendulum, we can use the following equation to find "g": ▪ What is a "radius of gyration", and why do we care? ▪ Conceptually, the radius of gyration is the distance that, if the entire mass of the object were all packed together at only that radius, would give you the SAME moment of inertia. ▪ That is, if you were to take the entire mass of the pendulum then pack it into two spots which were exactly "k" away from the center of mass of the pendulum, the pendulum would behave similarly, torque- and rotation-wise ▪ We care because we can compare the radius of gyration for the moment of inertia about the pendulum's axis of rotation, and ALSO its center of mass ▪ What did we actually DO during this part of the experiment? ▪ Firstly, we did a bunch of derivations (see above) to arrive at a formula which related T (period), b (distance between center of mass and axis of rotation), g (the gravitational constant), and k (the radius of gyration for the center of mass, also a constant): ▪ This allowed us to measure T for various values of b (we just change the set- up so that the pendulum rotates on an axis closer or further away from the center of mass (which is of course in the middle of the pendulum) ▪ Then we put all these values on a graph where we plot T2b against b2 -- which, when we compare it to the general formula of a graph y = mx + b, looks like this: ▪ The slope of the graph is "m" or ( ), and so if we determine the slope by inspection we can solve for g ▪ After knowing g, there are various ways in which we can find k ▪ Notably, k should be the same as what we calculated earlier, and so we may find that there was some experimental error DataStudio ▪ What was the general purpose of this part of the experiment? ▪ We used a computer program to GRAPH the movement of the linear air track (recall first part of the experiment) ▪ It is possible to make a position vs. time, velocity vs. time, and acceleration vs. time graph (the computer generates them all for you) ▪ Then we examined the graphs to see how closely their characteristics adhered to expected ones (see below for more details) ▪ Answer the following questions which were in the lab manual, and/or describe how we got the answers. ▪ How do you find the amplitude of the curves, and from there how do we get the angular frequency? ▪ Well the computer program tells us what the minimum and maximum amplitudes were (although of course they should be the same, in a model system) ▪ We find the average (for both the velocity and the acceleration) and then use the following formulas to find "w": ▪ How do we get the frequency of the "sine fit", and how should it compare to the angular frequency from above? ▪ OK, firstly you have to understand that the "sine fit" is just a "best fit" line for the points on the acceleration graph ▪ The program will tell us what the frequency of the best/sine fit line is ▪ We just then use w = 2pi f to find the angular frequency ▪ With respect to the position, when does the maximum acceleration occur? Does this make sense? ▪ It should occur when the position is at either extreme ▪ Find the value of the acceleration when the velocity is zero. Is the acceleration value a maximum or a minimum? Explain why this is the expected result. ▪ Acceleration should be a maximum at this point, because v = 0 when the mass is at either extreme (same as previous question) ▪ Would the frequency change if the amplitude were changed? ▪ No -- this is a property of SHM ▪ What are some formulas for position, velocity, and acceleration which would provide justification for the answers to these questions? Show how they might be used. FORMULAS: Part A: Following this, 1/keff1/k +11/k In2series! K eff + 1 2 In parallel!!! Part B: Ic= (mL )/12 Ic= mk following this, √ 2 ( ) where slope = 4pi /g Lab 3 Saturday, February 10, 2007 10:30 AM Lab 3: Standing Waves on a Wire Introduction ▪ Explain the equipment setup for this experiment, and how it allowed us to perform the different tests. ▪ The idea was that we had a wire clamped at both ends ▪ One end was passed over a pulley and attached to a weight, meaning that we could control the t
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