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Physics 2111 Laboratory Manual Edited by: Brian Cudnik, Mahmudur Rahman, and Shani Williams Spring 2012 Table of Contents The following is a list of experiments prepared for Physics 2111, General Physics Laboratory I which are performed in room 307 of the E. E. O‘Banion Science Building. Along with most of the experiments are suggested pre-lab activities that are meant to accompany each lab. The purpose of the pre-lab is to get students thinking ahead to the subsequent experiment, so as to arrive better prepared on the day of the experiment. A set of 10 experiments is used in the ―standard list‖ of experiments, but the individual instructor may switch one or two of them for another on the list. Many of these labs involve use of PASCO computer interface equipment which works with the Dells that are stationed in the lab. For the activities that include a computer assisted component, the computerized version is presented along with the traditional version, giving the class a choice of whether or not to use the computer interface equipment for that particular lab. Help for using the computers and the computer-interfaced equipment is located in a separate manual in room 305. At the end of this laboratory manual are suggested ―capstone‖ exercises to be done in place of the traditional laboratory final exam. Course and Manual Introduction (the labs and this document) .......................... 3 0. Safety Protocols for the Physics I Laboratory Environment ............................ 8 1. Introduction to graphing and measurement / the calculation of density ........ 10 2. Vectors on a Force Table ............................................................................. 15 3. 1D and 2D Motion: Free-fall and Projectile Motion ....................................... 18 4. Static and Kinetic Friction ............................................................................ 23 5. Centripetal Force .......................................................................................... 28 6. Hooke‘s Law (and the Conservation of Energy) ........................................... 34 7. Simple Pendulum ......................................................................................... 40 8. Conservation of Linear Momentum (Ballistic Pendulum or Air Table) .......... 45 9. Torque, Equilibrium and the Center of Gravity ............................................. 50 10. Rotational Inertia of a Disk and a Ring (With the Rotary Motion Sensor) ... 55 11. Standing (Transverse) Waves Using Vibrating Strings ............................... 58 12. Computer Experiment Simulation (Classical Mechanics) ........................... 63 Some Suggested Pre-Lab Activities ................................................................. 64 2 Introduction (to this Document and the Labs) Introduction This laboratory manual has descriptions of the laboratories which you will be doing this semester. It also explains some of the concepts required to be understood in order to successfully complete this course and provides examples from everyday life illustrating the concepts. This laboratory manual is required reading material for this course. The student will be learning to apply the scientific method. Science is the study of the interrelationships of natural phenomena and of their origins. The scientific method is a paradigm that uses logic, common sense, and experience in the interpretation of observations. The underlying basis of the scientific method is understanding through repeatable experiments. No theory is held to be tenable unless the results it predicts are in accord with experimental results. A major problem is: how does one quantify data so that experiments can adequately be compared? Physicists try to apply a rigorous method of error analysis, and then compare results with respect to the inherent experimental errors. If two experiments produce results that are the same to within experimental error, then we say that the experiments have validated each other. Error propagation It is up to your instructor whether error analysis will be included in your lab assignments. It is recommended for the University Physics Laboratories (Calculus-based, for Physics and Engineering majors), but not necessarily recommended for the General Physics Laboratories (Algebra-based, for non Physics and non Engineering-majors). Since this is a manual for the General Physics Laboratory, the discussion on error analysis will be limited to the percent difference calculation (one may refer to the Physics 2511 Lab Manual for a more complete write up of error analysis). In physics we often do experiments where we wish to calculate a value that has a functional dependence on some measurable quantities, for example: y = f (x, z) or y = f (x , x ,…, x ) 1 2 n In some cases, we wish to determine how close our experimental value is compared to the published result. This is usually performed by finding the percent difference between the experimental value and the theoretical value. The percent difference is given by: (exp erimental theoretical) %diff100% theoretical NOTE: Technically the term ―percent difference‖ refers to the difference between a measured or experimentally determined value and a theoretical or reference value. 3 ―Percent error‖ is the difference between two measurements of the same value made by two different methods. In general, errors in measurement can readily be obtained in the following ways: 1. If only one measurement was taken, use ½ the smallest scale division of the measuring device. 2. If multiple measurements were taken: • use standard deviation function on your calculator. • or use the standard deviation formula: Experimental Errors Experimental errors arise for many sources and can be grouped in three categories:  personal: from personal bias or carelessness in reading an instrument (e.g., parallax), in recording observations, or in mathematical calculations.  systematic: associated with particular measurement techniques - improper calibration of measuring instrument - human reaction time - is the ―same‖ error each time. This means that the error can be corrected if the experimenter is clever enough to discover the error.  random error: unknown and unpredictable variations - fluctuations in temperature or line voltage - mechanical vibrations of the experimental setup - unbiased estimates of measurement readings - is a ―different‖ error each time. This means that the error cannot be corrected by the experimenter after the data has been collected. Accuracy: how close to the true value is the result Precision: how much spread is in the data - the more precise a group of measurements, the closer together they are - high precision does not necessarily imply high accuracy Significant Digits It is important that measurements, results from calculations, etc. be expressed with the appropriate amount of digits. A real-life example concerns the average weight of the football players on PV‘s football team. If someone expressed this value as 305.234353556 lbs., that would not make much sense. If it were expressed as 305 lbs, that would make more sense, since typical scales found in bathrooms and gymnasia measure to the nearest pound. Similarly, when you present your results for a given 4 experiment, the figures should contain no more digits than necessary. A useful rule to remember when presenting your results is the ―weakest link‖ rule: your results are only as accurate as your least accurate measurement. If you are able to only make one or more measurements to the nearest tenth of a unit, and the rest to the nearest hundredth of a unit, the result / answer should be expressed to the nearest tenth. In summary, the rules for significant digits are as follows:  exact factors have no error (e.g., e, π )  all measured numbers have some error or uncertainty - this error must be calculated or estimated and recorded with every final expression in a laboratory report - the degree of error depends on the quality and fineness of the scale of the measuring device  use all of the significant figures on a measuring device. For example, if a measuring device is accurate to 3 significant digits, use all of the digits in your answer. If the measured value is 2.30 kg, then the zero is a significant digit and so should be recorded in your laboratory report notes.  keep only a reasonable number of significant digits - e.g., 136.467 + 12.3 = 148.8 units - e.g., 2.3456 ± 0.4345634523 units → 2.3 ± 0.4 units - NOTE: hand-held calculators give answers that generally have a false amount of precision. Round these values correctly; again as a rule, the final answer should have no more significant digits than the data from which it was derived. Graphing Techniques The following graph is an example as to how you are to turn in a graph. Graphs are either to be done on a computer or on quadrille-lined paper, for example, engineering paper. Note that the graph has the following attributes: 1. Each axis has an informative title that contains the units of measurement. 2. There is a graph title. 3. The axes are computed such that the data nearly covers the complete graph. 4. There is a ―best fit‖ straight line that most nearly goes through all of the data points. 5. The graph is clearly linear because the data ―looks‖ straight, and is a good linear fit because all of the data points are near the best-fit straight line. 6. Since the data is linear it can be parameterized with the following equation: x = x + vt o 7. This equation is similar to the standard equation of a straight line: y = a + bx where a is the y-intercept and b is the slope. 5 The data for this graph contains a possible example of systematic error. Either all of the times are one second too large or the distances one meter too small as the best fit straight line does not extrapolate through the point 0, 0 as is expected if the measurement was started at zero. Laboratory Report Format The finer details of the Laboratory Report Format may vary from instructor to instructor, but each will use a format similar to that described below. In some cases, a blank template will be handed out for the each group to fill in; in others, the group may be asked to write their report, following the below format, on a computer or by hand on paper. The students will then hand in written or typed reports, either individually or as a group. If you type the report, but do not have access to a proper equation writer, then it is better to leave blank spaces and fill in the equations by hand. For example: √x + 2 is 2 not the same as , nor is x2 an acceptable substitute for x . Ambiguous equations are much worse than hand-written equations. Students are expected to use the format for the laboratory report found on the following page. 6 Group Number: Date: Group Members: Purpose: What is to be done in this experiment? Equipment: Apparatus used to perform the experiment. Theory: The calculation equations used along with meaning of the symbols and units used. Equations can be neatly hand written. Data: Raw data in tables should be placed in this section. Sample calculations should be shown. Error calculations should be shown. Results and Discussion: Include a discussion of some of the sources of experimental error or uncertainty. If appropriate, should also include a comparison of various experimental errors. For example: We found that our value of the density, within one 3 3 standard deviation, has a range of 2.68 to 2.78 ×10 kg /m . The quoted value of the density for aluminum falls within this range, and no other material densities fall within this range, so our cylinder appears to be made of aluminum. Conclusion: Short but comprehensive. Was the object of the experiment met? 3 3 For example: The density of the cylinder was found to be (2.73 ± 0.05) ×10 kg /m . We selected aluminum as the material composing our cylinder because the density of 3 3 aluminum, 2.70 ×10 kg /m , is within the experimental error of our calculated density. Safety Reminder It will be necessary to follow procedures to ensure safety in each lab. Most laboratory exercises do not present any significant danger, but some will require certain safety measures to be followed. The general recommendation is to follow all safety instructions, including those posted on the wall of the room you are in; if additional special safety guidelines are needed, they will be printed for each lab needing them. Each student, student assistant, and instructor that uses the lab is required to receive a safety briefing before beginning laboratory exercises. More details on laboratory safety for Physics II laboratories is provided in the next chapter of this Laboratory Manual. 7 0. Safety Protocol for the Physics I Laboratory Environment Safety in the laboratory is very important. The experiments performed in the laboratory are designed to be as safe as possible, but caution is always advised concerning the use of the equipment. When you arrive at the start of each class meeting, it is very important that you do not touch or turn on the laboratory equipment until it has been explained by the professor and permission has been granted to get started. The equipment for the labs are set up for you in advance, so resist the urge to play with the equipment when you arrive, as you may hurt yourself or others, or damage the equipment. While the experiments done in Physics I (classical mechanics) are generally safe, it is always important to be cautious when using equipment, especially if you are unfamiliar with the equipment. If you have any questions about the safety of a procedure or of the equipment, ask your instructor before handling the equipment. Some specific exercises in the laboratory do pose minor safety risks (e.g. the projectile motion lab and the centripetal force lab), and guidelines related to these are presented with the write up for these particular labs. In fact, each laboratory write-up contains a section on safety that should be read and followed carefully. Other hazards may come from broken glass or thermometers; in the event of such, these should be cleaned up by the laboratory assistant or the instructor. Safety is also important for the equipment as it tends to be expensive and oftentimes delicate. The equipment is tested and set up prior to the laboratory period, but if you have any doubts about the functionality of the equipment or the way that it is set up, it is important to ask the instructor prior to conducting the experiment. If a piece of equipment is broken during an experiment, promptly notify your instructor or laboratory assistant who will remove the broken apparatus to a designated place and replace it with functioning equipment. Do not try to fix the equipment yourself. All Laboratory Students, Assistants, Faculty, and Staff must abide by the following safety rules when using the Physics Laboratory. This list may be modified as deemed appropriate for specific situations.  Follow directions carefully when using any laboratory apparatus to prevent personal injury and damage to the apparatus.  The instructions on all warning signs must be read and obeyed.  Wear safety goggles for laboratory activities such as projectile motion, centripetal force, and other labs that involve rapid motion or acceleration of any kind. The goggles are provided by the department and each person in the lab must wear them.  Long hair and loose items of jewelry or clothing MUST be secured during work with rotating machinery.  Each student MUST know the use and location of all first aid and emergency equipment in the laboratories and storage areas. 8  Each student must know the emergency telephone numbers to summon the fire fighters, police, emergency medical service or other emergency response services.  Each student must be familiar with all elements of fire safety: alarm, evacuation and assembly, fire containment and suppression, rescue and facilities evaluation.  NEVER aim or fire a projectile motion device at a person.  When using the Air Tracks: 1. Do not let air track carts run away from the user. 2. Catch the cart before it crashes into the bumper or travels off from the table. 3. Do not let the cart hit the motion sensor.  Keep hands clear of any fan blades, moving parts, or projectile launchers (other than to pull the trigger).  Laboratory walkways and exits must remain clear at all times.  Glassware breakage and malfunctioning instrument or equipment should be reported to the Teaching Assistant or Laboratory Specialist. It is best to allow the Teaching Assistant or Laboratory Specialist to clean up any broken glass.  All accidents and injuries MUST be reported to the Laboratory Specialist or Faculty teaching affected lab section. An Accident Report MUST be completed as soon as possible after the event by the Laboratory Specialist.  No tools, supplies, or other equipment may be tossed from one person to another; carefully hand the item to the recipient.  Casual visitors to the laboratory are to be discouraged and MUST have permission from the Teaching Assistant, Faculty Instructor of the section in question, or Laboratory Specialist to enter. All visitors and invited guests MUST adhere to all laboratory safety rules. Adherence is the responsibility of the person visited.  No open-toed shoes are allowed in the laboratory (lab assistants and professor included), as weights or other objects may accidentally drop on people‘s feet; ordinary footware provides a measure of protection from such instances.  Location of PPE (Personal Protection Equipment):  Safety goggles are staged in the back in the drawer marked ―goggles‖  A first aid kit is available near the sink at the front of the lab In addition, the laboratory manuals contain elements of the above as they pertain to each particular experiment. 9 1. Introduction to Measurement / Calculation of Density Purpose The purpose of this experiment is to introduce the student to the laboratory environment. This is done in several ways: with an introduction to the Excel spreadsheet and use of this program to learn how to draw graphs; with an exercise to learn how to use Vernier calipers and a micrometer; and a short experiment that uses the former to calculate the density of four cylinders, comparing these values of density to a standard list to identify the material that makes up each cylinder. Introduction Physics is the foundation of science and describes how things work. To approach the study of the universe in the most effective way, scientists utilize the scientific method, which is a paradigm that guides how inquiry is carried out. The process starts with a hypothesis or educated guess on how something works. That hypothesis is tested through observation, experimentation, calculations, and simulations. If the hypothesis passes the tests, then it is elevated to a theory or model. The theory continues to undergo testing as new evidence appears or as our ability to collect new data increases. Theory The Instructor will introduce the Excel spreadsheet utility in class, as well as demonstrate the use of the Vernier calipers and the micrometer. The students will have a chance to practice with these in the lab. Density is defined as the mass of a substance divided by its volume. m  v 10 The volume of a cylinder can be expressed as:  2 2 Vr l d l 4 where: V = volume r = radius d = diameter l = length 22  = pi, the ratio of the circumference of a circle to its diameter (= ). 7 Substitute the expression for volume into the expression for density to obtain a formula in terms of the measurable quantities. 4m  2 d l Equipment Equipment Needed Qtty. Computer with Excel spreadsheet software 1 Lab balance 1 Vernier Caliper 1 Micrometer 1 Cylinders of four different types of metal 1 each Procedure 1. The Introduction (to this Document and the Labs) section of this manual contains a thorough introduction of graphing techniques to be used in class. We will start with a simple exercise using the Excel spreadsheet to graph a set of points and to draw a best-fit line through these points. Note that your Instructor may have a different exercise in mind; this one is presented as a recommended activity. One can also do this exercise in the graphing utility of DataStudio to get a feel for how that program works. a. Copy the following points into the Excel spreadsheet (likely default name ―Book1‖). This book will ultimately be saved as an xls file to be included with your lab report. x y 0.5 2.5 1.0 2.7 1.2 3.7 2.0 6.5 2.5 6.7 b. Having done that, click on the Chart Wizard button in the toolbar. A box entitled ―Chart Wizard - Step 1 of 4 – Chart Type‖ should appear. Click on ―XY Scatter‖ under ―Chart Type‖, then click the ―Next‖ button c. Click on the ―Series‖ tab, then the add button. Three editable boxes should appear at right, along with a thumbnail-type image of a graph in the upper 11 part of the box. Under ―Name‖, put ―Graphing Exercise‖, then in the ―X values:‖ box, click on the icon button found at the right end of the box. d. The Chart Wizard box should be minimized now, at which point you can go to the points and select which ones will be included as x-values. Click and hold on the first data point x-value and continue to hold as you slide the pointer (now appearing as a ―+‖ sign) down to include all of the x- values. The selected boxes containing the values should be highlighted with a box defined by a moving dashed perimeter e. When the points are selected, click the button icon on the minimized Chart Wizard box to maximize it, then repeat for the y values f. Click on ―Next‖ and add the chart title (―Graphing Exercise‖, if it is not already visible), and x and y values (under the ―Titles‖ tab, name them ―x‖ and ―y‖, respectively). You can also click on the other tabs and edit your graph accordingly (it is recommended that under the ―Legend‖ tab that you deselect the ―Show legend‖ box, this will make the plot area of the graph larger since less area has to be devoted to the legend box) g. Select whether you want the graph to show up as a new sheet or as an object within the current sheet, and click on ―Finish‖ h. Finally, with the graph complete, select the graph itself by clicking once on it (near the edge to select the graph and not just the interior), go to the ―Chart‖ button at the top and single-click on it to generate a drop-down menu. Select ―Add Trend line‖ from the list i. A box will appear entitled ―Add Trend line‖. Select ―Linear‖ as the chart type. Click on the Options tab and select ―Display equation on chart‖ and also enter ―0.5‖ for each of the two boxes under ―Forecast‖. Having done r that, write the equation of the line in your report. Also include the R value (an index of goodness of fit) in your report. j. Save your file as group-x-phys2111-P0y.xls (the x is your group number, and the y is the last digit of the course section number). 2. Next we will work with measurement and density. Measure the dimensions of the four cylinders. Use the Vernier calipers to take at least three measurements of each dimension (that is, each member of your group should measure each dimension) and then use the average of each dimension for your length and diameter values. 3. Use the balance to find the mass of the cylinder. What units did you measure the objects in? 4. Use the equation for density to calculate the density of each cylinder from the measurements that you have taken. Remember to convert the units of measurement to SI units for calculation. 5. Identify each cylinder using the density you calculated and the table on the next page. Using the reference density (―theoretical‖) and the actual density (―experimental‖) of each cylinder, calculate the % difference of each. (exp erimental theoretical) %diff100% theoretical 12 State your answer with respect to error using the correct number of significant digits. Of what type of matter is each cylinder composed? That is, do your calculated values of density match to within experimental error (less than 5%) the densities in the following table (below)? 3 3 3 Aluminum 2.70 × 10 kg / m = 2.70 g/cm 3 3 3 Chromium 7.19 × 10 kg / m = 7.19 g/cm 3 3 3 Copper 8.93 × 10 kg / m = 8.93 g/cm 3 3 3 Iron 7.86 × 10 kg / m = 7.86 g/cm 3 3 3 Steel 7.82 × 10 kg / m = 7.82 g/cm 3 3 3 Stainless Steel 7.70 × 10 kg / m = 7.70 g/cm 3 3 3 Brass 8.40 × 10 kg / m = 8.40 g/cm 3 3 3 Nickel 8.75 × 10 kg / m = 8.75 g/cm 6. Use of the micrometer. (For this section write all measured values in micrometers or µm.) - What is the smallest scale division? - What is the inherent error of measurement using a micrometer? - Measure the thickness of one page in a book, as T1. - Measure the thickness of 100 pages in the same book, as T2 . - Is T /100 = T ? 2 1 - How much error do you expect? - Why should this ratio hold? Why might it not? - (Optional) For additional practice with the micrometer, measure several additional small items. Each individual should measure the same object, then compare the measurements to see how close (or how far apart) the measurements are. The spread in several measurements is a reflection of the standard deviation of the sample of measurements. Addendum: Making Column Graphs in Excel (Courtesy of Bob Aikenhead and Albert Bartlett). Directions for the two extra steps (beyond those needed to make an ordinary line graph) are given in all CAPS. 1) Highlight both data columns with or without the titles. 2) Click the Chart Wizard; this brings up "Chart Wizard, Step 1 of 4" 3) Choose Chart Type XY(Scatter) and any Chart Sub-Type. 4) Click Next 5) The next window is "Chart Wizard, Step 2 of 4" 6) CLICK THE "SERIES" TAB The x and y data ranges now appear in the small windows in the lower right, "X Values" and "Y Values." 7) Click Next and the new window is "Chart Wizard Step 3 of 4" 8) Enter the titles and other data as called for on the tabs. 9) Click Next and the new window is "Chart Wizard Step 4 of 4" 10) Choose the place for the chart 13 11) Click Finish and you now have the XY (Scatter) graph in the Chart Sub Type that you chose in Step 3. 12) RIGHT CLICK IN THE CHART AREA and the "Chart Type" window drops down with the "X Y (Scatter)" choice highlighted. 13) CLICK "COLUMN" to highlight the "Column" choice 14) CLICK OK and you have your column graph ready to finish in the normal way. 14 2. Vectors on a Force Table Purpose The purpose of this lab is to experimentally understand vector operations. Introduction and Theory A vector is a mathematical object used to represent quantities, which have two (or more) independent dimensions, such as magnitude and direction. The rules of scalar arithmetic and algebra do not apply to vectors. Today we will examine the rules of algebraic vector addition. There are three methods that can be used: graphical, analytical, and experimental. In two dimensions a vector can be defined by one of the following orthogonal representations, Cartesian or polar coordinates: A = A î + Aĵ or A = A, θ x y A Where î, ĵ are unit vectors along the x and y-axes of a Cartesian coordinate system respectively. In polar coordinates, A is the magnitude of the vector (its length) and θA is the angle to the vector as measured counter-clockwise from the positive x-axis of a Cartesian coordinate system. Bold characters are vector quantities and non-bold characters are scalar quantities. The projections of the vector A upon the Cartesian axis are: (i.e., polar to Cartesian transformation) A = A·î = Acosθ , A = A·ĵ = Asinθ x A y A 15 To transform from the Cartesian representation to the polar representation: A  y 2 21  A A A ,  tan x y A  A  x The angle is measured counter-clockwise from the positive x-axis (which is defined to be at zero degrees). A second vector can be defined as: B = B î + B ĵ x y The sum of the two vectors is: R = R, θ = A + B= (Ax + Bx)î + (Ay + By) ĵ R To investigate the nature of vector addition experimentally we will use a force table to add two vectors by measuring the net effect of the forces when the system is at equilibrium. Newton‘s Second Law gives the following equation for forces acting on a point when the acceleration of the point is zero: n F ma 0  i i1 We are adding: R = A + B which, when Newton‘s second law is applied, yields: A + B − R = 0 This means that in order to achieve equilibrium on the force table, the resultant vector must be placed in its complementary position. Equipment Equipment Needed Qtty. Force Table with 4 pulleys 1 Four Weight hangers 1 Set of slotted weights (masses) 1 Including (at least) 3 of 50g and 3 of 100 g String and ring assembly that usually comes 1 with the force table Protractor 1 Ruler 1 Bubble Level 1 3 Sheets of Cartesian graph paper 3 16 Procedure 1. Given: A has an angle of 30° and a mass of 200 grams B has an angle of 120° and a mass of 200 grams  Change this information into the polar representation of a vector. It is necessary to convert the units to kilograms and then express this as a force vector, that is, in units of Newtons (Check with your instructor if you don‘t know how).  Find their vector sum by experimental, graphical, and analytical methods. Express the experimental result in polar coordinates. Express the graphical result in rectangular coordinates. Express the analytical result first in the basis vector representation then convert it to the polar representation.  Refer to the experimental hints below. 2. Compare the results from each of the three methods of part one. Which method do you think gives the more accurate results? Why? 3. For the experimental results of part 1 show that: ∑F = 0 i 4. Draw a picture of the force table top with all vectors used in part 1 labeled correctly. 5. What are some physical sources of experimental error? 6. Your instructor may give additional situations to work with experimentally on the force table. One possible exercise involves finding the resultant of three masses at three different angles. Experimental hints: 1. Put 200 grams on A and B and then add masses to the negative of the resultant and vary the angle of the resultant until the circle is balanced in the middle of the force table. Don‘t forget to include the mass of the weight hanger. When this occurs a state of equilibrium exists and the sum of the vectors should be zero. 2. Calculate the force on each vector in SI units. You will need to convert measured values in grams to kg, and then remember that A is a force vector and so must be in Newtons (A = mg). Values should be stated using 3 significant figures 17 3. 1D and 2D Motion: Free-Fall and Projectile Motion th (Adapted from Jerry Wilson, Physics Laboratory Experiments, 4 Ed., pp. 101-105, © 1994 Houghton Mifflin Company) Purpose The purpose of this lab is to (1) determine the value for g, the acceleration due to gravity and (2) determine the muzzle velocity of a projectile fired from a spring-loaded gun. Safety reminder Follow all directions for using the equipment. It is required that safety glasses be worn when doing all the procedures. When using the pendulum, be careful not to injure your hand when cocking the gun, and keep your fingers away from the projectile end of the gun. Background and Theory A free-falling object is an object which is falling under the sole influence of gravity. Any object which is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics which are true of free-falling objects:  Free-falling objects do not encounter air resistance. 2  All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s 18 Examples of objects in free fall include skydivers, an object dropped from the top of a cliff, an apple falling from a tree etc. (www.howstuffworks.com) A projectile is any object which once projected or dropped continues in motion by its own inertia ( the property of matter by which it retains its velocity so long as it is not acted upon by an external force) and is influenced only by the downward force of gravity. By definition, a projectile has only one force acting upon it - the force of gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile. An object which is thrown vertically upward is also a projectile. And an object which is thrown upward at an angle to the horizontal is also a projectile. Projectile motion is simply the motion of an object in a plane (two dimensions) under the influence of gravity. The trajectory describes an arc. The equations of motion describe the components of such motion and are useful to analyze projectile motion. In textbook problems, the initial velocity of an object is typically given, and the subsequent motion is described with equations of motion. The method used in this lab will be to determine the initial velocity of a projectile from range-fall measurements. If a projectile is launched horizontally with an initial velocity of magnitude v = v from a height of y, then the ox o projectile will travel a horizontal distance x (the range of the projectile) while falling the vertical distance y. You can apply this knowledge to all kind of sports such as football, basketball, baseball, and more. 19 (www.howstuffworks.com) Some of the examples pertaining to the projectile motion which include the general motion of the objects moving through air in two dimensions near the earth surface are golf ball, thrown or batted baseball, kicked football, speeding bullet and athletics doing a long or high jump. (www.howstuffworks.com) Consider a cannonball shot horizontally from a very high cliff at a high speed as shown in the above picture. Gravity will act downwards upon the cannonball to affect its vertical motion. Gravity causes a vertical acceleration. The ball will drop vertically below. Gravity is the downward force upon a projectile which influences its vertical motion and causes the parabolic trajectory which is characteristic of projectiles. The initial vertical velocity is v = 0, and the acceleration in the –y direction has a oy magnitude of a = g (acceleration due to gravity—usually taken as negative but taken y positive for convenience for this experiment). There is no horizontal component of acceleration (a = 0); the components of motion are described by x 1 2 x v t and y gt  2 20

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