Laboratory Manual and Workbook for PS-103 Technical Physics I SPRING 2010

Transcription

1 Laboratory Manual and Workbook for PS-103 Technical Physics I SPRING 010

2 Table of Contents Exp. 1: Understanding the difference between Systematic Error and Measurement Uncertainty 3 Exp. : Graphing & Understanding Experimental Uncertainties 10 Exp. 3: Projectile Motion 13 Exp. 4: Equilibrium of Forces 17 Exp. 5: Friction Between Surfaces 0 Exp. 6: Conservation of Momentum 3 Exp. 7: Equilibrium of Torques 8 Exp. 8: Buoyancy: Archimedes Principle 33

3 Experiment 1: Understanding the difference between Systematic Error and Measurement Uncertainty Introduction There exist two types of experimental error that we ll encounter on a regular basis, namely random uncertainties and systematic errors. The value of the uncertainty in a measurement represents the typical size of variations between repeated measurements. In other words, if you repeat the measurement of some quantity many times, you won t in general get precisely same answer each time. Rather, the measured value will vary depending on how precisely you read the scale on the measurement device be it a ruler, an airspeed indicator, altimeter, etc. The random uncertainties are determined by the limitations of a measuring device, how finely the measuring scale is divided, and your ability to read the scale. When you make a measurement you should always estimate the uncertainty in your measurements and you should usually write it down right next to the measurement in the following manner x= / m This notation indicates that the measured value was 5.01 meters but that one could expect that value to be uncertain by as much as 0.01 meters or 1 mm because that is the smallest division on the ruler. The uncertainty gives us a sense of the precision with which the measurement was made. The second kind of experimental error is systematic error. Unlike uncertainties, systematic error is an error that shows up in all measurements unless corrected. An example of a systematic error is a non zero-offset. For example, if the end of a meter ruler is worn down, then this ruler will always underestimate the length of an object because the scale starts at a number higher than zero. There is one systematic error you always correct for in aviation: the altimeter is always set before takeoff to the altimeter setting provided by ATIS. This adjustment corrects the altimeter for non-standard temperature and pressure. An obvious application of systematic error, manifested by your reaction time, is encountered in piloting your aircraft on the ground and in the air. From your study of kinematics, you know that if you re traveling at a constant speed v, then a distance traveled (displacement) is given by: x = v t where t is an elapsed time. In the context of this experiment, decisions made in the cockpit can be delayed by the pilot s reaction time t, during which an aircraft traveling at constant speed v will travel some distance x. After completing this lab, you will know your reaction time t, and will therefore be able to calculate the distance your plane travels during your reaction time. 3

4 Procedure In this experiment, you will be using a ruler to measure your reaction time. Your lab partner will hold the ruler vertically so that it hangs between your thumb and forefinger which should not touch the ruler. Your partner will then drop the ruler and you will try to catch it between your thumb and forefinger, without moving your hand up or down! By noting how far the ruler fell before you could catch it, you can calculate your reaction time. See Appendix A for an explanation of the kinematics and how to calculate your reaction time. Part a: in-class activity Have your partner suspend a ruler between your thumb and forefinger, so that the point where the ruler reads zero centimeters is centered between your thumb and forefinger. Your partner should let go of the ruler and let it drop between your thumb and forefinger. As soon as you see the ruler start to drop, try to grab it with your thumb and forefinger. Read the centimeter scale where your thumb and forefinger grabbed the ruler and use the formula above to calculate your corresponding reaction time. Repeat for a total of thirty trials, recording your results in Table 1. You measure the y distance and calculate the time using t = (see Appendix A). g Table 1: Reaction Times TRIAL DISTANCE [CM] REACTION TIME [S] TRIAL DISTANCE [CM] REACTION TIME [S] 4

5 Part b: Calculations Determine the average of your trials and record it here. Save this number for use in future experiments. Average Reaction Time: [s] The average of a set of numbers isn t always guaranteed to be the most likely, and hence representative, outcome. For example, skewed distributions feature an average value which is offset from the most likely (or most frequent) outcome. In these cases, the statistical mode, the value of the reaction time (bin) which occurs most frequently, is a better characterization of your reaction time than the mean, or average. Make a frequency plot, also known as a histogram, of your trials. See Appendix B. Be sure to correctly label the axes on the plot. Is your plot Normal (a.k.a. Gaussian, or bellshaped) or skewed? What is the mode of the plot? Reaction time mode: [s]. What is the mean reaction time, ie. the average of the 30 reaction times you calculated? Mean reaction time: [s]. The histogram you generated above gives you an easy way of identifying the uncertainty in the measurement of the mean reaction time. Basically, the uncertainty is given by the width of the histogram divided by two. You can read this directly off the histogram. You can also use the fact that the width/ of a Normal distribution is given by the Standard Deviation, σ, which is a statistical function that can be found on most calculators these days. What is the uncertainty in your measurements, using either the standard deviation or by taking the width of the histogram and dividing by two? Uncertainty in mean reaction time [s] If you are flying at a constant velocity of (130 knots = 67 m/s), how far, in meters, does the plane travel in your reaction time? Attach your frequency plot of reaction time to this paper and turn it in to your laboratory Teaching Assistant (TA) for grading. 5

6 Appendix A: Kinematics From kinematics, the displacement as a function of time for an object experiencing a uniform acceleration is: 1 y v t at = 0 +, (1) where y is the displacement of the object, v 0 is the initial velocity of the object, a is the acceleration acting on the object, and t is the time the object is in motion. In this case, the object we are considering is the ruler. Because we let go of the ruler and allow it to fall, the initial velocity of the ruler is zero. However, the ruler does experience an acceleration due to gravity as it falls. Thus, equation (1) reduces to: y 1 = gt, () where a has been replaced by g, the acceleration due to gravity at the surface of the earth. Thus, if we know the distance through which an object has fallen, we can calculate the object s time of fight by rearranging equation (). To get t by itself, divide both sides of equation () by 1/g: y t g =. (3) Thus, y t =. (4) g 6

7 Appendix B: Statistics and Frequency Plots Descriptive Statistics The most common descriptive statistics, i.e. the accepted quantitative methods of describing a data set, are: The average (or mean): a measure of the central tendency of the data, or the clustering of the data about a single value. The standard deviation: a measure of the spread in the data about the mean. The median: if a data set is ordered from smallest to largest value, the median is the value in the center (for an odd number of values) or the average of the two middle values (for an even number of values) and The mode: the value which appears most frequently In addition, there are two less common but important ways to characterize the shape of a data distribution, namely: The skewness: a measure of the degree to which a data set or distribution is nonsymmetric; the most common outcome is that the distribution has a long tail on one side of the center. and The kurtosis: a measure of the sharpness or flatness of the distribution, as compared to a bell-shaped (Normal or Guassian) curve. Frequency Distributions and Plots. The frequency of data answers the question how many times does this value (or similar values) show up in my data set?. Because it doesn t make sense to talk about the frequency of a single data point, in general (think about it the frequency of a unique data point is exactly one (1.0), if that particular value, out to some number of decimal places, only appears once), we can bin similar data points together to see how often a group of data, with similar values, shows up within the overall data set. As an example, consider a game where you throw pennies into a row of cups, with the intent of getting the penny in the middle cup as often as possible the person with the most pennies in the middle cup wins. The cups are lined up as shown below: Figure B-1: Throwing cups After throwing coins, you count the number of pennies in each cup. Stacking the coins, you observe the outcome shown in Figure B-. 7

8 Cup: (none) Figure B-: The number of coins tossed into each cup In this example, you have one penny in cups 1 and, two pennies in cup 3, four pennies in cup 4 (the middle cup), three pennies in cup 5, one penny in cup 6, and you missed cup 7 entirely. Each cup constitutes a bin in the statistical sense. The frequency is just the number of pennies in each bin. The frequency distribution is the outcome of the game, considering all the cups (the entire ensemble of data). You can construct a frequency table easily, as shown below. Bin (cup number) frequency Table B-1: The gaming example frequency distribution You can also easily construct or draw a frequency histogram, as shown below (MicroSoft Excel was used to construct this histogram using the Graph Wizard menu). Coin Tossing Game Example 5 frequency (coins per cup) cup number 8

9 Appendix C: Notes on propagating measurement uncertainties through calculations When adding or subtracting numbers, add the actual uncertainties in quadrature For example, if z = x 1 + x or z = x 1 - x then the uncertainty is z = ( x 1 + x ) When multiplying or dividing numbers, add the fractional uncertainties in quadrature For example, if z = x 1 x or z = x 1 / x then z/z = [ ( x 1 / x 1 ) + ( x / x ) ] So that the uncertainty is z = z [ ( x 1 / x 1 ) + ( x / x ) ] 9

10 Introduction Experiment : Graphing & Understanding Experimental Uncertainties Graphs are a common way to present information in a concise, compact format, and are commonly encountered in aircraft operations and maintenance manuals. Knowledge of how to properly construct and interpret graphs is therefore essential to students in their major field of study, as well as in this Physics class! In this lab, we ll review the construction of graphs, gather simple data, plot data on these graphs and, along the way, explore the concept of random error. As discussed in your first lab, there exists two types of error, namely systematic (or bias or calibration or offset) error and random error. The former can be virtually eliminated by a careful calibration of measurement devices, but the latter can only be minimized, never entirely eliminated. Experimental Procedure In this lab, we ll collect simple data by dropping a ball from a measured height and recording the height to which it rebounds. While there s a lot of physics which influences this behavior, e.g. conservation of energy and momentum, and energy loss mechanisms, today we ll concentrate on data collection, plotting, and Equipment Meter stick tape (optional) Ball bubble level (optional) Figure 1:Laboratory Set-up 10

11 In-class activity Place a meter stick vertically next to a table or next to a wall with the metric side visible; you may wish to use tape to secure the meter stick to a surface, and you may use a bubble level to ensure that the meter stick is truly vertical. Place the ball next to the meter stick. With the bottom of the ball at an arbitrary height, record this height and drop the ball. To the best of your ability, observe and record the height to which the ball bounces or rebounds. This should be the height of the bottom of the ball at the top of the rebound. Record these data in Table 1 and indicate which of the two heights recorded in each trial is the independent variable (the one you can control) and which one is the dependent variable. Table 1: single measurement data set TRIAL INITIAL HEIGHT BOUNCE HEIGHT 1 +/- +/- +/- +/- 3 +/- +/- 4 +/- +/- 5 +/- +/- Record all heights in meters, [m]. Plot Table 1 s bounce height, as a function of initial height. Plot the uncertainties as error bars on each point. Return to the initial heights you utilized in filling out Table 1. For each of these heights, drop the ball five times and record the bounce height for each trial in Table. When each set of trials is complete, compute the average bounce height for each initial height. Table : average (multiple bounce) data set INITIAL HEIGHTS (from Table 1) Trial 1 Trial Trial 3 Trial 4 Trial 5 Average* +/- +/- +/- +/- +/- *Calculate the uncertainty in each average by computing the Standard Deviation, σ; σ = Σ (average - x i ) / (N-1) where N is the total number of measurements, and x i are the individual measurements. 11

12 Plot Table s average bounce height, as a function of initial height, on the same graph as the previous one. Plot the uncertainties as error bars on each point. Draw a best fit line. Clearly indicate which best-fit line is which by using a different color, a different line thickness, or some other means which allows the reader to distinguish between the two lines. Please answer the following questions. 1) One way to minimize the effect of random error/uncertainty on a measured value is to take multiple readings of the dependent variable for the same independent variable value. This is what you did in Table, by performing multiple trails using the same initial height, then averaging the resulting bounce heights. Does this procedure actually affect the best fit line? Explain. ) Describe some sources of random error that you observed in the experimental procedure: 3) Refer to your best-fit line drawn using the averaged data set of Table. Pick two points on this line. The points should be towards the ends of the line you ve drawn on your graph. Determine the x coordinate of each point by drawing a vertical line from the point down to the x (horizontal) axis. Determine the y coordinate by drawing a horizontal line from the point to the y (vertical) axis. Point 1: x 1 : [m] and y 1 : [m] Point : x : [m] and y : [m] Compute the differences of each coordinate: y = y y 1 = [m] and x = x x 1 = [m] Compute the slope of your line: Slope m = y/ x = Compute the y-intercept of your best-fit line: y-intercept b = y 1 - mx 1 = [m] Write the equation of your best-fit line in the form y = mx + b = [m]. 1

13 Experiment 3: Projectile Motion Introduction Today we ll examine projectile motion which is the motion of an object under the influence of gravity. Kinematics of Projectile Motion In this laboratory experiment, our projectile consists of a metal ball bearing. We will observe two distinct dynamical cases in this experiment, namely: Phase 1 - Constant Velocity Phase II - Constant Acceleration (free fall) In Phase 1, the metal ball will be rolling freely across a desktop. The purpose of this segment of the projectile s motion is to allow the experimenter to measure the initial velocity of the metal ball. Denoting the horizontal direction as x and applying the kinematic equations, x = x + v t v x = Because the forces in the horizontal direction (rolling friction, air drag) are negligible, the horizontal accelerations are negligible, so that the horizontal velocity is constant, as indicated in Eq. (). The position of the ball increases linearly in time. In the Phase II, free fall, the only non-negligible force acting on the projectile is gravity. If we retain our definition of x as being in the horizontal direction and define y to be in the vertical direction, then the kinematic equations are: 0 v 0, x 0, x (1) () y v x v y x = = x 0 v + 0, x v 0, x t 1 = y0 g t = g t (3) (4) (5) (6) Please note that these equations are specifically tailored to this lab s experimental procedure and do not represent the generic case. Equipment Meter stick Stopwatch Carbon paper Steel balls 13

14 Experimental Procedure Simply roll the ball along the desktop. Try your best to roll it so that it achieves a constant velocity. If you roll it slowly, it will be easier to time it. x 0 in Eq. (1) is the initial position of the ball on the desktop and x is the final position on the desktop. The time the ball takes to travel along the desktop is elapsed time t, measured using the stopwatch. From distance and time, you can solve Eq. (1) to yield the velocity of the ball across the desktop: x x0 v0, x = (7) t If x 0 was set to 0, Eq. (7) reduces to: v x = 0, x t (8) The simplest way to measure the velocity is to time how long it takes the ball to roll a distance of 1m along the desktop. Consider the path of the ball after it leaves the tabletop with a horizontal velocity given by Eq. (7) or (8). If we place the origin of the floor coordinate frame on the floor directly below the edge of the table, then we can identify the variables in Eq. (3)-(6) with the following quantities (preferred units are in [square brackets]): and x 0 : 0 [m]; v 0,x = the initial velocity in the horizontal direction [m/s], computed from either Eq. (7) or (8); y 0 : the initial height of the ball [m], which is just the height of the tabletop, H; v 0,y : the initial velocity in the vertical direction, which equals 0 [m/s] because the ball left the table horizontally; g: the acceleration of gravity [m/s ]; t ff is the free-fall time [s], starting when the ball leaves the table; x is the horizontal position of the ball [m] at time t; and y is the vertical position of the ball [m] at time t. The free fall time, t ff requires a starting point and an endpoint to qualify as an elapsed time. The logical start time is when the ball leaves the tabletop, while an obvious endpoint is when the ball impacts the floor. This time is too short to be measured using a stopwatch, but it can be determined using Eq. (5): 14

15 y y 0 = 0 H + H = = 1 g t g t, so, so t = H g (9) The distance traveled horizontally, x, is the range R of the projectile given by Eq. (3). If we combine Eq. (3) and (9) by substituting Eq. (9), the time t, into Eq. (3), then the horizontal range, R, is given by: x x 0 = R = v 0, x H g (10) Notice that we have eliminated the common factor, t, from both equations everything in Eq. (10) is a distance, a velocity, or an acceleration. From Eq. (10), we can solve for the acceleration g to which the ball is subject during free fall: g = v0, H R x (11) We ll use Eq. (11) to estimate the acceleration due to gravity. If, allowing for random experimental error, this value is equal to the standard value of 9.8 [m/s ], then we have accomplished our objective. In class activity (1) Place a meter ruler on the desktop. This is the numerator of Eq. (7) or (8). () Launch the ball across the table top and use the stopwatch to determine the time it takes the ball to travel the 1 m distance to the edge of the table; this is the denominator in Eq. (8). (3) Use Eq. (8) to calculate the velocity v 0,x across the table top. Record these values in Table 1. (4) Carefully measure and record the height of the table H and record this value in Table 1. (5) Carefully measure the projectile s horizontal range, R after it leaves the table; this is the horizontal distance from the table top s edge to the impact point on the floor. You can more easily observe the impact of the projectile by placing carbon paper on the floor. When the ball strikes the carbon paper, it will leave a mark on the white paper below the horizontal range R is then just measured from the table s edge to this mark. 15

16 Table 1 Phase I motion: across the tabletop Phase II motion: free fall Table Height H [m] Trial x [m] t [s] v 0, x [m/s] R (m) g (m/s ) Average g [m/s ] Standard Deviation σ [m/s ] Final Answer g = +/- ( m/s ) Use Eq. (11) to compute the value for g using the initial velocity v o,x and the range R av and the height H of the table. Repeat for each of the 4 trials. Calculate the average value of g. Calculate the uncertainty in the average value of g by computing the Standard Deviation, σ; σ = Σ (average - x i ) / (N-1) Please answer the following questions. 1) What possible sources of random error did you observe in this experiment? List as many as you can think of in the order of their relative importance: ) Do you think that the kinematic equations are validated by this experiment given the value you measured for g? Explain. 16

17 Experiment 4: Forces Quantities which have both magnitude and direction are called vectors. Velocity, acceleration, and force are all vector quantities; whereas, time, temperature, and volume are not. The latter three quantities have only size, but no direction, and are called scalars. Vectors, like scalars, can be added, but not with ordinary arithmetic, because vectors have direction. The vector sum of two or more vectors is called the resultant, and there are two basic methods used to find the resultant: the graphical method and the component method. Graphical Method for Adding Vectors We begin by looking at the graphical method. Vectors are drawn as arrows, where the direction of the arrow gives the direction of the vector and the length of the arrow gives the magnitude of the vector as illustrated in Figure 1. F 1 = 30 [N] P F = 40 [N] Figure 1: Two Force Vectors Acting on a Point, P. In order to find the resultant of two vectors graphically, they must be drawn head to tail, as in Figure. The head of a vector is the end with the arrowhead, and the tail of a vector is the end without the arrowhead. Resultant =? [N] F 1 = 30 [N] θ R =? F = 40 [N] Figure : The Resultant using the Graphical Method. 17

18 On a sheet of graph paper, use a ruler to draw vector arrows head to tail for two forces F 1 = 30 N and F = 40 N. Make sure you label each. Use a length scale of 1 [cm] = [N] to draw the vectors and record your scale on your graph paper. The resultant vector is drawn from the tail of the first vector to the head of the second vector. Measure the length of the resultant vector using a ruler and record your result on the graph paper. Finally, using a protractor, you can measure the angle of the resultant vector to determine its direction. Mathematical Method for Adding Vectors In order to mathematically determine the resultant of two vectors, we must break each vector into it s x and y components. Let F y = 30 N and F x = 40 N Calculate the magnitude of the resultant vector, R using the Pythagorean Theorem: R = ( F x + F y ) Calculate the angle, θ R, R= θ R = Tan -1 ( F y / F x ) θ R = Force Table Method for Adding Vectors Equipment Force Table Pulleys String Protractor Masses Mass Hangers Ruler Bubble Leveler Figure 3: Laboratory Set-up 18

19 Procedure The apparatus you will be using in this experiment is called a force table, and consists of a round metal plate with angle markings mounted on a vertical rod. Three radial forces are applied to a small ring at the center of the table by masses hung over pulleys. Until the forces applied by the masses at the ends of the strings are in balance, the ring is held in position by a vertical pin in the center of the table. In this experiment, you will use the force table to verify your calculations. When summing forces, we are dealing with Newton s Second Law. If a system is at rest the vector sum, or resultant, of the forces acting on the system must be equal to zero. This state is called static equilibrium, since the system is at rest. When a system is in equilibrium, it has a vector called the equilibrant, which cancels the effects of the other vectors. The equilibrant has the same magnitude as the resultant but points in the opposite direction. When two vectors are in the opposite direction of one another, they are 180 apart. In this experiment, you will calculate the resultant force from two vectors on your vector force table, and then apply that magnitude force in the opposite direction to your force table. When you apply the equilibrant vector to your force table, the pin in the center of the table should be at the center of the ring. The magnitudes of your force vectors will be provided by the weights which you hang over the pulley. The direction of each of your force vectors is given by the angle scale on the round face of the table. For the purposes of this experiment, we will consider the positive horizontal direction (+ x) to be at an angle of 0, the negative horizontal direction (- x) to be at an angle of 180, the positive vertical direction (+ y) to be at an angle of 90, and the negative vertical direction (- y) to be at an angle of 70. Before beginning, make sure the force table is level, and adjust each string so that it is level with the force table and aligned correctly on its pulley. Check the center knot on each string to make sure that each string is pointing toward the center of the ring, and adjust this over the course of the experiment as need be. Finally, orient the force table with the 0 angle on the right side. Add a total of 400 [g] to one mass hanger which runs over a pulley at 0. Add a total of 300 [g] to another mass hanger which runs over a pulley at 90. Use the force you calculated mathematically to determine the amount of mass, M needed on the third mass hanger to cancel the effect of the resultant vector. Add this mass to the mass hanger over the third pulley. The direction should be opposite the angle you calculated for the resultant vector, or, equivalently, 180 o + θ R. The ring should now be centered providing empirical verification of your calculations. Write down the actual values for the mass you added and the angle that was required for the ring to be centered. mass, M = +/- [g] angle 180 = θ R = +/- [degrees] 19

20 Experiment 5: Friction Friction is the resisting force which opposes the motion or attempted motion of one body over another. The force needed to overcome friction depends on the nature of the materials in contact, and varies from one material to another. For example, the friction force between two smooth surfaces in contact, such as marble or tile, is less than the friction force between two rough surfaces in contact, such as tires on pavement. To illustrate this, imagine pushing a box across the floor. If the magnitude of the force, F, that you exert on the box is small, the frictional force, f, acting in the direction opposing F, will keep the box from moving, and the box remains in a state of static equilibrium. If the magnitude of the force, F, is gradually increased, the box will begin to move across the floor. The frictional force f s is related to the perpendicular normal force, N, by the relationship: f = N, (1) s, k µ s, k where N is the magnitude of the normal force between the box and the surface, and µ s (Greek lower case mu ) is the coefficient of static friction between the box and the surface. In this experiment, you will be determining the coefficient of static friction for a wood block on an inclined plane. Dividing both sides of equation (1) by the normal force, N, yields an expression for the coefficient of static friction, Equipment Inclined Plane f s µ = s N () Masses y x N F f m W W W // Figure 1 0

21 In this experiment, we will consider what happens when we have a situation like that depicted in Figure 1. In this case, the normal force, N, and the weight of the block, W, are not equal. Instead, the normal force is equal to the perpendicular component of the weight of the block, W = W cosθ, so that N = W cosθ (3) where is the angle at which the plane upon which the block rests is inclined. Just before the block slips, the frictional force f s just balances the parallel component of the blocks weight directed down the slope, W // = W sinθ, so that If we divide equations (3) and (4) we obtain; f s = W sin θ (4) f s / N = Tan θ Thus, by comparison with equation (), we see that the coefficient of static friction is given by: µ s = Tan θ. (5) Use equation (5) to calculate the coefficient of static friction, µ s. Take the average of all five values and calculate the standard deviation, σ; σ = Σ (average - x i ) / (N-1) Table 1: Coefficient of Static Friction on an Inclined Plane Trial Angle, Coefficient of Static Friction, µ s Average µ s +/ 1

22 Would you expect the frictional force to depend on the area of contact between the two surfaces? Investigate this hypothesis by repeating the experiment, but now with the block tipped on it s narrow edge. Table : Coefficient of Static Friction on an Inclined Plane - Block on it s edge Trial Angle, Coefficient of Static Friction, µ s Average µ s +/ Compare your average with the previous average and answer the following question: Does the frictional force depend on the area of contact between the two surfaces? Explain how you reached your conclusion: Are you surprised by your conclusion? Explain.

23 Experiment 6: Conservation of Momentum Collisions are a common experience in our lives, from the collisions of billiard balls to the collisions of two cars. Over the course of history, various natural philosophers have discussed the laws that govern collisions. As early as the seventeenth century, it was understood that collisions between hard bodies, such as billiard balls, behaved differently from collisions where the two bodies stick together after the collision, as in throwing spaghetti bolognaise against a wall. It is the concept of momentum which governs the mechanics of collisions. Momentum is a vector quantity, having both magnitude and direction, and is defined as an object s mass multiplied by its velocity: p = mv, (1) with units of [kg m/s]. If an object is assumed to have constant mass, then the object s change in momentum corresponds to a change in the object s velocity. Because a change of velocity over a time interval t, is the definition of an acceleration, we can consider Newton s Seconds Law, F = ma, rewritten in the form: ( m v) F =, () t where the quantity m v is the change in the object s momentum, or simply p. Multiplying both sides of equation (), we can write: F t = m v, (3) where the quantity m v is the change in the object s momentum. Thus, F t = p, (4) where the quantity F t is called the impulse. Thus, a force, F, acting on a body for some amount of time, t, causes a change in the object s momentum. When you kick a soccer ball, for example, your foot imparts a force to the ball during the time it is in contact with the ball. This imparted force causes a change in motion, and thus a change in momentum of the ball from its initial state before you kicked it. A simple system consists of two objects, A and B, colliding with each other. During the collision, the force F A acting on object A is equal and opposite the force F B acting on object B: F A = -F B. Because the time t during which the forces acting on each object are the same, it follows that the change in momentum of the two objects are equal and opposite, so that the total change in momentum of the system is zero. Thus, the momentum before a collision is equal to the momentum after a collision; p + p = p + p, A, i B, i A, f B, f 3

24 where the left hand side of the equation indicates momentum before the collision, and the right hand side indicates momentum after the collision. A collision is called totally elastic if both momentum and energy are conserved in the collision process. A collision is called totally inelastic if the two bodies stick together after the collision. Momentum is conserved but Energy is not. It is inelastic collisions that we are going to explore in this lab. Equipment Air track carts Lab Pro + photo-gates Triple Beam Balance Bubble Leveler Figure 1. Laboratory Equipment Figure. Laboratory Set-up 4

25 Procedure In this experiment, you will observe conservation of momentum for inelastic collisions using carts gliding on an air track. An air track is used in order to minimize the main external force which is friction between the carts and the track. In order to calculate the momentum of the carts, we need to know only their masses and their velocities before and after the collision. The velocities will be measured by a computer when each cart passes through a pair of photo-gates. 1. Choose two reference points along the air track before and after the place where the collision will occur. For example, if the collision will occur at about the 50 [cm] mark, choose the 0 [cm] mark and the 40 [cm] mark before the collision and the 60 [cm] mark and the 80 [cm] mark after the collision. Place the photo-gates at these locations.. Using the LabPro computer, make a note of the initial velocity of cart A, (cart B is stationary) and the final velocities of carts A and B. 3. You can obtain an estimate of the velocity uncertainty by sending a single cart along the track through both pairs of photo-gates. In an ideal situation, the velocity should remain constant. In actual fact, the velocity will change slightly because of friction and air resistance. Make a note of the difference in the two velocities and adopt that as a measure of the uncertainty in the velocity. Write your velocity uncertainty here m/s Part I: Inelastic Collision, Equal Masses Figure 3: Cart Configuration for Inelastic Collisions 5

26 Record data from Part I here: Mass of Cart 1 (m 1 ): Mass of Cart (m ): Table 1.1 Before the collision (Note: Cart is stationary, so it s initial momentum = 0 kg m/s) Cart 1 Before Collision (Hint: Use Appendix C to calculate the uncertainties in derived quantities) Trial Initial Velocity[m/s] Initial Momentum [kg m/s] 1 +/- +/- +/- +/- 3 +/- +/- Table 1.: After the Collision (Note: Cart 1 and are now joined together, so the combined mass, m 1 + m = kg ) Cart 1 + Cart After the Collision (Hint: Use Appendix C to calculate the uncertainties in derived quantities) Trial Final Velocity [m/s] Final Momentum [kg m/s] 1 +/- +/- +/- +/- 3 +/- +/- 6

27 Table : Conserving Momentum Trial 1 initial momentum = +/- final momentum = +/- final initial = +/- Trial initial momentum = +/- final momentum = +/- final initial = +/- Trial 3 initial momentum = +/- final momentum = +/- final initial = +/- What should the final initial momentum be for each trial? Is momentum conserved for your experiment, within the measurement uncertainties? If not, please explain why not. 7

28 Experiment 7: Moments (Torques) and Equilibria When considering rotational motion, there exist rotational analogs to all the quantities discussed in linear motion and the one examined in the following experiment is that of torque, the rotational equivalent of a force. In order to produce rotational motion, several factors come into play. When using a wrench, for example, the distance from your hand to the bolt is important, as is the direction you push and the magnitude of the force you apply. A torque is a vector quantity, just as a force is, and therefore, a torque, or the tendency of a force to produce a rotation, is defined as: τ = Fr, (1) where the quantity r is called the lever arm, and is the perpendicular distance from the applied force to the center of rotation, and is the Greek lower case letter tau, the symbol used to indicate torque. Thus, a larger force applied at a farther distance from the center of rotation produces greater rotation than does a smaller force applied at a lesser distance from the center of rotation. Because force is expressed in units of [N], and distance in units of [m], torque has units of [N m], which is similar to the units of energy or work. However, the two represent completely different concepts. While work denotes a change in energy through or over some distance, torque denotes a rotation produced as a result of a force acting at some distance. Just as we did for forces, we can use Newton s Second Law with torques to say that the sum of the individual torques acting on a system is equal to the net torque acting on a system: τ = τ net. () It is necessary to consider torque, rather than force, in rotational motion, because two equal but opposite parallel forces on an object would produce no translational motion since they sum to zero. It is possible, however, for two equal but opposite parallel forces to produce a net torque that is not equal to zero. Consider the following diagram, where two parallel forces of equal magnitude, but opposite direction, are applied to a rectangular block that is allowed to rotate about a vertical axis. The arrows labeled ω indicate the direction of rotation caused by the application of both forces. If we curl our fingers in the direction of rotation, the direction our thumb points is the direction of the torque vector. In Figure 1, because both forces cause rotation in the same direction, both torques are in the same direction, and though the net force on the block is zero, the net torque is not. 8

29 Axis of Rotation ω F = 5N ω F 1 = 5N 1 R/ Figure 1: Rotational Disequilibrium In order to distinguish a positive torque from a negative one, we use the right hand rule, where the fingers are curled in the direction of rotation. When your fingers curl in a counterclockwise direction, we call this a positive rotation, and thus, a positive torque. Similarly, when your fingers curl in a clockwise direction, we call this a negative rotation, and thus, a negative torque. In each case, your thumb indicates the direction of the torque. In Figure 1, both torques are negative. All of Newton s laws of motion apply to rotational motion. Thus, an object which is not rotating will continue not to rotate as long as the net torque on it is zero, and likewise for an object rotating with uniform speed. Equipment R Meter Stick Clamp Support Stand Mass Hangers Masses Procedure In this experiment, you will consider torques in static rotational equilibrium. The apparatus you will use is depicted in Figure. It consists of a meter stick with a clamp on a support stand, and masses hanging from moveable mass hangers. 9

30 m 1 m Figure : Rotational Equilibrium Apparatus In this case, the forces on the meter stick producing the torques are the weights of the masses, mg, as shown in Figure 3. Thus, equation (1) can be written as: τ = mgx. (3) where we have not only replaced F = W = mg, but we have also replaced r with x, where x is the distance from the mass hangers to the balance point of the meter stick. x 0 x 1 x ω 1 ω m 1 m W 1 W Figure 3: Rotational Equilibrium Once again, the rotations caused by the forces are indicated with gray arrows, while the torque vectors are indicated using the following symbols: indicating out of the page and indicating into of the page. Note that throughout this experiment, the torque 30

31 vectors will be pointing either toward or away from you, as here, where 1 is out of the page, and is into the page. Before you begin, you will need to balance the meter stick. Once done, record the balance point, x 0, in your data tables. Also, measure the mass of the mass hangers and include them in your masses. Part I: Two Unequal Masses in Equilibrium 1. Hang a 100 [g] mass, m 1, at a position approximately 45 [cm] from x 0 on either side of the meter stick. Record this position, x 1, in your data table.. Hang a 00 [g] mass, m, on the opposite side of the meter stick from m 1. Adjust the position of m until static equilibrium is found. Record the position, x, of m from x Using equation (3), calculate the torques, 1 and, around x 0 due to the weights of m 1 and m, and record them in your data table. Remember, torque is a vector, so the torques on one side of the ruler are going to have opposite sign to the torques on the other side. Choose a sign convention and stick to it. Table 1: Two Unequal Masses in Equilibrium Balance Point, x 0 = [m] Mass of hanger 1, [kg] Mass of hanger, [kg] Mass, m 1 +hanger [kg] Mass, m +hanger [kg] Distance of x 1 from x 0 [m] +/- Distance of x from x 0 [m] +/- Torque, 1 [Nm] (Hint: Use Appendix C to calculate the uncertainties in derived quantities) +/- Torque, [Nm] (Hint: Use Appendix C to calculate the uncertainties in derived quantities) +/- 1 + = +/- What should the sum of the torques be when the ruler is balanced? 31

32 Part II: Three Unequal Masses in Equilibrium 1. Hang a 50 [g] mass, m 1, and a 100 [g] mass, m, at two different positions on the same side of the meter stick. Record their positions from x 0 as x 1 and x respectively.. Hang a 00 [g] mass, m 3, on the opposite side of the meter stick from m 1 and m. Adjust the position of m 3 until static equilibrium is found. Record the position, x 3, of m 3 from x Using equation (3), calculate the torques, 1 and, around x 0 due to the weights of m 1 and m, and record them in your data table. Remember, torque is a vector, so the torques on one side of the ruler are going to have opposite sign to the torques on the other side. Choose a sign convention and stick to it. Table : Three Unequal Masses in Equilibrium Balance Point, x 0 = [m] Mass of hanger 1, [kg] Mass of hanger, [kg] Mass of hanger 3, [kg] Mass, m 1 +hanger [kg] Mass, m +hanger [kg] Mass, m 3 +hanger [kg] Distance of x 1 from x 0 [m] +/- Distance of x from x 0 [m] +/- Distance of x 3 from x 0 [m] +/- Torque, 1 [Nm] +/- Torque, [Nm] +/- Torque, 3 [Nm] +/ = +/- What should the sum of the torques be when the ruler is balanced? 3

33 Experiment 8: Archimedes Principle Introduction Archimedes principle states: the buoyant force on a body in a fluid is equal to the weight of the fluid displaced by the body. We will examine this principle in today s laboratory for objects that sink when immersed in a fluid. If we have an object of mass m and volume V, its density ρ (the Greek lower case letter rho ) is given by: m = (1) V In air, the objects weight, w, will be equal to m g, where g is the acceleration due to gravity. If the object is weighed when immersed in the fluid, it will apparently weigh less because the object is subject to a buoyant force, equal to the weight of the fluid displaced. The apparent weight is, w' = m' g, where m' is its apparent mass. According to Archimedes we can write the equation: m g m' g = m f g () The difference in the weight of the object out of the fluid and the weight of the object while immersed in the fluid is equal to the weight of the fluid displaced, m f g, where m f is the mass of the fluid displaced. If the fluid has a density, ρ f = m f /V f, we can substitute ρ f V f for m f in equation (), obtaining: m g m' g = ρ f V f g (3) However, since the volume if the fluid displaced is equal to the volume of the submerged object, we can replace V f with m/ρ from equation (1). m m g m g = ρf g (4) ρ Diving both sides of equation (4) by g, and solving for ρ, we get: m ρ = ρf (5) m m 33

34 Therefore, if we have a fluid of known density, and we can measure the mass of the object m and its apparent mass while immersed in the fluid m', we can use Archimedes principle to find the object s density. Equipment Graduated Cylinder Metal Cylinder Ruler, Meter Stick, or Vernier Calipers Triple Beam Balance Wood Cylinder Ring Stand Figure 1: Laboratory Set-up Procedure 1. Verify Archimedes principle by using a graduated cylinder or beaker to determine how much water is displaced by the small metal cylinder provided. Read the level of the water before and after the object is submerged and determine the amount of water displaced. Assuming the density of water to be 1 x10 3 kg/m 3, calculate the weight of the water which is displaced. Use the scale to determine the mass of the metal cylinder and its apparent mass when submerged in water. Find the buoyant force supplied by the water and compare to the weight of the water displaced. 34

35 Table 1: (Hint: Use Appendix C to calculate the uncertainties in derived quantities) Mass of object in air (kg) Apparent mass of object in water (kg) Mass Difference (kg) Bouyant Force (N) +/- +/- +/- +/ Initial volume of water (m 3 ) Final volume of water (m 3 ) Volume difference (m 3 ) Mass of water displaced (kg) Weight of displaced water (N) +/- +/- +/- +/- +/-. Use Archimedes principle to determine the density of the object using equation (5). Density of object = +/- kg/m 3 3. Use the ruler provided to determine the length of the object and the vernier calipers to measure the diameter. Use the mass of the object from Table 1 and calculate the density using equation (1). Note: the volume of a cylinder is given by: V = r h where h is the height of the cylinder or it s length and r is its radius. Table : (Hint: Use Appendix C to calculate the uncertainties in derived quantities) Length of object (m) Diameter of object (m) Volume of object (m 3 ) Mass of object (kg) Density of object (kg/m 3 ) +/- +/- +/- +/- +/- 4. Compare the results of step 3 to those of step. Do your results verify Archimedes principle? Please explain your answer: 35