5. Newton's 211d Law Lab Purpose The purpose of this lab is to calculate the acceleration of a block using both kinematic and dynamic equations of motion. Newton's Second Law Figure 1 below shows a simplied schematic of the experimental setup to help with the discussion. A block is sitting on a table with a string attached to it. At the opposite end of the table is a grooved board with the groove side down and a textbook atop the board to hold it steady. The string then runs through the opening created by the groove, over the edge of the table and a mass is attached to the hanging end of the string. Figure 2 shows the end view of the setup. More detailed gures are included in the step by step portion of this lab. Friction Block ./ lunar-:1 baud SOgmass Figure 1 Side View of Lab schematic Figure 2 end view Newton's Second Law states that acceleration is directly proportional to the Net force acting on an object and is inversely proportional to the mass of the object. In order to determine the Net Force (Fm) on the Friction block it is helpful to draw a Force Diagram. A force diagram shows all the forces acting on an object. Let's create a force diagram for the friction block sitting on a table. What forces are acting on the block? two forces as an action-reaction force pair... because both forces are acting on the same object they aren't action reaction force pairs. You will learn more about this in later chapters). Next looking in the horizontal direction, there are also two forces acting on the friction block, the string pulling the block to the right and 'iction opposing the motion of the block. If the block is allowed to move freely, the force due to the string will be larger than the force of friction and the block will accelerate to the right. Figure 3 below shows these forces acting on the friction block. Note the size of each arrow shows the relative magnitude of the force and the arrow shows the direction the force acts. Support Force Friction Force Forcc due to the string Force of Gravity Figure 3 Force Diagram for the friction block sitting on a table We also know the force of gravity is the weight of the object and can be found by the equation Force of Gravity = W = mg Where: 0 W is the weight of the object o m is the mass of the object and o g is the acceleration due to gravity = -9.81 m/s2 (the negative sign means it is downward or toward the center of the Earth) In this lab the friction block (referred to as A) is pulled by a string running under a board, over the edge of a table with a hanging mass (referred to as B) attached to the end of the string. The weight of the hanging mass (B) causes the 'iction block (A) to accelerate. The acceleration of the 'iction block can be predicted using dynamics theory; specically Newton's 2'\" Law of Motion (F = ma). Since both the friction block and the hanging mass are being accelerated by the force of gravity pulling on the hanging mass we must treat this problem as a single system. The total mass being accelerated is the combined mass of the friction block and the hanging mass (m + m3). Both the friction block and the hanging mass will have the same exact acceleration as long as the string doesn't stretch or break and the hanging mass doesn't hit the oor. The A + B system has an acceleration we can call asysnaiu and a mass of mSYSTEM. Using Newton's second law we know that the acceleration of a system is equal to the net force divided by the mass of the system. In order to use dynamics we have to ignore friction. Once we ignore friction we know: Fnet asystem _ msytem We know the net force acting on the system is simply the weight of the hanging mass, (since the weight of the friction block cancels with the support force acting on the friction block and we are ignoring friction). Putting it together we get that the acceleration of the system is equal to the weight of the hanging mass divided by the total mass of the system or in equation form: Fnet mBg msystem mA + m3 asystem This equation gives us the ability, in theory, to predict the acceleration of the A + B system if we know the masses of both the friction block and the hanging mass. Since our prediction is based on a theory, let's call this acceleration the Theoretical Acceleration. . "1 So now we can write: aTheorm-ml = m f: where g = 9.81 m/s2 A B One Problem; remember this equation ignores friction. If there isn't any friction then there is no problem. The odds are though, there will be some friction. If so, the acceleration of the A + B system will be less than the amount predicted using our dynamics (Newton's 211d Law) equation. We can experimentally determine the acceleration of the A + B system by making measurements and using kinematics (which doesn't involve forces). From kinematics we know the distance (d) traveled by an object starting from rest & undergoing a constant acceleration (a) is given by the equation: d = 1/2 at2 Solving for acceleration gives us: a = 2d/t2. (Also a kinematics equation) Thus by measuring the distance over which the friction block accelerates and by measuring the time it spends accelerating we can calculate an experimental value for the actual as the string doesn't stretch or break and the hanging mass doesn't hit the oor. The A + B system has an acceleration we can call asysnaiu and a mass of mSYSTEM. Using Newton's second law we know that the acceleration of a system is equal to the net force divided by the mass of the system. In order to use dynamics we have to ignore friction. Once we ignore friction we know: Fnet asystem _ msytem We know the net force acting on the system is simply the weight of the hanging mass, (since the weight of the friction block cancels with the support force acting on the friction block and we are ignoring friction). Putting it together we get that the acceleration of the system is equal to the weight of the hanging mass divided by the total mass of the system or in equation form: Fnet mBg msystem mA + m3 asystem This equation gives us the ability, in theory, to predict the acceleration of the A + B system if we know the masses of both the friction block and the hanging mass. Since our prediction is based on a theory, let's call this acceleration the Theoretical Acceleration. . "1 So now we can write: aTheorm-ml = m f: where g = 9.81 m/s2 A B One Problem; remember this equation ignores friction. If there isn't any friction then there is no problem. The odds are though, there will be some friction. If so, the acceleration of the A + B system will be less than the amount predicted using our dynamics (Newton's 211d Law) equation. We can experimentally determine the acceleration of the A + B system by making measurements and using kinematics (which doesn't involve forces). From kinematics we know the distance (d) traveled by an object starting from rest & undergoing a constant acceleration (a) is given by the equation: d = 1/2 at2 Solving for acceleration gives us: a = 2d/t2. (Also a kinematics equation) Thus by measuring the distance over which the friction block accelerates and by measuring the time it spends accelerating we can calculate an experimental value for the actual acceleration of the friction block. Since this value is determined experimentally, we can call this acceleration aexp. 2d dexp 72 So now we have two equations for acceleration: Eq. 1) aTheoretical = mBXg Eq. 2) aexp = 2d/t2 mAtmB In this lab you will first determine the Theoretical acceleration (which ignores friction), then determine the experimental acceleration (which includes friction). From this we can determine how large the force of friction is Again starting with Newton's second law, remember the Net Force on the system is equal to the force due to the hanging mass (which is the weight of the hanging mass) minus the force of friction. Fr If we calculate the Net force on the system using the experimental acceleration, you can rearrange to solve for the Force of friction. Net Force = Weight of hanging mass - Force of friction. Rearranging we get Force of friction = Net Force - Weight of hanging mass Where the Net Force is the mass of the system times the experimental acceleration Net Force = msystemdexp The Step by Step procedures for this lab are on the remaining pages below. Gather the following materials Student Supplied HOL Supplied Digital stopwatch* Wood friction block Wood friction block Grooved board Heavy object or book string** (optional) painters or masking tape 50 g mass Digital scale, precision Measuring tape *Either a physical stopwatch or computer/phone app may be used as long as it records 0.00 seconds. ** you will use the string in later experiments so be sure to save the portion of the string used during this experiment as well as the remaining string.Procedure 1. Measure the mass of the wood friction block and the mass of the 50 g mass using the precision digital scale and record to two decimal places in Data Table 1 in the worksheet. 2. Add the mass of the friction block to the mass of the 50 g and record as the total mass of the system in Data Table 1. 3. Convert the masses in grams to mass in kg using the conversion 1 kg = 1000 g and record in Data Table 1. 4. Use the equation below and the masses in kg to calculate the theoretical acceleration of the block and record in Data Table 1 in the worksheet. _ m5*9 _ 2 aTheoreticaI mA'l'mB, Where g 9.81 III/S 5. Find a very smooth surface with an overhang to use as your work surface: a wood table is best, but a desk or counter top will work. You need enough room on top of the surface for the friction block to move between a half a meter and a meter distance before hitting the grooved board, and enough height for the hanging mass to drop without hitting the oor. 6. Place the grooved board on the surface with one edge even with the edge of the table. 7. Use steps 8 - 10 below to determine how long to cut the string. 8. Lay the uncut string out across the length of the table, on top of the grooved board with one end hanging over the edge of the surface. (Note Figure 4 to the left shows the nal setup, not with the string on top of the grooved board.) 9. Pull this hanging end down until it is just above (but not touching) the oor. 10. Trim the portion of the string on the surface (not the portion hanging over the edge) about 10-15 cm from the opposite edge of the grooved board. This will leave you enough string to make loops without letting the mass hit the ground. 11. Next make a small loop at the end of the string that is hanging over the end of the table. An easy way to do this is to wrap the end of the string around one nger with enough excess string on the end to tie a double knot. Tie a double knot and make sure the knot is tight while keeping a small loop. 12. Repeat step 11 to make another small loop at the other end of the string. 13. Use Figure 5 - Figure 8 below and step 14 to attach the string to the 'iction block. / Figure 5 loops 1 Figure 6 loops 2 Figure 7 loops 3 Figure 8 loops 4 14. Slide one of the loops through the hoop on the friction block (Figure 5), then slip the other loop through the rst loop (Figure 6) and tighten so the string is attached to the hoop (Figure 7 and Figure 8) 15. Place the grooved board over the string so that the string is inside the groove. Pull the string back and forth to ensure it moves freely inside the groove. You should feel when it is completely inside the groove as it will move more eely back and forth. 16. Place a mark (or tape) on the smooth at surface at the end of the grooved board, see Figure 9 below. This will be your starting grooved board position. The board may slide during the experiment so you need a consistent mark to use to rest the grooved board before each Trial. Figure 9 marking grooved board position 17. Attach the 50 g mass to the loop on the end of the string overhanging the table. Be sure the hanging mass does not touch the oor when the loop of the friction block is up against the grooved board. 18. Pull the friction block away from the grooved board until the hanging mass is near the top of the surface, but still hangs freely. Mark the surface at the front edge of the hoop of the friction block. See Figure 10. Notice how the front edge of the tape is even with the front edge Figure 10 5\"\" {mg position of the hoop on the iction block. (This is the marking point of the 'iction block that strikes the grooved board) This will be the starting point for the friction block for each trial 19. Double check that the hanging mass does not hit the oor when the friction block is up against the grooved board 20. Measure the distance between the front edge of the grooved board (as marked by the tape) and the front edge of the friction block) also marked by tape. Be sure to use the correct edge of each tape. Record the distance in cm to two decimal places in Data Analysis Table 2. 21. Convert the distance to meters using the conversion factor 1 m = 100 cm, and record in Data Analysis Table 2 with the correct number of sig g. 22. Practice starting the stopwatch while simultaneously releasing the friction block and stopping the stopwatch when the friction block rst touches the grooved board. Practice until you get consistent times. Once you have consistent times begin Data collection. 23. At the beginning of each trial, ensure the grooved board is lined up with its marking, and the hoop of the Friction block is aligned with the starting position marking. Release the friction block while simultaneously starting the stop watch. Stop the stopwatch when you hear the block hit the grooved board. Record the time in seconds to two decimal places in Data Table 2. 24. Repeat step 23 four more times for a total of ve trials. 25. Compute the average time the block spends accelerating by adding the ve trial times and dividing by 5. Record the average time in Data Analysis Table 2. 26. Compute the experimental acceleration using the distance in m and the equation below: aexp = 2d/t2 and record in Data Analysis Table 2. 2?. Calculate the weight of the hanging mass by multiplying the mass of the hanging mass times the acceleration due to gravity W = mhanging mass * y where g = 9.81 rn/s2 and record in Data Analysis Table 3. 28. Use Newton's Second Law (F = ma) to compute the Net Force (experimental force (F EXP by multiplying the total mass of the system (in kg) by the experimental acceleration from Data Analysis Table 2 and record in Data Analysis Table 3. 29. Calculate the 'iction force by nding the difference between the weight of the hanging mass and the experimental force. Force of Friction = Weight of hanging mass Fm, Record your calculated Force of friction in Data Analysis Table 3. 30. Calculate the percent difference between your theoretical acceleration and your experimental acceleration using the equation latheo _ aexpl X 100% (atheo + aexp) 2 Percent difference = Remember the easiest way to calculate this is to nd the average of the two acceleration values, write that number down. Find the difference between the two accelerations, if it is negative drop the negative side. Write that number down. Divide the difference by the average, then multiply by 100. Record the percent difference in the worksheet. 31. In the vast majority of circumstances, the percent difference should far exceed the benchmark of S 10% due to data collection errors. If your value is greater than 10% explain what you believe accounts for the disparity (i.e. what factors were included or excluded in the two calculations). Record your answer in the worksheet. 32. Another way to calculate the Friction force is to multiply the difference between the theoretical acceleration and the experimental acceleration by the mass of the system, using the equation: Force of Friction = m5y5t9m(ameo new) and record in the worksheet. (Note this should be similar or identical to the value you calculated in step 29. 33. In this experiment you used Newton's Second Law and kinematic equations to find the force of friction acting on a sliding friction block. Thinking about the possible sources of data collection errors, what might be done to improve the accuracy of this experiment? Note: changing the surfaces (or the amount for friction) doesn't improve the accuracy, only changes the values of the 'iction force. Record your answer in the worksheet. 34. Return all items to the kit for lture labs. Ffriction = Msystem (atheo - aexp) N What do you think might be done to improve the accuracy of this experiment? Add footerData Table 1 Mass of Mass of 50 *Theo. Acceleration of the block Friction Block gram mass (g) g) g) m/$2 31.00 50.00 81.00 (kg) kg) (kg) 0.0310 0.0500 0.0810 Data Table 2 * Note: use g =9.81 m/s when solving for the acceleration Trial Time (s) 0.40 2 0.38 3 0.43 0.40 5 0.43 AVERAGE 0.41 Data Analysis Table 2 Distance (cm) Distance (m) Average Time(s) Exp. Acceleration of the block (m/s?) 60.96 0.6100 0.41 Data Analysis Table 3 Weight of hanging mass Exp. Force [F exp = m aexp] (N) Friction = Weight of hanging mass - Fexp (N) Calculate the percent difference between your theoretical acceleration and your experimental acceleration. Percent Difference in acceleration: % This percent difference should far exceed experimental error. What do you think accounts for the disparity? uselly higher than the experimental value due to various factors such as friction, air reistance, and measurement errors