Newton's laws Olympic College, Physics 110 Background Newton's laws In the 1600s, Isaac Newton discovered three...

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Newton's laws Olympic College, Physics 110 Background Newton's laws In the 1600s, Isaac Newton discovered three laws governing interactions between objects: I. An object's velocity does not change unless the object is acted on by a force. II. The acceleration of an object directly proportional to its mass, and inversely proportional to the vector sum of all forces acting on the object. L m III. When one object exerts a force on another, the second object exerts a force with the same strength but in an opposite direction on the first object. The first law can be seen as a consequence of the second-if there is zero force acting on an object, it will have zero acceleration. When you study the quantity called momentum, you'll see that the third law is a consequence of momentum conservation. Mathematically, the second law can be represented as Fnet = ma This gives us a quantitative tool that can be used to analyze an object's motion. Note that acceleration, which is one way of describing the motion of an object, is an effect of all forces acting on the object. There may be many forces acting on an object; along with the object's mass, these forces determine how an object's motion will change. Atwood's machine George Atwood was a physics instructor from the late 1700s who was well-known for developing effective demonstrations. One of these devices is now simply called Atwood's Machine, and a simplified version can be used to investigate Newton's laws. (For more information, ask Professor Roth for a copy of a short article about Atwood and his machine.) Atwood's Machine consists of two masses connected by a light string, hung over a pulley, as shown below. If one side has more mass than the other, the system will accelerate. Atwood machine: two masses, connected by a string, hang over a pulley. In this lab, you will primarily be investigating Newton's second law. Newton's third law shows up briefly in part of the analysis. Calculating acceleration Since most people do not own motion detectors that measure acceleration directly, you'll need to calculate it. For an object that starts from rest and falls a distance h in a time t, the magnitude of the acceleration is given by a = 2h t I will not present the derivation here, but this equation is derived using kinematics, the mathematical description of motion. Experimental design: isolating variables In the Atwood machine, there are two principle variables we'll be looking at: the total mass hanging over the pulley; and the difference between the two hanging masses. When doing experiments, it is important to change as few variables at a time as possible. This allows us to isolate just one relationship for examination. For example, when you're considering the relationship between acceleration and the difference between the two masses, you don't want the total mass to have any impact. In other words, you want to keep the total mass the same, and only change the difference between the two masses. Similarly, when you're considering the relationship between acceleration and total mass, you don't want the difference between the masses to have any impact. So, you want to keep the difference between the two masses the same, and only change the total mass. Materials You will be building your own Atwood machine to do this experiment. It is best if you happen to have pulleys and somewhere to hang them, but a string hanging over a rod or broom handle works just fine. Hanging a string over a doorknob is not recommended; I've found the string gets stuck behind the doorknob, and the masses have a tendency to slide against the door itself, introducing extra friction. For the two hanging masses, you'll have the best results if you use a collection of objects similar masses. Coins work very well because the mass is standardized: a penny has a mass of 2.5 g, a nickel has a mass of 5.0 g, a dime has a mass of 2.268 g, and a quarter has a mass of 5.67 g (source: the US mint). Other decent choices are nuts, washers, or similarly-sized decorative glass beads or pebbles. You will want to use a bag or pouch to hold the objects. For measurements, you'll need something to measure length, something to measure time, and, if you're using masses other than coins, something to measure mass. (Note: if you're using a scale that reads in ounces or pounds, it is giving you a measurement of weight, not mass. You will need to do the appropriate unit conversions to find the mass.) Procedure You will be investigating two relationships: Changing mass difference 1. Set up your Atwood machine as described above. 2. Put different amounts of mass in each bag. 3. 4. Release from rest and record the time and distance (as described above). Move mass from one bag to the other This will change the difference in mass between the two bags, but the total mass hanging over the pulley will remain unchanged. 1. Repeat the procedure until you have 15 data points. You may use the table below to record your data and calculations. Trial Mass 1 Mass 2 Mass difference 1 2 3 4 5 6 7 8 9 10 11 12 13 Screenshot(Alt + A) 14 15 Total mass Time Height Acceleration 1. 2. How the acceleration depends on the difference in mass hanging from either side. How the acceleration of the system depends on the total mass hanging over the pulley. Both investigations will use the same basic Atwood machine setup. Changing total mass 1. Set up your Atwood machine as described above. Atwood machine basic setup 2. Put different amounts of mass in each bag. 3. 1. Tie two bags together with a string. Make sure the string is long enough that one bag will reach the ground before the other reaches the pulley. Release from rest and record the time and distance (as described above). 4. Add the same amount of mass to each bag 2. Hang the string over a pulley or rod. 3. Add mass to the bags. For example, say you are using washers; you would add the same number of washers to each bag. 4. Measure the distance from the heavier bag to the ground. 5. Release the bags from rest. 1. This will change the total mass involved, but keep the difference between the two bags constant Repeat the procedure until you have 15 data points. 6. Time how long it takes the bag to reach the ground. You may use the table below to record your data and calculations. 4. Trial Mass 1 Mass 2 Mass difference Total mass Time Height Acceleration 1 The tension pulling up on each mass is the same (this is due to Newton's third law, as alluded to in the overview section of this handout). The acceleration (a) of each mass is also the same. Set your two equations equal to each other and solve for a. You will ultimately find the expression a = m1m2 -g m1 + m 10 11 12 13 14 15 23456789222345 Data analysis Theoretical analysis It is common to compare your experimental data to a theoretical analysis of the situation, which often involves considering an ideal case. (The meaning of "ideal" will be explored in one of the followup questions.) These instructions will walk you through the process of finding the acceleration in terms of the two masses. At first, you will be working out a purely symbolic expression; you will not be using numbers until after you have found an expression for the acceleration a in terms of the two masses m and m. 1. Draw a free body diagram for each hanging mass. Label the coordinate axes with the +y direction pointing up. Attach an image (scan or picture) of your free body diagrams here: 2. Apply Newton's second law (Fy = may) to each mass separately. Note that for the heavier mass, the acceleration is negative: ay = a. This negative sign is crucial to an accurate analysis. 3. Solve each equation for tension. That is, follow the appropriate algebraic steps to get the tension FT by itself on one side of the equals sign, and all other quantities (m, a, and g) on the other. or something that is similar but mathematically equivalent. Note that m - m is the difference between the two masses, and m + m2 is the total mass. Show your work for all the mathematical steps, starting from Newton's second law, below: 5. Using this relationship, calculate the theoretical acceleration for each trial. Changing mass difference Changing total mass Trial Theoretical acceleration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Trial Theoretical acceleration 1234567 12 13 14 15 "2345 8 9 10 11 Graphs On each graph you will plot two sets of data: the experimental data and the theoretical data. This will give you a visual comparison between the two. The data points from the theoretical analysis should be distinct in some way from the experimental data; use a different shape and/or color for the icon on the graph. A legend should be included to indicate which is which. Changing mass difference Make a graph of acceleration vs. mass difference from your data. That means the acceleration is on the vertical axis, and the mass difference is on the horizontal axis You do not need to find a line of best fit; we will be making qualitative evaluations (instead of quantitative) Do not have the data points connected in any way. Even though this is the default setting in some graphing programs, it is not considered good scientific presentation of data. Include your graph below: Changing total mass Make a graph of acceleration vs. total mass from your data. Followup questions 1. Two quantities are directly related when increasing results in increasing the other. Two quantities are inversely related when increasing one decreases the other. Look at your two graphs, and answer based on the experimental data: 2. 3. a. Are acceleration and mass difference related directly, or are they related inversely? b. Are acceleration and total mass related directly, or are they related inversely? m1-m2 m+m2 Does the theoretical expression for acceleration, a = g, predict the same behavior that you observed, in terms of direct and indirect relationships? Explain your reasoning. Now compare your theoretical results to the experimental results: are you experimental results consistently higher than the theory predicts? Are they consistently lower? What could explain discrepancies, if there are any? Explain your reasoning. Specifications Your submission for this lab must meet the following specifications: Check-in See the assignment in Canvas for details. 1. Successful completion of check-in 1 on time. Successful completion entails: Responding to all prompts in the assignment. Reporting any required data or calculated results with appropriate units. Showing all work for any analysis and/or calculation. Any writing required is typed in complete sentences, using spelling and grammar conventions appropriate for scientific communication. Data & analysis 1. 3. 4. 5. All data from the changing mass difference experiment are included. The total mass remains constant while the mass difference changes. All units are included. Acceleration is calculated from the experimental height and time data. All data from the changing total mass experiment are included. The mass difference remains constant while the total mass changes. All units are included. Acceleration is calculated from the experimental height and time data. All data are plotted on the correct graphs. Each graph is labeled with a descriptive title and axis labels. Axis labels include units. Data are presented professionally: each data set on a graph is distinct from the other; data points are not connected point-to-point. Derivation of theoretical acceleration in terms of mass is correct, including both FBDs, and the symbolic derivations of tension and acceleration. All work is shown, either written by hand or typeset with an equation editor. Calculations of theoretical acceleration values are consistent with the data presented. That means acceleration is on the vertical axis, and the total mass is on the horizontal axis 2. You do not need to find a line of best fit. Do not have the data points connected in any way. Even though this is the default setting in some graphing programs, it is not considered good scientific presentation of data. Include your graph below: Followup questions 1. 2. 3. 4. Responses to question 1 are consistent with the experimental data presented. Response to question 2 is consistent with the theoretical data presented. Response is explained fully and clearly explained. Response to question 3 is consistent with the data presented. Response is complete and reasoning is explained clearly. Responses to questions 2 and 3 are written in complete sentences following spelling and grammar conventions appropriate to professional science communication. Newton's laws Olympic College, Physics 110 Background Newton's laws In the 1600s, Isaac Newton discovered three laws governing interactions between objects: I. An object's velocity does not change unless the object is acted on by a force. II. The acceleration of an object directly proportional to its mass, and inversely proportional to the vector sum of all forces acting on the object. L m III. When one object exerts a force on another, the second object exerts a force with the same strength but in an opposite direction on the first object. The first law can be seen as a consequence of the second-if there is zero force acting on an object, it will have zero acceleration. When you study the quantity called momentum, you'll see that the third law is a consequence of momentum conservation. Mathematically, the second law can be represented as Fnet = ma This gives us a quantitative tool that can be used to analyze an object's motion. Note that acceleration, which is one way of describing the motion of an object, is an effect of all forces acting on the object. There may be many forces acting on an object; along with the object's mass, these forces determine how an object's motion will change. Atwood's machine George Atwood was a physics instructor from the late 1700s who was well-known for developing effective demonstrations. One of these devices is now simply called Atwood's Machine, and a simplified version can be used to investigate Newton's laws. (For more information, ask Professor Roth for a copy of a short article about Atwood and his machine.) Atwood's Machine consists of two masses connected by a light string, hung over a pulley, as shown below. If one side has more mass than the other, the system will accelerate. Atwood machine: two masses, connected by a string, hang over a pulley. In this lab, you will primarily be investigating Newton's second law. Newton's third law shows up briefly in part of the analysis. Calculating acceleration Since most people do not own motion detectors that measure acceleration directly, you'll need to calculate it. For an object that starts from rest and falls a distance h in a time t, the magnitude of the acceleration is given by a = 2h t I will not present the derivation here, but this equation is derived using kinematics, the mathematical description of motion. Experimental design: isolating variables In the Atwood machine, there are two principle variables we'll be looking at: the total mass hanging over the pulley; and the difference between the two hanging masses. When doing experiments, it is important to change as few variables at a time as possible. This allows us to isolate just one relationship for examination. For example, when you're considering the relationship between acceleration and the difference between the two masses, you don't want the total mass to have any impact. In other words, you want to keep the total mass the same, and only change the difference between the two masses. Similarly, when you're considering the relationship between acceleration and total mass, you don't want the difference between the masses to have any impact. So, you want to keep the difference between the two masses the same, and only change the total mass. Materials You will be building your own Atwood machine to do this experiment. It is best if you happen to have pulleys and somewhere to hang them, but a string hanging over a rod or broom handle works just fine. Hanging a string over a doorknob is not recommended; I've found the string gets stuck behind the doorknob, and the masses have a tendency to slide against the door itself, introducing extra friction. For the two hanging masses, you'll have the best results if you use a collection of objects similar masses. Coins work very well because the mass is standardized: a penny has a mass of 2.5 g, a nickel has a mass of 5.0 g, a dime has a mass of 2.268 g, and a quarter has a mass of 5.67 g (source: the US mint). Other decent choices are nuts, washers, or similarly-sized decorative glass beads or pebbles. You will want to use a bag or pouch to hold the objects. For measurements, you'll need something to measure length, something to measure time, and, if you're using masses other than coins, something to measure mass. (Note: if you're using a scale that reads in ounces or pounds, it is giving you a measurement of weight, not mass. You will need to do the appropriate unit conversions to find the mass.) Procedure You will be investigating two relationships: Changing mass difference 1. Set up your Atwood machine as described above. 2. Put different amounts of mass in each bag. 3. 4. Release from rest and record the time and distance (as described above). Move mass from one bag to the other This will change the difference in mass between the two bags, but the total mass hanging over the pulley will remain unchanged. 1. Repeat the procedure until you have 15 data points. You may use the table below to record your data and calculations. Trial Mass 1 Mass 2 Mass difference 1 2 3 4 5 6 7 8 9 10 11 12 13 Screenshot(Alt + A) 14 15 Total mass Time Height Acceleration 1. 2. How the acceleration depends on the difference in mass hanging from either side. How the acceleration of the system depends on the total mass hanging over the pulley. Both investigations will use the same basic Atwood machine setup. Changing total mass 1. Set up your Atwood machine as described above. Atwood machine basic setup 2. Put different amounts of mass in each bag. 3. 1. Tie two bags together with a string. Make sure the string is long enough that one bag will reach the ground before the other reaches the pulley. Release from rest and record the time and distance (as described above). 4. Add the same amount of mass to each bag 2. Hang the string over a pulley or rod. 3. Add mass to the bags. For example, say you are using washers; you would add the same number of washers to each bag. 4. Measure the distance from the heavier bag to the ground. 5. Release the bags from rest. 1. This will change the total mass involved, but keep the difference between the two bags constant Repeat the procedure until you have 15 data points. 6. Time how long it takes the bag to reach the ground. You may use the table below to record your data and calculations. 4. Trial Mass 1 Mass 2 Mass difference Total mass Time Height Acceleration 1 The tension pulling up on each mass is the same (this is due to Newton's third law, as alluded to in the overview section of this handout). The acceleration (a) of each mass is also the same. Set your two equations equal to each other and solve for a. You will ultimately find the expression a = m1m2 -g m1 + m 10 11 12 13 14 15 23456789222345 Data analysis Theoretical analysis It is common to compare your experimental data to a theoretical analysis of the situation, which often involves considering an ideal case. (The meaning of "ideal" will be explored in one of the followup questions.) These instructions will walk you through the process of finding the acceleration in terms of the two masses. At first, you will be working out a purely symbolic expression; you will not be using numbers until after you have found an expression for the acceleration a in terms of the two masses m and m. 1. Draw a free body diagram for each hanging mass. Label the coordinate axes with the +y direction pointing up. Attach an image (scan or picture) of your free body diagrams here: 2. Apply Newton's second law (Fy = may) to each mass separately. Note that for the heavier mass, the acceleration is negative: ay = a. This negative sign is crucial to an accurate analysis. 3. Solve each equation for tension. That is, follow the appropriate algebraic steps to get the tension FT by itself on one side of the equals sign, and all other quantities (m, a, and g) on the other. or something that is similar but mathematically equivalent. Note that m - m is the difference between the two masses, and m + m2 is the total mass. Show your work for all the mathematical steps, starting from Newton's second law, below: 5. Using this relationship, calculate the theoretical acceleration for each trial. Changing mass difference Changing total mass Trial Theoretical acceleration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Trial Theoretical acceleration 1234567 12 13 14 15 "2345 8 9 10 11 Graphs On each graph you will plot two sets of data: the experimental data and the theoretical data. This will give you a visual comparison between the two. The data points from the theoretical analysis should be distinct in some way from the experimental data; use a different shape and/or color for the icon on the graph. A legend should be included to indicate which is which. Changing mass difference Make a graph of acceleration vs. mass difference from your data. That means the acceleration is on the vertical axis, and the mass difference is on the horizontal axis You do not need to find a line of best fit; we will be making qualitative evaluations (instead of quantitative) Do not have the data points connected in any way. Even though this is the default setting in some graphing programs, it is not considered good scientific presentation of data. Include your graph below: Changing total mass Make a graph of acceleration vs. total mass from your data. Followup questions 1. Two quantities are directly related when increasing results in increasing the other. Two quantities are inversely related when increasing one decreases the other. Look at your two graphs, and answer based on the experimental data: 2. 3. a. Are acceleration and mass difference related directly, or are they related inversely? b. Are acceleration and total mass related directly, or are they related inversely? m1-m2 m+m2 Does the theoretical expression for acceleration, a = g, predict the same behavior that you observed, in terms of direct and indirect relationships? Explain your reasoning. Now compare your theoretical results to the experimental results: are you experimental results consistently higher than the theory predicts? Are they consistently lower? What could explain discrepancies, if there are any? Explain your reasoning. Specifications Your submission for this lab must meet the following specifications: Check-in See the assignment in Canvas for details. 1. Successful completion of check-in 1 on time. Successful completion entails: Responding to all prompts in the assignment. Reporting any required data or calculated results with appropriate units. Showing all work for any analysis and/or calculation. Any writing required is typed in complete sentences, using spelling and grammar conventions appropriate for scientific communication. Data & analysis 1. 3. 4. 5. All data from the changing mass difference experiment are included. The total mass remains constant while the mass difference changes. All units are included. Acceleration is calculated from the experimental height and time data. All data from the changing total mass experiment are included. The mass difference remains constant while the total mass changes. All units are included. Acceleration is calculated from the experimental height and time data. All data are plotted on the correct graphs. Each graph is labeled with a descriptive title and axis labels. Axis labels include units. Data are presented professionally: each data set on a graph is distinct from the other; data points are not connected point-to-point. Derivation of theoretical acceleration in terms of mass is correct, including both FBDs, and the symbolic derivations of tension and acceleration. All work is shown, either written by hand or typeset with an equation editor. Calculations of theoretical acceleration values are consistent with the data presented. That means acceleration is on the vertical axis, and the total mass is on the horizontal axis 2. You do not need to find a line of best fit. Do not have the data points connected in any way. Even though this is the default setting in some graphing programs, it is not considered good scientific presentation of data. Include your graph below: Followup questions 1. 2. 3. 4. Responses to question 1 are consistent with the experimental data presented. Response to question 2 is consistent with the theoretical data presented. Response is explained fully and clearly explained. Response to question 3 is consistent with the data presented. Response is complete and reasoning is explained clearly. Responses to questions 2 and 3 are written in complete sentences following spelling and grammar conventions appropriate to professional science communication.