VIEWS: 19 PAGES: 44 POSTED ON: 3/11/2010
Instructor’s Notes – Physical Science Activities Measuring Velocity This first activity is about data collection and simple mathematical calculations. It has also been designed to introduce the scientific way of approaching a problem (by asking questions), to show the practice of testing a hypothesis (how many of your other lab experiments actually do that?), and to demonstrate the concept of constant velocity from data analysis, not as a verbal explanation to be taken on faith. In this activity, your Early Childhood Education (ECE) students measure distances with a meter stick and time with a stop watch, and then calculate speed by dividing distance by time. However, what may seem obvious and easy to us can be quite intimidating them. But DON‟T express any displeasure or dismay if your students have trouble making „simple‟ measurements and performing „simple‟ calculations. It‟s not that simple to them! It is important to set the appropriate perspective for all the remaining activities in the course by NOT explaining very much at all about the data collection. Get out of the way and let them struggle. Give general, evasive answers and walk away when they ask for assistance. But before the end of class time, call everyone together and have a thorough, detailed discussion of the data and results. Call on someone to volunteer their data and walk through the calculations, step by step, allowing time for everyone else to calculate or re-calculate their numbers along with you. The underlying goal of this activity is to encourage your students to take chances in the face of uncertainty – something that is anathema to this personality type. By following this advice, you can use Measuring Velocity to build the necessary trust between you and your students for the remainder of the course. The message you are sending is that it‟s okay to be uncertain and struggle and not know all the answers immediately, because it will not be held against you – or your grade. Acceleration Acceleration is such a difficult concept for most students to grasp, so it‟s important to have a lab activity. This lab also builds on the data collection skills introduced in Measuring Velocity. Many of your students will not understand how and where to use the two different time measurements: time down the ramp and time across the floor. Either circumvent the problem by explaining clearly before they begin data collection, or be prepared to deal with the issue when reviewing the data together in the large group. Acceleration deliberately returns to the ramp and golf ball of the previous activity. Students are confronted with the fact that the motion of the ball on the ramp is qualitatively different from the motion of the ball rolling across the floor. The difference is acceleration (the ball is speeding up); therefore, velocity and acceleration can’t be measuring the same thing. Most of your ECE students will never thoroughly grasp the difference between velocity and acceleration, and this activity will not suddenly make the concept clear to them. Instead, the goal of this activity is to engage your students in the concept of acceleration by making them calculate one. It is harder to ignore a concept after you have been directly confronted by it. [Quick side note: Isaac Asimov wrote a story about a man who invented an anti-gravity device. No one would believe him because the Science Administration said anti-gravity was impossible. He solved his problem by floating in an administrator‟s office and claiming that he was not floating! The bureaucrat had to confront the reality before his eyes, instead of just debating a proposition.] Graphs of Motion This activity introduces the Vernier (or Pasco) data collection system. It also shows the relationship between types of motion and graphs of motion. Students will very quickly understand and be able to manipulate the Logger Pro software and its capabilities. I have used graph matching with students ranging from the 5th through post- college, and everyone is impressed by it. Introducing the Vernier system with the graph matching activity will make computer-based data collection less intimidating. Graphs of Constant Acceleration The main purpose of this activity is to show students that the slope of a velocity vs time graph is a measure of acceleration, which will be a fundamental concept for the Newton’s Second Law activity. Depending on how deep you want to get into concepts of motion, you can use this activity to introduce or stress the difference between positive and negative acceleration, or that the acceleration during up-and-down motion may be constant, including at the point where the object is momentarily at rest. The Practicing Physics exercises on free fall can be a good follow-up to this activity (pp 3-4). Equilibrium and Vector Addition This is an „extra‟ activity. It has been included to link the textbook content to actual practice. It also corresponds to the first practice page, Vectors and Equilibrium. (This activity can be extended to also test the hypotheses advanced on this practice page.) Hewitt explains how he became motivated to study physics while investigating the equilibrium rule (page 23 of the text). This activity is designed to reproduce (approximately) what he did – to test these ideas. The second part of the activity corresponds to Section 3.5 of the text on vectors and the accompanying Math Connection on vector components. My goal is to engage students in the key concepts and principles of physical science. I find that reading about science is never good enough. I claim that if a student can‟t write an original sentence explaining or describing what he or she has learned, then nothing has been learned. I don‟t think many students will have an understanding about these ideas without active engagement, so either skip these sections of the text or include an activity about them. This activity relates directly to exercises 8-12 of Chapter 2 of the text. Newton‟s First Law If Newton‟s Laws are worth introducing, then it‟s worth having an activity for each one of them. Newton‟s first law is both qualitative and complicated. (It has three separate parts: that‟s complicated to a non-scientist just introduced to these ideas.) Therefore, this activity is designed to be quick and qualitative. In the first part of the activity, students create demonstrations showing each of the three conditions set forth by the law. Follow-up questions help students understand that we don‟t always understand the law of inertia because of the prevalence of frictional forces in the real world. The activity concludes with two applications of the first law. Newton‟s Second Law I consider this activity to be one of the most important ones in the course. This activity is a clear demonstration to non-science students that they can both understand and demonstrate real science concepts. If a motion detector is used, it is helpful to have completed Graphs of Constant Acceleration first, to introduce the idea that the slope of a velocity vs time graph is a measure of the acceleration. If the Vernier or Pasco system is not available, acceleration could be calculating the time it takes (t) the weight hanger to drop from its rest position until it strikes the floor (d), and use a = 2d/t2. Newton‟s Third Law Like Newton’s First Law, this activity is both quick and qualitative. It can be used to stress the concept that a null result is acceptable and valuable one in science, since there is no way to create a situation where the magnitude of one force can be greater than the other. The Force of Friction This is an „extra‟ activity that introduces the coefficient of friction. It also is another example of the graphical manipulation of data to demonstrate a mathematical relationship between two quantities. Momentum and Collisions This activity is my „lecture‟ on conservation of momentum. I do this as a class lab – although each table has its own experimental set up, we complete each other the eight trials at the same time and check the math and results of all groups before moving on to the next one. As in other activities, I like to make sample calculations using real data from a group for each trial. Because there are eight trials, I move from group to group for data. This method includes all groups, and it makes all groups work at data collection since no group knows when they will be called on to provide their data. Energy Conversion This activity provides practice calculating three different forms of energy: kinetic, elastic, and gravitational potential energy. Each part is a different conversion: gravitational to kinetic; elastic to gravitational; and elastic to kinetic. The experimental design presumes that conservation of energy is true. The next activity demonstrates energy conservation. Energy Conservation This activity uses the Vernier or Pasco data collection system. It requires the most advanced instructor preparation, but I encourage its inclusion as an activity, if only as a single whole-class demonstration. It is not designed as an in-depth examination if energy conservation, only a quick “hey, wow, that‟s neat” look at the principle. A good follow up question for Bounce is, what happened to the missing energy with each bounce? And the vibrational motion can be referred to again in the chapter on vibrations, waves and sound. Comparing Methods I like this activity because it is the only one I have in the set that includes the calculation of work and of impulse, and because it explicitly shows that Newton‟s laws, impulse-momentum, and work-energy theorems may all be used to calculate kinematic quantities – it‟s just that some methods are more useful than others, depending on the circumstances (see the next activity). Ballistic Pendulum This is the only activity I have for Chapter Five of Hewitt‟s text. By this point in the course, the students are really sick of mechanics. The ballistic pendulum is a nice capstone, including energy, momentum, and a bit of projectile motion. For my physics students, the ballistic pendulum is a “you figure this out without my help” lab. For the ECE students, I talk them through it, one step at a time, but I still avoid explaining all the details of the experimental method and design to them. The goal is not the final number, it‟s the steps that got them there. I want them to see how different ideas and principles from mechanics can come together to solve a real problem. I also want them to see how the calculated speed of the bullet is accepted as accurate not because I say so, but because the calculations are internally consistent with the principles of physics developed and observed previously in the course. Measuring Velocity Opening Questions 1) In your own words, explain the difference between something that is moving and something that is at rest. 2) In your own words, explain how you could show that one object is moving faster than another. 3) One way to determine how fast something is moving is to calculate its speed. What measurements must be made first in order to calculate a speed? What do you do with those measurements to determine the speed? 4) Speed and velocity are similar, but not the same thing. How is velocity different from speed? Introduction The purpose of this activity is to measure velocity. If you took the time to answer the Opening Questions before reading this sentence (and I hope you did!), then you have already figured out that in order to calculate velocity, it will be necessary to measure both a distance traveled and the time it takes to travel that distance. Also, to be more accurate (which is a common characteristic of scientists), we will be more concerned about the speed of the object than its velocity. To figure out why that is the case, look at Question 4 again. But there is a second goal to this activity. Based on observations you will make, we will have reason to believe that the velocity of the object is (mostly) constant. It will be the job of you and your partner to test this hypothesis, and present data to support or refute it. Equipment A golf ball, board, masking tape, books, meter stick, and stop watch. You will need to set up the board on a smooth, hard surface, preferably a linoleum floor. An index card is optional Set Up Make a ramp out of the board by placing one end against a stack of books. The ramp does not need to be very steep – the golf ball shouldn‟t bounce when leaving the ramp. It may be helpful to tape an index card to the end of the board and the floor if the golf ball bounces when it hits the floor. Use the masking tale and meter stick to mark off 0, 50, 100, 150, and 200 centimeters from the end of the board. Assume the end of the board is the origin (zero distance). Also place a piece of tape near the top of the board to serve as a consistent starting point for releasing the golf ball. Initial Observations Release the golf ball from the starting point and from rest a number of times and observe its motion down the ramp and across the floor. Describe what you see in complete sentences on paper. Making Measurements Use the stop watch to measure the time it takes the golf ball to travel from the zero mark to the 100 cm mark. Make at least three trial runs and average the time. From the distance traveled and the time of travel, calculate the average speed of the golf ball. Next, use the stop watch to measure the time it takes the golf ball to travel from the 50 cm mark to the 150 cm mark. Once again, make at least three trial runs and calculate the average speed of the golf ball. Now measure the travel time between the 100 cm and the 200 cm mark and find the average speed, as before. Does your data support the hypothesis that the speed of the ball is constant on the smooth, flat surface? Making Predictions PREDICT the time it will take the golf ball to travel from the 0 cm mark to the 200 cm mark. Use the equation that defines speed to do this. After making your prediction, collect data to test your result. PREDICT the time it will take the golf ball to travel from the 0 cm mark to the 50 cm mark, then test your prediction. PREDICT the travel time for the golf ball between the 100 cm and 150 cm mark. What does the data just collected suggest about the calculation of speed? Tricky Question Suppose you and another group set up similar ramps at the same angle. Would you expect your calculated speeds to be the same? Would you expect your calculated velocities to be the same? What extra information would you have to supply to report a velocity, as opposed to a speed for the golf ball? Acceleration Opening Questions 1) An object traveling with a constant speed has zero acceleration. Identify two ways that an object could have an acceleration based on speed. 2) Acceleration is defined as a change in velocity over a period of time. Recall the difference between speed and velocity. What is a third way that an object can have an acceleration? 3) Did the ball in the previous activity have an acceleration on the ramp? How do you know? Introduction We examined the motion of a ball on a smooth, flat surface in Measuring Velocity; we now want to focus on ball as it travels down the ramp. If acceleration is a change in velocity over a period of time, we can calculate the acceleration of the golf ball if we know three things: the initial velocity of the ball; the final velocity of the ball; the time it took the ball to change from its initial to its final velocity. We know the initial velocity of the ball at the top of the ramp is zero (why?). We know the final velocity of the ball at the bottom of the ramp is the same as its velocity as it travels across the flat surface, and we know how to calculate that. We can measure with a stop watch the time it takes the ball to travel down the ramp. Equipment and Set Up Same as the previous activity: a golf ball, board, masking tape, books, meter stick, and stop watch; a ramp made by placing one end of the board against a stack of books. Making Measurements You decide what measurements need to be made and how you will make them in order to calculate the average acceleration of the ball on the ramp. The data table on the answer sheet may help you. You will also need to decide how many trials to take for accurate results. You may need more columns than given on the answer sheet. IMPORTANT NOTE: In this experiment, you will be measuring two different times. One measure will be the time it takes the ball to travel down the ramp. The other measure will be the time it takes the ball to travel a certain distance across the floor. Don‟t confuse the two different times in your calculations! Extra Challenge We have reason to believe that the acceleration on the ramp is constant. How could you test this hypothesis? Think about this, and be ready to discuss it with the rest of the class. Another Extra Challenge Observation suggests that a ramp at a steeper angle will result in a larger acceleration. Is there a way to use the class data to check this hypothesis? Graphs of Motion Introduction One of the main purposes of this activity is to become familiar with the Vernier data collection system. The system consists of hardware (the LabPro) and software (LoggerPro). Instead of detailed written instructions on how to set up the LabPro and manipulate data in LoggerPro, the instructor will talk you through that during the activity. A second important purpose of this activity is to recognize that there are many different ways to represent motion – equations, motion diagrams, and graphs – and that all of these methods are interconnected with each other. Basic Concepts The instructor will explain how to connect the Vernier LabPro to the computer and start the LoggerPro software. LoggerPro will recognize that a motion detector has been plugged into the Dig/Sonic 1 port and display two graphs: the top graph shows position versus time, and the second shows velocity vs. time. Choose a group member to be the test subject. Have that person stand in front of the motion detector. (Note: the motion detector cannot properly measure distances closer than about 40 cm in front of its face. Also, it cannot measure side-to-side motion, only back-and-forth motion.) Predict what a graph of position vs. time will look like if the test subject stands still in front of the motion detector. Sketch your prediction on the answer sheet, then test your prediction by pressing the green „Collect‟ button. If the correct answer is significantly different from your prediction, include it on the sketch as a dashed line. Continue to predict, sketch, and test the following motions: 1) moving away from the motion detector at a slow but steady rate 2) moving away from the motion detector at a faster but steady rate 3) moving toward the motion detector at a slow but steady rate 4) moving toward the motion detector at a faster but steady rate Graph Matching The instructor will show how to open the file 01b Graph Matching. Your task will be to move in such a way as to match the line shown on the graph. IMPORTANT CONDITIONS 1: The test subject may not see the screen while data is being collected. 2: Group members may not speak or give directions to the test subject while data is being collected. HELPS AND HINTS 1) Note the actual times and distances on the two axes. Distances are measured from the front of the motion detector. Use a meter stick and masking tape to mark distances on the floor. 2) Practice counting of seconds to yourself before you begin (one- thousand-one, one-thousand-two, etc.). 3) Don‟t have the person running the computer say, „Go!”. Instead, just listen for the clicks from the motion detector to start your „clock‟. Discuss together how a person must move to match the graph before anyone makes an attempt. Also, let every member of the group try to match the graph. After matching this motion, open file 01c Graph Matching and try that one. Print-Out Save at least one example of the first and second graph matches and print them. Make sure to choose „landscape‟ under Page Setup. Also, add a footer to the graph with the names of your group members and the date. Attach a copy of the graphs to your answer sheet. Graphs of Constant Acceleration Introduction Science and mathematics are not the same thing, but mathematics can be very useful for understanding scientific concepts and principles. For example, consider the definition of acceleration: a= Δv = vf – vi = v – v0 = v – v0 Δt tf – ti t–0 t The first ratio is the definition of acceleration. The second ratio explicitly includes the meaning of Δ as the final value minus the initial value. The third ratio assumes some general final time and corresponding velocity and an initial velocity v0 at time t=o. We can rearrange a = (v – v0 )/t into an equivalent form: v = v0 + at This equation assumes that the acceleration is constant. If the velocity increases by the same amount each unit of time – for example, every second – then a plot of velocity vs. time will be a straight line, and the slope of the line will be the constant acceleration. But the position vs. time graph will not be a straight line, because the object is moving faster and faster, so it is gaining a greater distance over the same time interval. The correct mathematical relationship between position and time for constant acceleration is one based on the square of the time. A graph of a square relationship ( d = mt2) is a parabola. Equipment and Set Up For this activity, we will use the computer-based data collection system with a motion detector. We will record the motion of a cart starting from rest and traveling down an inclined track. Procedure Use a few books to set up the track at an incline. Set the motion detector at the top of the track and at least 40 cm from the back of the cart. It may be helpful to attach and index card to the cart to create a larger target for the motion detector to „see‟. The time of each run will not be long, so it will be possible to wait a few moments after hearing the „clicks‟ before letting go of the cart. Data Collection It only takes a few seconds to record data , so get a „good‟ run. Does the plot of position vs time look like a parabola? Does the plot of velocity vs. time look like a straight line? Note the time on the graph when the cart first begins to move. Record the position and velocity of the cart. Record position and velocity for four or five more equally spaced times from this starting point: for example, if the cart begins to move at 1.35 seconds, record position and velocity at 1.45, 1.55, 1.65, and 1.75 seconds. These are four data points equally spaced from the starting point of 1.35 seconds every 0.1 seconds. Is the change in velocity approximately the same for each change in time? For example, is (v2 – v1) = (v3 – v2)? Is the total distance traveled seem to be proportional to the square of the travel time? For example, is the distance traveled (d2 – d0) approximately four times the distance traveled (d1 – d0), since the time of travel is twice as long? This relationship works exactly like this only if at time t=0 the cart is at rest, v0 =0. Other Trials 1. Suppose you set the motion detector at the end of the track instead of at the top? How would the graphs change? Would the acceleration be any different? Try it, but be sure to stop the cart before it hits the motion detector. 2. Next, push the cart up the track, let it slow down, stop, the move faster down the track. Save and plot the graphs of position vs time and velocity vs time. Do these graphs makes sense? Are they reasonable? How does the acceleration up the track compare to the acceleration down the track? Equilibrium and Vector Addition Introduction In Section 2.5 of Conceptual Integrated Science, Paul Hewitt explains how he became interested in physics when he was a sign painter in Florida. The equilibrium rule states that if an object is at rest, then the sum of all forces acting in one direction must be equal to the sum of all forces acting in the exact opposite direction. Section 3.5 of the text considers the properties of vectors. Understanding the properties of vectors and how to manipulate them is very important in physics, because so many important quantities are vectors, such as force, acceleration, velocity, displacement. It is rare in the real world that all the vectors happen to be acting in the same direction. Luckily, mathematicians and scientists have developed a complete set of rules for vector mathematics, including addition, subtraction, and multiplication of vectors. This activity will give you some hands-on practice working with the equilibrium rule and vector addition. Although we will add together only force vectors, the same mathematics applies to other vectors, such as velocity and acceleration. Equipment A horizontal bar; two spring scales; about 40 cm of string with loops on both ends; a meter stick; and 100 and 200 gram weights. Part One: A single weight For accurate results, „zero‟ the spring scale first and re-zero it when changing the orientation of the scale. 1: Attach a 100 gram weight to a string. Attach the other end of the string to a spring scale. (For the purposes of this activity, we can use the gram measurements on the spring scale.) Record the spring scale reading on the answer sheet. The spring scale is measuring the tension in the string. The tension in the string is equal to the weight of the mass in this case. 2: Loop the string over the horizontal bar. The scale is now upside down, so it may need to be re-zeroed. Hang the weight (still attached to the string) from the other end of the spring scale. What does the spring scale read now? Can you explain why? (Hint: the spring scale is measuring the tension in the string. Has the tension in the string changed? Why or why not?) Part two: A scaffold Create a scaffold from the meter stick (similar to Figure 2.11 in the text) by hanging two spring scales from the horizontal bar, attaching strings to the other ends of each scale, and tying the meter stick to the strings. Re-zero both scales to eliminate the weight-effect of the meter stick. The strings should be attached as close to the ends of the meter stick as possible. 3: If you hang a 200 gram weight from the center of the meter stick, what should be the reading of each spring scale? Make a prediction then test it. Write down both the prediction and the result on the answer sheet. 4: If you hang a 200 gram weight from the 66 cm mark of the meter stick, what should be the reading of each scale? Predict then test. Can you explain your prediction? 5: If you hang a 100 gram weight from the 25 cm mark and a 100 gram weight from the 75 cm mark, what do you predict will be the two scale readings? 6: Suppose you replace one of the two 100 gram weights with a 200 gram weight. How should each scale reading change? Give a qualitative answer, and a reason for it. 7: Hang three weights from three different locations on the meter stick. Verify that the equilibrium rule still holds. Part three: The Equilibrium Rule in two dimensions Leave the scales hanging from the horizontal bar, but loop a single string between them. Hang a 200 gram mass on the string, as shown in the figure. 1: Slide the scales along the bar to create 60 degree angles between the bar and each spring scale. Use a protractor to get accurate estimates of the angles. Record the scale readings. 2: Slide the scales further to create two 45 degree angles and record the spring scale readings again. 3: Finally, slide them further until the angles are approximately 30 degrees and record the readings. Vector Addition For each case, find the vector components and show that the sum of the horizontal components add to zero, and the sum of the vertical components add to zero. Use the Useful Triangles page to do the calculations FOR EXAMPLE: Suppose the scale reading for the left-hand scale was 145 grams for the 60 degree angle. Then the horizontal vector component will be 145 g * (0.5) = 74 g and the vertical component will be 145 g * (0.866) = 126 g Note that the three magnitudes in each case (the two scale readings and the mass of the weight) will never add to zero, no matter which numbers you call negative and positive. Vectors do not follow the rules of arithmetic and algebra, except when all vectors are acting along the same line (in the same direction). Question Given the fact that there are no horizontal forces acting, as in the case of the scaffold, can you see why the scale reading must be the same, if the weight is hanging symmetrically from the two strings? USEFUL TRIANGLES Trigonometry is the study of the relationships between the three sides of a triangle, especially right triangles. We don‟t need to consider all of trigonometry; we can limit ourselves to three useful cases: The 45-45-90 triangle: If one leg of the triangle has a length of one unit, the other leg is also one unit long, and the hypotenuse (longest side) is √2 = 1.414 units long. The 30-60-90 triangle: If the shortest leg of the triangle has a length of one unit, the hypotenuse is two units and the third leg is 1.73 units long. The 3-4-5 triangle In this triangle, then lengths of the sides are specified instead of the three angles. The smaller angle is 37 degrees and the larger angle is 53 degrees. It is now possible to find the relative magnitudes of the components of vectors that match one of these three triangles by setting up ratios. For example, Suppose a vector with a magnitude of 10 units matches the 30- 60-90 triangle. Then the component corresponding to the short side (call it A) will be: A/10 = 1/2 = 0.5 A = 10*(0.5) = 5 units And the other side, B, will be: B/10 = 1.73/2 = 0.866 A = 10*(0.866) = 8.66 units Newton’s First Law Introduction One version of Newton‟s First Law states: An object at rest stays at rest, or travels in a straight line with constant speed, unless impelled by a force to do otherwise. The law consists of three phrases, separated by commas. In this activity, we will try to understand what this law means by examining these three parts. Part One: An object at rest Place a golf ball on a smooth flat surface at a point where it is at rest. Observe the ball for about half a minute. What does the ball do? Record your observations on the answer sheet. Do your observations verify the first phrase of the law? Part Two: An object in motion Give the golf ball a quick push and observe it‟s motion immediately after the push. Record your observations on the answer sheet. Do your observations support both conditions of the second phrase of the law? Part Three: Unless Repeat Part Two, only now place a piece of cloth or carpet a few feet in front of the ball. Observe the ball‟s motion on the carpet and record your observations on the answer sheet. What caused the ball to change from initially at rest to moving? What explanation can you give for the ball‟s behavior on the carpet? Why was the ball‟s behavior on the carpet different from the ball‟s behavior on the smooth floor? The ball‟s motion on the smooth floor was probably not ideal. To what can you attribute the imperfect motion of the ball on the smooth floor? Applications 1. Place an index card on top of a glass or cup, and a coin on top of the index card. With your finger, quickly flick the edge of the card, causing it to move horizontally. On the answer sheet, record the behavior of the index card, the coin, and the cup. Carefully explain the behavior of each object separately in terms of Newton‟s First Law of motion. 2. Get two pieces of a light cotton string and a heavy (at least 200 gram) mass with two hooks or loops on it. Tie one string to each loop and hang the mass from one of the two strings. When performing the next two tasks, be careful to keep your feet or other body parts out of the way of the falling mass. Quickly pull the lower string until one of the two strings breaks. What do you observe? Reattach the broken string to the mass. Slowly pull the lower string until one of the two strings breaks. What do you observe? Carefully explain both observations using Newton‟s First law. Newton’s Second Law Introduction (long explanation) Newton‟s Second Law states: The acceleration of an object is proportional to the net force acting on the object and inversely proportional to its mass. Proportional means as one variable increases then so does the other by the same factor. For example, if the force is doubled, then the acceleration also doubles; if the force is cut to a tenth of its previous value, then the acceleration is also a tenth of what it used to be. Of course, these results assume that the value of the mass has not changed in the meantime – the mass is constant. Inversely proportional means that as one variable increases, the other decreases by the same factor. For example, if the mass is tripled, then the acceleration is only one third of its previous value, assuming the same force is applied. Newton‟s second law can be stated as an equation: F = ma, where F is the applied force, m is the object‟s mass, and a is the object‟s acceleration. Newton‟s second law states that the product of an object‟s mass and acceleration is proportional to the applied force. If we choose to measure acceleration in m/s2, mass in kilograms, and force in the unit of the Newton, then the proportionality constant is 1. (Think about converting between feet and inches; it is necessary to multiply or divide the measurement by 12 to get the new correct value. The unit of force, the Newton, is defined as 1 Newton = 1 kg-m/s2 , so the conversion factor between the unit of force and the product of mass and acceleration is 1.) The second law does not mean that force is the same thing as mass times acceleration. Forces exist independently of the accelerations they cause. Forces still exist even when there is no acceleration. If the relationship between two variables is proportional, then a graph of one versus the other is a straight line. The general form of a straight line is y = mx + b Compare this to F = ma If force is plotted on the Y-axis and acceleration is plotted on the X-axis while the mass is held constant, then we expect the graph to be a straight line with the slope of the straight line equal to the mass of the object. Note that the y- intercept is zero. This makes sense, since according to Newton‟s first law, if there is no net force, then the acceleration is zero. Experimental Design We will measure the acceleration of a system under six different applied forces and plot the data. If the data suggests a straight line on a graph of force vs. acceleration, then we have reason to believe that Newton‟s was correct and acceleration is proportional to the applied force. We will also be able to compare the slope of the line to the value of the constant mass. We will also check Newton‟s second claim, that acceleration is inversely proportional to mass, by doubling the mass of the system for one of the six applied forces and seeing if the acceleration is approximately halved. Equipment and Set Up Our system will consist of a cart attached to a weight hanger by a sting. The weight hanger will be suspended in the air by passing the string over a pulley. The weight hanger will provide the applied force to cause the system to accelerate. Adding more mass to the weight hanger will increase the applied force. We will measure the acceleration of the cart using a motion detector and the computer-based data collection system Procedure The weight hanger has a mass of 50 grams. We will add ten grams to it at a time (60, 70, 80, 90, 100) until the total mass is 100 grams, or twice the initial value. (What do you predict the acceleration should be in that case, compared to the first measured acceleration?) If the mass is 50 grams, then the weight is approximately 0.5 Newtons. A 100 gram mass has a weight of approximately 1.0 Newtons. The cart is being accelerated by the hanging mass. But the weight hanger (plus any extra mass on it) is also being accelerated at the same rate. Therefore, to keep the mass of the system constant, we will start with two 20- gram masses and one 10-gram mass on the cart, and transfer 10 grams at a time to the weight hanger for each trial. This means that the total mass of the accelerating system is the mass of the cart plus 100 grams. The total, constant mass can be measured. The applied force can be calculated from the known mass on the weight hanger. The acceleration will be measured using a motion detector. Set up the computer-based data collection system with a motion detector attached. Place the motion detector at the far end of the track, away from the pulley. Be sure to start the cart at least 40 cm away from the detector. It may be helpful to tape an index card to the back of the cart to create a larger target. Each run will be only a few seconds, so it will be possible to wait a few moments after the clicking has started before letting go of the cart. Be sure to catch the cart at the other end of the track, before it smashed through the pulley and off the track. Measure the acceleration for each trail by finding the slope of the velocity vs time graph created by the motion detector. Highlight the straight-line section of the graph and choose the R= button. The slope of the line will be reported. Round the value to the closest 0.01 m/s/s. Record the data in the data table. After all data is collected, make a graph of force vs. acceleration. Use the R= function again to find the slope of the straight mine, and compare the slope to the total mass of the system (about 600 g = 0.600 kg). Make one more run, this time adding a 500 gram mass to the cart, and using 100 grams hanging from the pulley. Compare the measured acceleration to the value for Trial 6. Does the data support Newton‟s contention that mass and acceleration are inversely proportional? Newton’s Third Law Equipment Two spring scales – one for each partner. Part One: Spring scales at rest Hook the two spring scales together, hold them horizontally, and zero them. Each partner should be holding one spring scale. Pull on the scales, but keep them at rest. How do the two scale readings compare? Pull harder, stretching the springs more. How do the two scale readings compare? Part Two: Spring scales moving at constant speed Move the spring scales back and forth at a constant speed. How do the two scale readings compare? Move faster back and forth, but at a greater speed. How do the two scale readings compare? Part Three: Spring scales accelerating Now accelerate the spring scales as they are hooked together. At any given instant of time, how do the two scale readings compare? Challenge Is there any way you can move the spring scales while they are hooked to each other so that the magnitude of one scale reading is different than the magnitude of the other scale reading at any single instant of time? If so, demonstrate to the instructor how it may be done. The Force of Friction Introduction Friction is so prevalent in our personal experience that it is the main reason why most of us find Newton‟s laws of motion counter-intuitive. In our experience, the natural tendency of objects is to slow down and stop, not to keep moving with a constant velocity. But once the culprit of friction is identified, Newton‟s first law seems to be more reasonable. Basic Facts of Friction Friction is actually a very complicated and detail subject, but scientists have been able to develop some useful approximations so that we can work with friction in real-world situations. Static Friction prevents things at rest from starting to move. Static friction always acts opposite to the intended direction of motion. Static friction has a magnitude equal to the size of the applied force, up to the „breaking point‟ – then the object begins to move. Kinetic Friction makes moving things slow down and stop. Kinetic friction always acts opposite to the direction of motion. It is logically necessary that the magnitude of static friction is always larger than (or at best, equal to) the magnitude of kinetic friction. (Why?) Both kinetic friction and static friction are proportional to the size of the force pressing on the object perpendicular to the surface where the object is located. Perpendicular forces are called normal forces, from an old-fashioned word meaning „perpendicular‟. The proportionality constant converts from the normal force to the size of the frictional force. The scientific name for the constant is „the coefficient of friction‟. A coefficient is a ratio and has no units: it converts from a force measured in Newtons to a force measured in Newtons. The symbol is the Greek letter „mu‟: μ. If the coefficient of static friction was 0.50, and the normal force was 10 Newtons, that would mean that static friction would react with a force to balance the force trying to make the block move up to a maximum value of 5 Newtons, and then static friction would „break‟. If the normal force was 15 Newtons, then the maximum static frictional force would be 7.5 Newtons. If the coefficient of static friction was 0.25, and the normal force was 10 Newtons, then while the object was moving, a constant 2.5 Newton frictional force would be acting on it trying to make it stop moving. If the normal force increased to 15 Newtons, then the frictional force would be 3.75 Newtons while it was moving. Equipment Wooden block with an eye hook, masses, and a spring scale Procedure Find the mass of the block and convert it to Newtons. (100 grams = 1 Newton) Hold the spring scale horizontally and zero it. Attach it to the eye hook of the block. Slowly pull on the block until it begins to move. Note the maximum value of the force on the spring scale before this happened. Do this a number of times and record the best average value for this force in the Static Friction column of the answer sheet. Pull the block across the surface at a steady speed and record the average value of the spring scale force in the Kinetic Friction column of the answer sheet. Place more mass on top on the block, 100-200 grams at a time, and repeat the procedure for four or five more trials. Make a graph of Static and Kinetic frictional force on the Y-axis vs the Normal force (= weight of the block and masses) on the X-axis. Is the Kinetic friction data a straight line? If so, find the slope of the line, which is the kinetic coefficient of friction. Is the Static friction data a straight line? If so, find the slope of the line, which is the static coefficient of friction. Is the coefficient of static friction larger than the coefficient of kinetic friction? Momentum and Collisions Introduction Newton‟s laws of motion always apply in all situations. Scientists use the laws to calculate where an object will be located and how fast it will be moving as a function of time. But Newton‟s laws are not always easy to apply. For example, when a bat strikes a baseball, the force is large and constantly changing during the interaction, and the time of action is very short. Instead, conservation of impulse-momentum may be used instead to describe the interaction. Impulse is defined as the product of force acting over time. If there are no external forces acting (as in this activity) then the impulse is zero. Momentum is defined as the product of mass and velocity: p = mv. Like velocity, momentum is a vector; direction makes a different. Conservation of impulse-momentum states that the sum of all momenta at a point before an interaction, plus and impulses caused by external forces, must be equal to the sum of all the momenta at a later point in the interaction. We are going to collide to carts into each other. For some trials, one or both carts may start from rest. In some trials, the carts may have the same mass; in others, the masses will be different. Some times, the carts will stick together; at other times, the carts will bounce off each other. For each different trial, the question is the same: does the data suggest that the sum of all momenta before the collision is equal to the sum of all the momenta after the collision? Equipment and Set Up Two carts, one with a spring-loaded bumper; a 500 gram weight, the computer-based data system with either two photogates or two motion detectors. The instructor will sow how to set up and use the photogates for data collection if they are used instead of motion detectors. Procedure The answer sheet lists eight different collisions. If the trial is an explosion (both carts start at rest) then push in the plunger and use a meter stick to quickly release the spring. If the trial is a collision, then push one cart towards the other. If motion detectors are used to measure velocity, find the slope of the straight line portion(s) of the position vs time graph. Each motion detector will record the speed of the cart it is facing. If photogates are used, position the gates as close to the index cards attached to the carts as possible without measuring the motion during the interaction for the most accurate results. For each trial, determine which direction will be positive and which will be negative. Use plus and minus signs accordingly for each velocity for that trial. (If a cart is moving in the positive direction, the velocity is positive; if a cart is moving in the negative direction, the velocity is negative.) Note that for inelastic trials where the carts stick together, there is only one final mass and one final velocity (not two). Energy Conversions Introduction Energy can come in many different forms. Energy, like momentum, is never measured directly. There is no „momentum meter‟ or „energy meter‟; instead, energy and momentum must be calculated from other measured quantities. The purpose of this activity is to practice calculating different forms of energy, and to verify that energy is conserved, in the absence of external forces working on the system. Here are three forms of energy and the simple equations used to calculate their values: Kinetic Energy is the energy that an object possesses because it is moving. The equation for kinetic energy is: KE = ½ m v2 where m is the mass of the object and v is its speed Because mass is always positive and the square of a number is also positive, kinetic energy is always positive. Gravitational Potential Energy is the energy that an object possesses near the surface of the earth based on it‟s vertical height. The equation for gravitational potential energy is: PEg = m g h where m is the mass of the object, g is the acceleration of gravity, and h is its vertical height. The reference point for the height is arbitrary: a table top, the floor, or any other height can be labeled „zero height‟. An object located below the reference point would then have negative energy. Elastic Potential Energy is the energy stored in a rubber band or spring. The equation for elastic (or spring) potential energy is: PEs = ½ k x2 where k is the spring constant and x is the amount of stretch or compression from the spring‟s equilibrium length. The spring constant is related to the stiffness of the spring; t measures how much force it takes to stretch or compress the spring by a certain amount. The equilibrium length of the spring is measured when no forces are acting on the spring at all. The spring constant can be found as the slope of the straight line on a graph of applied force vs spring length. To measure the spring constant, it is necessary to pull or push on the spring with a known force and measure the extension or compression of the spring. Best results are achieved if a number of data points are collected and a graph is constructed. Slope is spring constant F o r c e Stretch Part One: Gravitational Potential to Kinetic Energy To measure the conversion of gravitational potential energy to kinetic energy, we will roll a ball or cart down a ramp again. Data can be collected using meter sticks and stop watches (as in the Measuring Velocity or Acceleration activities) or using a cart on a track with a motion detector (as in the Graphs of Constant Acceleration activity.) The choice is yours. Measure the mass of the cart or ball, and the vertical height of the object above the level surface at the base of the ramp or track. Convert the height in cm to meters, and the mass in grams to kilograms. Use 10 m/s2 as the acceleration of gravity, g. Calculate the gravitational potential energy at the top of the ramp. The unit of energy is the Joule, J. Measure or calculate the speed of the cart or ball at the base of the ramp or track. Calculate the kinetic energy of the object, in Joules. Are the two values close? Part Two: Elastic to Gravitational Potential Energy Get a spring and calculate its spring constant. Measure and record the length of the unstretched spring in the data table on the answer sheet. Use a spring scale to pull on the spring. Record at least four different pulling forces and corresponding spring lengths. Find the spring constant from a graph of applied force vs. stretch length. We are going to launch the spring from a rod or meter stick vertically into the air. We want to stretch the spring to the right length so that the spring just brushes the ceiling in the room. Measure the vertical distance from the launch point to the ceiling. Calculate the gravitational potential energy at the ceiling using PEg = mgh, where m is the mass of the spring and h is the distance just measured. Make sure mass is measured in kilograms and height is measured in meters. Set the gravitational potential energy equal to the elastic potential energy of the spring: PEg = PEs mgh = ½ k x2 SOLVE for x, the stretch necessary to make the spring just reach the ceiling. Launch the spring using that stretch and see if conservation of energy worked. Part Three: Elastic Potential to Kinetic Energy Set up a cart with a spring-loaded bumper. It takes a force of 25 Newtons to push the spring bumper in 3 cm. From this data, estimate the spring constant for the bumper. Press in and lock the bumper. Place it against the back end of the track. Release the bumper and measure the speed of the car after release – either using a motion detector or a meter stick and stop watch. Compare the stored elastic potential energy when the cart was at rest to the kinetic energy of the cart after the spring was released. Are the energies close? Energy Conservation Introduction The principle of energy conservation is so very useful because it is only necessary to identify two particular points in the interaction – two „snapshots‟ – in order to calculate useful quantities like a distance or a speed. In this activity, we will see that energy is conserved at all times of an interaction. We are going to use the motion detector to collect a continuous set of data over a few seconds for two different systems: a mass bouncing up and down on a spring; and a ball bouncing on the floor. Experiment files have already been created and saved for you that include calculations of elastic, gravitational, and kinetic energy. But it will be necessary to edit the definitions, adding the particular spring constant and mass of your system. The instructor will show you how to do that. Part One: Bounce Attach a motion detector to the Lab Pro and open the experiment file called Bounce.cmbl. Hold the motion detector one meter off the floor, facing downward. Use a meter stick to obtain a good approximation of this distance. Hold a ball 50 cm away from the floor, or halfway between the motion detector and the floor. Start the data collection and release the ball. Follow the ball as it bounces (if needed) for at least four bounces. After data collection, use the Graph Options to examine the gravitational potential, kinetic, and total energy of the bouncing ball. Print the graph and attach it to your answer sheet. Describe what you see, and what it means. The instructor can show you how to edit the definition of the potential energy to achieve better results if the detector was not exactly one meter from the floor, or the ball was not released exactly 50 cm from the floor. Part Two: Stretch Attached a motion detector to the Lab Pro and open the experiment file called Stretch.cmbl. Hang a spring from a rod and attach a weight hanger to the other side of the spring. Choose a total mass to add to the hanger to make vibrations on the spring. If you don‟t know the value of the spring constant, you will need to find it. Edit the definitions of the elastic potential energy and the kinetic energy to include the value of your spring constant (in Newtons per meter) and your mass (in kilograms). Place the motion detector on the floor, facing upward and directly underneath the weight hanger. „Zero‟ the motion detector when the weight hanger is at rest. Slightly nudge the weight hanger to start it moving. Large amplitude vibrations do not necessarily lead to good results. After data collection, use the Graph Options to examine the elastic, kinetic, and total energy of the vibrating mass. Print the graph and attach it to your answer sheet. Describe what you see, and what it means. Comparing Methods Introduction Newton‟s laws of motion, conservation of impulse-momentum, and conservation of work-energy are three mathematically equivalent methods for understanding how and why things move. In this activity, we are going to collect data for one single trial and demonstrate that if the product of mass and acceleration is numerically equal to the applied force; then the impulse acting on the system is equal to the change in the system‟s momentum; and the work done by the applied force acting over a distance is equal to the change in klinetic energy of the system. Equipment and Set up The activity used the same equipment and set up as Newton‟s Second Law. Procedure Collect data using a motion detector for one „good‟ run of mass on a weight hanger accelerating a cart down a track. Newton’s Laws: Dynamics Newton‟s second law states that the product of mass and acceleration is equal to the applied force. The applied force is the weight of the mass hanging over the pulley. The mass is the mass of the cart, plus weight hanging mass. The acceleration of the system can be found from the slope of the straight-line portion of the velocity vs time graph. Is the magnitude of the net force approximately equal to the product of the mass and acceleration? Conservation of Impulse-Momentum Impulse is the product of a force acting over time; momentum is the product of mass and velocity. We have already calculated the magnitude of the constant force acting during the trial run. Use the X= button to locate a data point on the straight line portion of the velocity vs time graph near the point in time when the cart first began to move. Record both the time and velocity. Locate a second point near the end of the straight-line portion of the graph and record the time and velocity. Multiply the force by the change in time to find the impulse. Multiply the mass by each velocity to find the initial and final momenta associated with this impulse, then find the change in momentum. Are the numerical values for the impulse and the change in momentum approximately the same? Conservation of Work-Energy Work is the product of a force acting over distance; kinetic energy is hjalf the product of mass and the square of the velocity. Use the X= button to find the positions of the cart at the two times chosen for the previous part from the position vs time graph. Multiply the force by the change in distance to find the work. Also calculate the initial and final kinetic energies associated with these two positions. Are the numerical values for the work and the change in kinetic energy approximately the same? Important Note Because these three methods can be shown to be mathematically equivalent, and because we are using the same data for all three methods, the uncertainties should be the same. For example, if the force was only 80% of the product of mass and velocity, we would expect the other two comparisons to also be off by about 20%. Ballistic Pendulum Introduction Here is an opportunity to employ all three methods for understanding motion – Newton‟s laws, conservation of momentum, and conservation of energy – in an interesting, real life problem. We are going to find the speed of a bullet immediately after it leaves a gun by two different approaches. One approach will apply conservation of momentum and conservation of energy as the bullet strikes a second object; the second method will apply Newton‟s laws of motion to a trajectory. If the two calculations of the „muzzle velocity‟ are close, then we have reason to believe that all three methods for understanding how things move are scientifically accurate. Instead of just telling you what to do, a series of questions will help you piece together the scientific principles needed to calculate the speed, the measurements that must be made, and the calculations necessary to find the bullet‟s speed using the two methods. Set Up A brass ball will serve as our bullet on a spring loaded gun. For the first calculation of its speed, we will shoot the ball into a pendulum. The force of impact will cause the pendulum to rise to a maximum height, and a locking system will hold it in place. From the vertical rise of the pendulum and the measured masses of the pendulum and bullet, we will be able to calculate the speed of the ball when it left the spring-loaded gun. For the second calculation, the pendulum is removed and the ball is shot off the table. The vertical distance of the ball from the ground and the distance it traveled forward until it hit the ground are the only two measurements we need to calculate the speed of the ball by this method. The Impact Method – Questions 1) The Impact Method consists of two parts: the bullet strikes the pendulum, and the pendulum rises in the air. Conservation of energy and conservation of momentum are always true, but in this case, one method is more useful for the bullet striking the pendulum, and the other method is more useful for the pendulum rising in the air. Which method should be used with which part? (Hint: the bullet strikes the pendulum.) 2) Consider the pendulum and imbedded bullet at the highest point of its motion, when it is momentarily at rest. What kind of energy does the system possess at this point? 3) Consider the pendulum and imbedded bullet at the lowest point of its motion, immediately after the collision. (We can assign this point to be zero height.) What kind of energy does the system possess at this point? 4) According to the principle of conservation of energy, all of the energy of the pendulum plus bullet at the lowest point of the pendulum arc, immediately after the collision, must equal all of the energy of the pendulum plus bullet at the highest point of the arc, when the pendulum and bullet are momentarily at rest. (We will assume that external forces are very small, so that no work is done by outside forces and no energy is lost.) Use your answers to questions 2 and 3 to write an equation based on conservation of energy. 5) What quantities must be measured to calculate the energy of the bullet and pendulum at the highest point of the arc? Measure these quantities and calculate the energy. 6) What quantities must be measured to calculate the energy of the bullet and pendulum at the lowest point of the arc, immediately after the collision? One of these quantities will be unknown. Set up the equation and solve for the unknown. 7) The value calculated in Question 6 is not the speed of the ball as it left the gun. Use the principle of conservation of momentum to compare the momentum of the bullet-and-pendulum after the collision with the momentum of the bullet and the momentum of the (at rest) pendulum before the collision. 8) Use the definition of momentum, p = mv, to expand the equation in Question 7 to include measurable quantities: masses and velocities. All of the masses can be measured. One of the velocities was calculated in Question 6 (which one?). Solve for the one unknown velocity – the velocity of the ball as it left the spring-loaded gun. Trajectory method – Questions 1) Write down the definition of the horizontal velocity of the ball after it leaves the gun. What two quantities must be measured to calculate this speed? 2) One of those two quantities can be measured directly. Do that and record the value on the answer sheet. 3) As the ball travels forward horizontally, it also falls vertically. The initial vertical velocity is zero. 4) Write down the equation describing the vertical motion of the ball as it falls. (Hint: it can be found on page 32 of the text, under the section on Hang Time.) 5) Two of the three variables on the equation from question 4 are either known or can be measured. Solve the equation for this value. (Hint: it is the same calculation as the second equation in the section on hang time.) 6) The quantity you just calculated for Question 5 is the same unknown value that is needed in the equation from Question 1 to solve for the horizontal speed of the ball. Use it to find the speed. Final Result Compare the two calculated speeds of the ball as it left the spring-loaded gun. Are they close?
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