UNIT 5 SESSION 3: FORCE, MASS AND ACCELERATION

Transcription

1 Name Date Partners UNIT 5 SESSION 3: FORCE, MASS AND ACCELERATION "... equal forces shall effect an equal change in equal bodies..." I. Newton OBJECTIVES To develop a definition of mass in terms of an object's motion under the influence of a force. To understand the relationship among the force applied to an object, the mass of the object and its motion. To find a mathematical relationship between the acceleration of an object and its mass when a constant force is applied--newton's Second Law. To examine the quantitative relationship between force, mass and acceleration--newton's Second Law--in terms of the SI units (N for force, kg for mass, and m/s/s for acceleration). To pull all of the observations together and state Newton's First and Second Laws of Motion for motion in one dimension, along a straight line for any number of forces acting on an object Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE) Modified for SFU by N. Alberding, 2005, 2007.

2 Page 5-42 Real Time Physics: Active Learning Laboratory SFU 1077 OVERVIEW In this lab you will continue to develop the first two of Newton's famous laws of motion. You will do this by combining careful definitions of force and mass with observations of the mathematical relationships among force, mass and acceleration. You have seen that the acceleration of an object is directly proportional to the combined or net force acting on the object. If the combined force is not zero, then the object will accelerate. If the combined force is constant, then the acceleration is also constant. These observations are part of Newton's Second Law of Motion. You have also seen that for an object to move at a constant velocity (zero acceleration) when friction is negligible, the combined or net force on the object should be zero. The law which describes constant velocity motion of an object is Newton's First Law of Motion. Newton's First and Second Laws of Motion are very powerful! They allow you to relate the force on an object to its subsequent motion. Therefore, when the nature of the force(s) acting on an object is known, then Newton's laws of motion allow you to make mathematical predictions of the object's motion. In Investigation 1 of this lab you will study how the amount of "stuff" (mass) you are accelerating with a force affects the magnitude of the acceleration. What if the mass of the object were larger or smaller? How would this affect the acceleration of the object for a given combined force? In Investigation 2 you will study more carefully the definitions of the units in which we express force, mass and acceleration. Finally, in Investigation 3, you will examine the motion of an object caused by a force applied to it when friction is large enough so that it cannot be ignored Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

3 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-43 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws INVESTIGATION 1: MEASURE MASS In previous activities you have applied forces to a cart and examined its motion. Up until now, you have always used a cart with the same mass. But when you apply a force to an object, you know that its mass has a large effect on the object's acceleration. For example, compare the different accelerations that would result if you pushed a 1000 kilogram (metric ton) automobile and a 1 kilogram cart, both with the same size force! It would be nice to be able to do a mechanics experiment in one part of the world and have scientists in another part of the world be able to replicate it or at least understand what actually happened. This requires that people agree on standard units. In 1960 an international commission met to agree upon units for fundamental quantities such as length, time, mass, force, electric current, pressure, etc. This commission agreed that the most fundamental units in the study of mechanics are length, time, and mass. All other units including force, work, energy, torque, rotational velocity, etc. which you encounter in your study of mechanics can be expressed as a combination of these basic quantities. The fundamental International System or SI units along with the standard unit for force are shown below. FUNDAMENTAL UNITS FOR MECHANICS Length: A metre (m) is the distance travelled by light in a vacuum during a time of 1/299,792,458 second. Time: A second (s) is defined as the time required for a cesium-133 atom to undergo 9,192,631,770 vibrations. Mass: A kilogram (kg) is defined as the mass of a platinum-iridium alloy cylinder kept at the International Bureau of Weights and Measures in Sèvres France. It is kept in a special chamber to prevent corrosion. Suppose you want to find the mass of an object in kilograms. You need to compare it to the one kilogram platinum-iridium alloy cylinder at the International Bureau of Weights and Measures in France. It would be nice to have a standard kilogram in your laboratory. You could go to France, but it is unlikely that they would let you take the standard home with you! Suppose, however, that you accelerate the standard mass with a constant force and measure the force and also the resulting acceleration as accurately as possible. Next you would need to make a cylinder that seemed just like the standard one and add or subtract stuff from it until it undergoes exactly the same acceleration with the same constant force. Then within the limits of experimental uncertainty this new cylinder standard and the bureau standard Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

4 Page 5-44 Real Time Physics: Active Learning Laboratory SFU 1077 would have the same mass. If the comparison could be made to three significant figures, then the mass of your new standard would be m std = 1.00 kg. Suppose you head home with your standard mass. You wish to determine the mass of another object. You could apply the same constant force, F, on the standard and on the other object, and measure both accelerations. Then, according to Newton's Second Law, F = ma, m std = 1.00 kg = F/a m other = F/a other Since the constant force, F, applied to both masses was the same, m other = 1.00 kg a/a other In this investigation you will explore the mathematical relationship between acceleration and mass when you apply the same constant force to a cart while changing the mass. You will need: Lab Pro software RealTime Physics Mechanics Experiments folder Force probe Motion detector Lab Pro interface Low-friction cart standard bar mass Force and Motion track Fan accessory + four batteries. Two dummy batteries in their holders Variety of additional masses (e.g., tennis ball, steel ball) Activity 1-1: Acceleration and Mass You can easily change the mass of the cart by attaching masses to it, and you can apply the same force to the cart by using using the fan accessory. By measuring the acceleration of different mass carts, you can find a mathematical relationship between the acceleration of the cart and its mass, with the applied force kept constant. 1. Set up the track, cart, fan, motion detector. Be sure that the track is level. Use the fan accessory with all the batteries installed The motion detector should be plugged into DIG/SONIC First measure the acceleration of the cart and fan without any additional mass. Record the velocity vs. time graph, select a region of the graph after release and before crash and that is relatively linear Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

5 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-45 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws Choose Linear Fit. The acceleration is the slope of the graph acceleration of cart+fan = Copy this number to the appropriate cell of the table. 3. Place the mass bar on the cart, under the fan, and again measure the acceleration, as before, with the additional mass acceleration of cart+fan+bar Copy this number to the appropriate cell of the table. 4. Calculate the mass of the cart+fan in terms of the standard bar mass and enter the mass into the mass cell in the first row. Use the relationship m 1/m 2 = a 2/a 1. In this way we are using inertia as a way of measuring mass. Comment: Physicists call the quantity you have just calculated--the ratio of combined (net) force on an object to its acceleration--the inertial mass of the object. mass of cart+fan = bars 5. Assume that the bar mass is calibrated to be exactly kg. Determine the mass of the cart+fan in kg. mass of cart+fan = kg Prediction 3-1: Suppose that you place another standard bar on the cart, and accelerate it with the same applied force. Predict the acceleration and enter the predicted value after the p: in the table Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

6 Page 5-46 Real Time Physics: Active Learning Laboratory SFU Test your prediction. Place another bar on the cart (borrowing one from a neighbour if necessary) and measure the acceleration. Enter the measured value in the table after m: and compare the value with your prediction. Is it significantly different? (That is, considering the accuracy of your measurement, is any difference between the predicted value and the measured value real and reproducible?) Question 1-1: Did the acceleration agree with your prediction? If it s different, is it significantly different? (That is, considering the accuracy of your measurement, is any difference between the predicted value and the measured value real and reproducible?) Explain Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

7 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-47 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws 7. Take off the extra bar and place another object that will fit on the cart cause a significant increase in the total mass. Measure the acceleration again and determine the mass of the extra object that you just put on the cart. For next week s activities you will need to know the masses of a tennis ball and a steel ball. You can measure both of these masses now Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

8 Page 5-48 Real Time Physics: Active Learning Laboratory SFU 1077 INVESTIGATION 2: MEASURING FORCE So far you have been measuring force in standard units based on the pull exerted by a spring scale calibrated in newtons. Where does this unit come from? We need a way to define the unit of force without having a spring scale that is already calibrated. That can be done if we accept the relation F = ma We know how to measure acceleration by observing the motion of an object. The mass of something is measured by comparing accelerations when the same force is exerted on it and on a standard mass. The unit of force called a newton is that amount of force that would accelerate one kilogram one m/s/ s. In the following experiment we shall find the force that the fan exerts on the cart by measuring the acceleration of known masses. You will need the following equipment: Logger Pro software RealTime Physics Mechanics Experiments folder Force probe Motion detector LabPro interface Spring scale with a maximum reading of 5 N Low friction cart Fan Accessory Force and Motion Track THE FORCE UNIT EXPRESSED IN TERMS OF LENGTH, MASS, AND TIME Force: A newton (N) is defined as that force which causes a 1 kg mass to accelerate at 1 m/s/s. Activity 2-1: Calculating the Fan Force in Newtons 1. You have already enough data to calculate the force of the fan cart when it has four batteries. Use Newton s second law to calculate the force exerted by the fan cart. Calculate the Fan force in each case and enter the values in the table. Comment on whether the values are consistent with each other or not. If you don t think they are consistent, why? Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

9 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-49 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws Question 2-1: Each of the forces calculated were the result of the fan accessory and four batteries. There are probably differences in the calculated values obtained, even though the force was expected to be the same. Comment on whether you think any differences are due to random experimental errors, or whether the differences represent real variations of the force. You could continue to determine and compare masses by accelerating them and taking force to acceleration ratios, but this process is pretty tedious. A simpler approach is to use an electronic scale or a mechanical balance that has already been calibrated in kilograms using a standard mass by somebody who is intelligent and knowledgeable! (The details of why such devices can give us correct masses in kg will not be easy to understand fully until after gravitational forces are studied in Lab 7.) 3. Compare your inertial mass measurements with the values you get by placing your cart on an electronic or mechanical scale. Record these values in the last column of the table. Question 2-2: If you change the number of batteries and place an object of unknown mass in the cart, can you determine either the value of the force in newtons or the value of the mass in kilograms? Comment: In your experiments, you have seen that the physical quantities force, mass and acceleration are related through Newton's Second Law. In the activity you have just done, you have used this relationship to define force in terms of standard units of mass, length and time. This is a good logical definition of inertial mass Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

10 Page 5-50 Real Time Physics: Active Learning Laboratory SFU 1077 It would also be possible to use standard units of force, length and time and then define mass. Question 2-3: What would be the units of mass if the standard units were newtons, metres and seconds? Activity 2-2: Determine the force of the fan cart with fewer than four batteries. 1. Set up the track, cart, fan accessory, and motion detector as in Activity 1-1. (See page 6-4.) Place known masses on the cart and replace two batteries with the dummy slugs. Place the unused batteries in the holders where the slugs were. 2. Measure the acceleration produced by the fan in this case. Fill in the next row in the table and calculate the force. 3. Change the mass and repeat step 2 and fill in the next row. 4. If you have time you can also measure the force produced by using three batteries or even one single battery. Question 2-4: Do you think that the relationship between number of batteries and the force is linear? Question 2-5a: A force of 5.4 N is applied to an object, and the object is observed to accelerate with an acceleration of 3.0 m/s/s. If friction is so small that it can be ignored, what is the mass of the object in kg? Show your calculation Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

11 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-51 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws Question 2-5b: An object of mass 39 kg is observed to accelerate with an acceleration of 2.0 m/s/s. If friction is negligible, what is the force applied to the object in N? Show your calculation. Comment: The main purpose of this unit has been to explore the relationship between the forces on an object, the object's mass, and its acceleration. You have been trying to develop Newton's First and Second Laws of Motion for one-dimensional situations in which all forces lie in a positive or negative direction along the same line. Activity 2-3: Newton's Laws in Your Own Words Question 2-6: Express Newton's First Law (the one about constant velocity) in terms of the combined (net) force applied to an object in your own words clearly and precisely. Question 2-7: Express Newton's First Law in equations in terms of the acceleration vector, the combined (net) force vector applied to an object, and its mass. If ΣF = then a = and v = Question 2-8: Express Newton's Second Law (the one relating force, mass, and acceleration) in terms of the combined (net) force applied to an object in your own words clearly and precisely Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

12 Page 5-52 Real Time Physics: Active Learning Laboratory SFU 1077 Question 2-9: Express Newton's Second Law in equations in terms of the acceleration vector, the combined (net) force vector applied to an object, and its mass. If ΣF 0 then a = Comment: The use of the equal sign does not signify that an acceleration is the same as or equivalent to a force divided by a mass, but instead it spells out a procedure for calculating the magnitude and direction of the acceleration of a mass while it is experiencing a net force. What we assume when we subscribe to Newton's Second Law is that a net force on a mass causes an acceleration of that mass. Beware, the equal sign has several different meanings in text books and even in these labs. In order to reduce confusion a few of the meanings of and symbols used for the "equal" sign are summarized below: = the same as (e.g., 2+2 = 4) = also means equivalent to (e.g., four quarters is equivalent in buying power to one dollar) = can be calculated by (e.g., in physics for a body undergoing constant acceleration v = vo + at) = is replaced as (e.g., in a computer program x= x+1 means replace x with the value x+l), ~ is approximately the same as or equivalent to is defined as (e.g., vavg (x 2 -x1)/(t2-t1)) Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

13 Real Time Physics: Lab 6: Force, Mass and Acceleration Page 5-53 Authors: David Sokoloff, Ronald Thornton & Priscilla Laws FINAL COMMENTS ON FORCE MASS AND ACCELERATION You started your study of Newtonian Dynamics in Unit 5-1 by attempting to develop the concept of force. Initially when asked to define forces most people think of a force as an obvious push or pull such as a punch to the jaw or the tug of a rubber band. By studying the acceleration that results from a force when little friction is present, we came up with a second definition of force as that which causes acceleration. These two alternative definitions of force do not seem to be the same at all. Pulling on a hook attached to a wall doesn't seem to cause the wall to move. An object dropped close to the surface of the earth accelerates and yet there is no visible push or pull on it. The genius of Newton was to recognize that he could define net force or combined force as that which causes acceleration and that if the obvious applied forces did not account for the degree of acceleration then other "invisible" forces must be present. A prime example of an invisible force is the gravitational force--the attraction of the earth for objects. Finding invisible forces is often hard because some of them are not active forces. Rather, they are passive forces which come up in response to active ones. They are not only invisible, but they only crop up to oppose active forces. Friction is an invisible passive force. The passive nature of friction is obvious when you think of an object like a block being pulled along a rough surface at constant velocity. There is an applied force (active) in one direction and a friction force in the other direction which opposes the motion. If the applied force is discontinued the block will slow down to rest but it will not start moving in the opposite direction due to friction. This is because the friction force is passive and stops acting as soon as the block comes to rest. CHALLENGE: In your attempts to measure mass using the fan cart you probably noticed that the Inertial Mass measurement with the fan cart differed from the Gravitational Mass as measured by the scale. Later we will study friction and find that the strength of the frictional force increases with the weight of the object. Speculate on how our neglect of frictional forces could have affected the inertial mass measurement. Would it have made the measurement too low or too high? Would the affect of friction on acceleration under a constant force have the same mass dependence as inertia? Dickinson College, Tufts University, University of Oregon Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)

14 Page 5-54 Real Time Physics: Active Learning Laboratory SFU 1077 Force, Mass and Acceleration Table Source of force Object Acceleration (m/s 2 ) Mass (kg) Force (N) fan & 4 batteries cart+fan fan & 4 batteries cart+fan + ¼kg bar fan & 4 batteries cart+fan + 2 ¼kg bars pred: meas: fan & 4 batteries cart+fan + fan & batteries cart+fan + fan & batteries cart+fan + fan & batteries cart+fan + fan & batteries cart+fan + fan & batteries cart+fan Dickinson College, Tufts University, University of Oregon, modified for SFU by N. Alberding, 2005, Supported by National Science Foundation and the U.S. Dept. of Education (FIPSE)