Laboratory Exercise. Newton s Second Law
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1 Laboratory Exercise Newton s Second Law INTRODUCTION Newton s first law was concerned with the property of objects that resists changes in motion, inertia. Balanced forces were the focus of Newton s first law. Newton s second law explores what happens to objects when the forces acting on it are imbalanced. What he found was that the ForceNet = Mass x Acceleration. He saw that when an imbalanced force acts on an object the object accelerates in the direction that the sum (net) of all forces is applied. In the first exercise in this series you will use a Force Sensor and an Accelerometer to measure the force on a cart simultaneously with the cart s acceleration. The total mass of the cart is easy to vary by adding masses. Using these tools, you can determine how the net force on the cart, its mass, and its acceleration are related. This relationship between F, M, and A is Newton s second law of motion. The second exercise has you explore Newton s second law using a Force Sensor, Photogate, and Picket Fence. Objectives Collect force and acceleration data for a cart as it is moved back and forth. Compare force vs. time and acceleration vs. time graphs. Analyze a graph of force vs. acceleration. Determine the relationship between force, mass, and acceleration. computer Vernier computer interface Logger Pro Materials Vernier Low-g Accelerometer Vernier Force Sensor dynamics cart w/0.5 kg mass Page 1 of 8
2 Making predictions - your hypothesis 1. When you push on an object, how does the magnitude of the force affect its motion? Make a sketch of Force verses Velocity for an object moving under the influence of a continuous positive net force. Describe what this graph shows.. Make a sketch of your prediction for the Force vs. Acceleration graph for an object moving under the influence of a continuous positive net force. Describe what this graph shows. 3. Finally make predictive graph showing what will happen to the acceleration of an object of double the mass of the mass in predictions one and two when under the influence of the same applied net force. Describe what this graph shows. Figure 1 Force vs. Velocity Figure Force vs. Acceleration Figure 1 Force vs. Acceleration PART I Procedure 1. Connect a Dual-Range Force Sensor to Channel 1 on the Vernier computer interface. Connect the Low-g Accelerometer to Channel on the interface.. Open the file 09 Newton s Second Law from the Physics with Vernier folder. 3. Attach the Force Sensor to a dynamics cart so you can apply a horizontal force to the hook, directed along the sensitive axis of your particular Force Sensor. Next, attach the Accelerometer so the arrow is horizontal and parallel to the direction that the cart will roll. Orient the arrow so that if you pull on the Force Sensor the cart will move in the direction of the arrow. Find the mass of the cart with the Force Sensor and Accelerometer attached. Record the mass in the data table. 4. Place the cart on a level surface. Make sure the cart is not moving and click. Check to make sure both sensors are highlighted and click. Trial I 5. You are now ready to collect force and acceleration data. Grasp the Force Sensor hook. Click and take several seconds to move the cart back and forth on the table. Vary the Page of 8
3 motion so that both small and large forces are applied. Make sure that your hand is only touching the hook on the Force Sensor and not the Force Sensor or cart body. 6. Note the shape of the force vs. time and acceleration vs. time graphs. Click the Examine button,, and move the mouse across the force vs. time graph. When the force is maximum, is the acceleration maximum or minimum? To turn off Examine mode, click on the Examine button,. 7. The graph of force vs. acceleration should appear to be a straight line. To fit a straight line to the data, click on the graph, then click the Linear Fit button,. Record the equation for the regression line in the data table. 8. Save the graphs to include in your web site. Trial 9. Attach the 0.5 kg mass to the cart. Record the mass of the cart, sensors, and additional mass in the data table. 10. Repeat Steps 5 8. Trial Use this apparatus as a way to measure mass. Place an unknown mass on the cart. Measure the acceleration for a known force and determine the mass of the unknown. Compare your answer with the actual mass of the cart, as measured using a balance. Report data and % error in your web site. Data Trial I Mass of cart with sensors (kg) Regression line for force vs. acceleration data Trial Mass of cart with sensors and additional mass (kg) Regression line for force vs. acceleration data Page 3 of 8
4 Analysis Part 1 Include answers in the analysis portion of your web site. 1. Compare the graphs of force vs. time and acceleration vs. time for a particular trial. How are they different? How are they the same?. Are the net force on an object and the acceleration of the object directly proportional? Explain, using experimental data to support your answer. 3. What are the units of the slope of the force vs. acceleration graph? Simplify the units of the slope to fundamental units (m, kg, s). 4. For each trial compare the slope of the regression line to the mass being accelerated. What does the slope represent? 5. What general equation relates all three variables: force, mass, and acceleration? PART This exercise is similar to the forst in that a net force is applied to a mass. In this experiment, however, the acceleration will be due to gravity rather than you pulling the cart. Mr Ballog will show you the apparatus for this experiment. It is called a modified Atwood s machine. A weight is connected to a cart by a string over a pulley. When the weight is allowed to fall, observe the motion of the cart. A force sensor mounted on the cart enables you to measure the force acting on the cart when it is moving. A photogate will be used to measure the velocity of the cart. Attach an Ultra Pulley to the Pulley Bracket; attach this assembly to one end of the Dynamics Track. Make sure that you have adjusted the track to make frictional forces negligible.. Connect the Dual-Range Force Sensor (DFS) and a Photogate to the interface; then start the data-collection program. You will need only graphs of force vs. time and velocity vs. time, so you can delete others and then re-size graphs to make them easier to see. 3. Attach the force sensor to the cart. 4. Attach the photogate to the track using a bracket so that the spokes on the pulley interrupt the beam, as shown in Figure 1 (the bracket used in class will not match the diagram). Figure 1 Page 4 of 8
5 5. Discuss with your team what range of masses to use to apply the force that will accelerate the cart. Determine the total mass of your cart, force sensor, and any additional mass you may be instructed to use. 6. Set up data collection. Using Logger Pro a. Choose Set Up Sensors Show All Interfaces from the Experiment menu. b. Click the image of the Photogate, select Set Distance or Length, then select Ultra Pulley (10 Spoke) Inside from the list of devices. 7. Disconnect the hanging mass from the force sensor, then zero the sensor. 8. Re-connect the hanging mass to the force sensor. Position the cart so that the hanger has about 30 cm to fall. Use something to cushion the hanging weight when it strikes the floor. 9. Begin collecting data, then release the cart. Stop data collection after the weight hits the cushion on the floor. 10. To determine the force acting on the cart, select the portion of the force vs. time graph corresponding to the interval during which the cart s velocity was changing smoothly. Find the statistics for this interval. Manual scaling of your graph is more helpful for doing this than Autoscaling. 11. To determine the acceleration of the cart, perform a linear fit on the portion of the velocitytime graph during which the velocity is changing smoothly. Be sure to record the values of force and acceleration for this hanging mass in your lab notebook. 1. Repeat Steps 9 11 until you have three pairs of force vs. acceleration data that are reasonably consistent for that mass. 13. Increase the hanging mass and continue until you have acceleration-force data for at least five different hanging masses. 14. Save all data and graphs to include in your web site. Analysis Part 1. To evaluate the relationship between acceleration and force, disconnect the sensors and choose New from the File menu.. Even though you investigated how acceleration responded to changes in the force, in order to facilitate your analysis of data, plot a graph of force vs. acceleration. Compare this to your previous predictions. 3. If the relationship between force and acceleration appears to be linear, fit a straight line to your data. If possible, print a copy of your data table and graph. 4. Write the equation that represents the relationship between the force, F, acting on the cart and the cart s acceleration, a. Record the value of the mass of the cart, sensor and any additional masses you used and report in your web site. 5. Write a statement that describes the relationship between the force acting on the cart and Page 5 of 8
6 the cart s acceleration. 6. Compare your results to those obtained by others in class. What relationship appears to exist between the slope of the graph of F vs. a and the mass that was accelerated? 7. Assuming that the conclusion you reached in Step 6 is correct, express the SI unit of force, N, in terms of the fundamental units of mass, length and time. 8. Write a general equation that expresses the relationship between the force, mass and acceleration. PART 3 When you solve physics problems involving free fall (such as in the second exercise), often you are told to ignore air resistance and to assume the acceleration is constant and unending. In the real world, because of air resistance, objects do not fall indefinitely with constant acceleration. One way to see this is by comparing the fall of a baseball and a sheet of paper when dropped from the same height. The baseball is still accelerating when it hits the floor. Air has a much greater effect on the motion of the paper than it does on the motion of the baseball. The paper does not accelerate very long before air resistance reduces the acceleration so that it moves at an almost constant velocity. When an object is falling with a constant velocity, we prefer to use the term terminal velocity, or v T. The paper reaches terminal velocity very quickly, but on a short drop to the floor, the baseball does not. This takes Newton s second law one step further by considering two forces acting on the same object. Air resistance is sometimes referred to as a drag force. Experiments have been done with a variety of objects falling in air. These sometimes show that the drag force is proportional to the velocity and sometimes that the drag force is proportional to the square of the velocity. In either case, the direction of the drag force is opposite to the direction of motion. Mathematically, the drag force can be described using: Fdrag = bv or Fdrag = cv The constants b and c are called the drag coefficients that depend on the size and shape of the object. When falling, there are two forces acting on an object: the weight, mg, and air resistance, bv or cv. At terminal velocity, the downward force is equal to the upward force, so mg = bv or mg = cv, depending on whether the drag force follows the first or second relationship. In either case, since g and b or c are constants, the terminal velocity is affected by the mass of the object. Taking out the constants, this yields either v T m or v T m If we plot mass versus vt or vt, we can determine which relationship is more appropriate. In this experiment, you will measure terminal velocity as a function of mass for falling coffee filters and use the data to choose between the two models for the drag force. Coffee filters were chosen because they are light enough to reach terminal velocity in a short distance. You can also experiment with toy parachutes. Page 6 of 8
7 Making predictions - your hypothesis 1. Hold a single coffee filter in your hand. Release it and watch it fall to the ground. Next, nest two filters and release them. Did two filters fall faster, slower, or at the same rate as one filter? Explain your reasoning.. What kind of mathematical relationship do you predict will exist between the velocity of fall and the number of filters? 3. If there was no air resistance, how would the rate of fall of a coffee filter compare to the rate of fall of a baseball? Do heavier things really fall faster? What if the coffee filter weighed the same as the baseball? 4. When the filter reaches terminal velocity, what is the net force acting upon it? 5. Predict the sketch of a graph of the velocity vs. time for one falling coffee filter. Include answers to these questions and your predictions in your web site. 1. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface.. Support the Motion Detector about m above the floor, pointing down, as shown in Figure Open the file 13 Air Resistance from the Physics with Vernier folder. 4. Place a coffee filter in the palm of your hand and hold it about 0.5 m under the Motion Detector. Do not hold the filter closer than 0.15 m. 5. Click to begin data collection. When the Motion Detector begins to click, release the coffee filter directly below the Motion Detector so that it falls toward the floor. Move your hand out of the beam of the Motion Detector as quickly as possible so that only the motion of the filter is recorded on the graph. Figure 6. If the motion of the filter was too erratic to get a smooth graph, repeat the measurement. With practice, the filter will fall almost straight down with little sideways motion. 7. The velocity of the coffee filter can be determined from the slope of the position vs. time graph. At the start of the graph, there should be a region of increasing slope (increasing Page 7 of 8
8 velocity), and then it should become linear. Since the slope of this line is velocity, the linear portion indicates that the filter was falling with a constant or terminal velocity (v T ) during that time. Drag your mouse pointer to select the portion of the graph that appears the most linear. Determine the slope by clicking the Linear Fit button,. 8. Record the slope in the data table (a velocity in m/s). 9. Repeat Steps 4 8 for two, three, four, and five coffee filters nested together. Data Number of filters Terminal Velocity v T (m/s) (Terminal Velocity) v T (m /s ) Analysis Part 3 1. To help choose between the two models for the drag force (v T m or v T m ), use LoggerPro to plot terminal velocity v T vs. number of filters (mass). On a separate graph, plot v T vs. number of filters. Scale each axis from the origin (0,0). Page of the experiment file is already prepared for you. Include all graphs with this lab report.. During terminal velocity the drag force is equal to the weight (mg) of the filter. If the drag force is proportional to velocity, then v T m. Or, if the drag force is proportional to the square of velocity, then v T m. From your graphs, which proportionality is consistent with your data; that is, which graph is closer to a straight line that goes through the origin? 3. From the choice of proportionalities in the previous step, which of the drag force relationships (v T m or v T m ) appears to model the real data better? Notice that you are choosing between two different descriptions of air resistance one or both may not correspond to what you observed. 4. How does the time of fall relate to the weight (mg) of the coffee filters (drag force)? 6. Given the relationship you selected, if one filter falls in time, t, how long would it take four filters to fall, assuming the filters are always moving at terminal velocity? Page 8 of 8
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