Name: Partner s Name: EXPERIMENT 500-2 MOTION PLOTS & FREE FALL ACCELERATION APPARATUS Track and cart, pole and crossbar, large ball, motion detector, LabPro interface. Software: Logger Pro 3.4 INTRODUCTION In this lab we will explore the motion of objects, and how to efficiently describe this motion. Namely, we will learn how to create motion plots (position, velocity or acceleration as a function of time), and how to extract kinematic information from these diagrams. In short, we want to be able to cast the motion of any object in the real, physical world into a two-dimensional plot that captures the essential parameters of the motion. Of course, we can and will also do the opposite: translate the plot into a real, physical motion of an object. To display the motion plots quickly, we will use a motion detector connected to a computer. The motion detector will measure the position of moving objects as a function of time. From the position data, the computer will derive the values of the object s velocity and acceleration as a function of time. The motion detector uses sonar to measure the distance to an object. It emits a short burst of ultrasound and measures the time for echoes to return to it. Since the velocity of sound is known, the distance can be determined from the echo delay. This is the same principle that bats and submarines use in order to navigate. The average velocity of the object as a function of time can be determined from the position data using the definition, v average Δx = = Δt i+ ) x( ti ) This is the average velocity of the ball in the time interval from ti to ti+. Similarly, once the velocity as a function of time is known, the average acceleration of the ball in the time interval from ti to ti+ is defined to be The computer will perform these calculations for you. In practice, the computer also does some averaging of the data as well, to reduce the noise. x( t t i+ t i a average = Δv v( t = Δt t i+ ) v( ti ) i+ t i
SETUP. Plug in the LabPro interface, and connect the motion detector to the port Dig/Sonic. The LabPro translates the electrical signals from a variety of measuring devices into the digital language of the computer. Put the motion detector on the lab table facing the room. 2. Launch Logger Pro by clicking the icon on the desktop that shows a picture of a vernier caliper. Logger Pro collects, displays and analyzes data from the LabPro interface. Let s adjust some of the settings, to get a feeling for how Logger Pro works. a. Use a thermometer to determine the room temperature in centigrade. Then under Experiment go into Set up sensors, click on the motion detector and select set temperature. Enter the temperature in the appropriate field. Logger Pro needs to know this, since the speed of sound in air depends on temperature. Finally, click OK. b. Click on the button that looks like a clock. Make sure that the mode is set to time based. That tells Logger Pro that time is one of the variables in your experiment. Click on the tab marked Sampling. Set the experiment length to 5 seconds, and the sampling speed to 20 samples/second. In the future, you will use pre-programmed setup files, so you won t always have to go through this procedure; but now you should have a better idea of how Logger Pro works. 3. Make sure you have three motion plots on the screen: Position, Velocity, and Acceleration. (If the Acceleration plot isn t there, select the menu item Insert / Graph. Then hit CTRL+R to auto-arrange it.) 4. Check out the motion detector. Click the Collect button. When the motion detector starts to click, place an object (such as your hand) in the ultrasound beam to see if it gives sensible readings. You should get correct position readings for any object more than 40 cm away from the detector. The beam of ultrasound makes a cone of about 5 ; make sure the object whose position you are measuring is within this cone, and that there are no other objects in the cone that are closer to detector. The motion detector may also be confused by multiple echoes from hard surfaces such as the ceiling and floor. If you get nonsensical distance readings, make sure the motion detector is aimed in the right direction, and that there are no other objects in the way. The problem of echoes can be reduced by reducing the sampling speed, but you don t want to do that if you don t have to. Ask the lab instructor for help if you are unable to get sensible readings from the motion detector. 2
PART I MOTION PLOTS Motion plots are plots of position, velocity or acceleration as a function of time. They display the essential parameters of the motion of an object. In this lab, a motion detector plugged into a computer will create these plots for you. The horizontal axis exhibits the independent variable as usual, in this case time. The origin of the time axis, i.e. the moment when t = 0 s, corresponds to the moment when you pressed the Collect button to begin taking data. The vertical axis will represent either position or velocity of an object. Note that both quantities change in time, i.e. are functions of time, and therefore dependent variables. A. Position versus time Let s first explore position versus time motion plots. What does the computer display on the vertical axis? Position is spatial, i.e. somehow in the room. Where is the position coordinate axis in the space of the physics lab? We have to measure position relative to something; there has to be an origin and a notion of both positive and negative directions. Follow the procedure listed below to familiarize yourself with the motion sensor and answer the questions. Of course, you are expected to go beyond a simple "yes" or "no" response and provide justification for your answers.. Put the motion detector down on a table or counter so that you are not moving it. While recording data, have a member of your group walk toward and away from the detector. Do this multiple times if you need to. Where is the position axis origin located? Where are the positive and negative values of position located? 2. Next figure out how the graph tells you which way the person was moving and whether they were moving slowly or quickly. Record your answers here: 3
3. Do your answers change if the person stands still and you move the motion detector back and forth? If so, in what way or ways do you they change? Feel free to explore this question by playing with the detector. 4. Using the apparatus you have and just one person moving, can you create a position versus time graph that has the shape of a circle? Why or why not? Matching a position versus time plot with actual motion You should now be able to analyze a plot and write a description of motion that might have created the plot. Also, you should be able to move in such a way that you create the plot, which is what we ll do next. 5. Keeping the motion detector stationary, walk so that you match the graph below. 3 Position (m) 2 0 0 2 4 6 8 t (s) Write a description of how you had to move to create this graph. Include qualitative descriptions using words such as: toward, away, slowly, quickly, gradually, etc. 4
6. Keeping the motion detector stationary, walk so that you match the graph below. 3 Position (m) 2 0 0 2 3 4 5 6 7 t (s) Write a description of how you had to move to create this graph. Include qualitative descriptions using words such as: toward, away, slowly, quickly, gradually, etc. B. Velocity versus time Let s now explore velocity-versus-time motion plots. Velocity implies a vector quantity so it must convey both speed (magnitude) and direction. In one-dimensional motion the direction is indicated by assigning a positive or negative sign to the speed. This changes the scalar speed into a velocity vector. Keep this in mind as you work through the exercises that follow. 5
Matching a velocity versus time plot with actual motion 7. Walk so that you match the graph below. (You may have to adjust the scale on the velocity vs. time graph to make it look like this.) Velocity (m/s) 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-0 2 3 4 t (s) Write down a description of how you had to move to create this graph. Include qualitative descriptions. 6
8. Walk so that you match the graph below. 0.8 0.6 Velocity (m/s) 0.4 0.2 0-0.2-0.4-0.6-0.8-0 2 3 4 t (s) Write down a description of how you had to move to create this graph. Include qualitative descriptions using words such as: toward, away, slowly, quickly, gradually, etc. 7
Connecting position and velocity graphs By now you should have a good understanding of how a person's motion in front of the motion detector translates into a velocity versus time graph on the computer. We will now connect the position-time graph to its corresponding velocitytime graph. 9. Go back to the position-time graph of question 5. Draw a velocity-versus-time graph that matches this motion on the axes to the right. PART II CART ON TRACK AT CONSTANT ACCELERATION Now we will move to some actual physics, namely the motion of objects under the influence of gravity. We are still only describing motion, i.e. asking how something is moving, and not asking why it moves. That is, we are dealing with kinematics, not dynamics. Since we are not allowed to talk about forces yet, we can cast the influence of gravity entirely into a single, simple statement: objects under the influence of gravity move at constant acceleration. In free fall nothing is holding the objects back, so they are subject to the maximal acceleration, which is a = g = 9.8 m/s 2. This is a large acceleration, and accordingly, objects move very fast typically too fast for humans to pay attention to precisely how they fall. (Do they speed up? How?) Back in the day, Galileo didn t even have a watch at his disposal, so he slowed things down in an ingeniously simple way: he put objects on a very slightly inclined plane. This is basically what we did in the last activity, but this time we will use the encoder carts on a slanted track.. Change to the motion encoder: a. Disconnect the motion detector. b. Connect the track transponder to the Lab Pro instead. 8
c. Go to Experiment / Set up Sensors / Lab Pro:. Click on DIG/SONIC and select the Motion / Motion Encoder Cart d. Turn on the cart itself by pushing the button. It should shine blue when on. 2. Starting from the low end of the track, give the cart a push, so that after you let it go it rolls up, turns around roughly in the middle of the track, and rolls back. 3. Print out the three motion diagrams and staple these motion diagrams to the back of your lab. Describe how they look, focusing on the interesting part, i.e. the time when the cart was rolling without being touched. 4. Determine the slope of the velocity plot with the method we learned in the previous lab. What is its connection to the value of the (constant) acceleration plot? 5. Make the incline steeper by raising the high end, i.e. increase the angle of inclination. Push the cart so it goes half-way up again and produce a new set of motion diagrams. Staple this diagram to the back of your manual as well. Please label each diagram (i.e. steep slope and gradual slope ). Describe what changes and what does not (shapes, maxima, slopes, intercepts, etc.) 6. Predict what is going to happen if we increase the incline indefinitely, i.e. the inclination angle become 90 degrees. 9
PART III FREE FALL ACCELERATION In this part you will measure the gravitational acceleration at the earth s surface by observing the acceleration of a freely falling object (in this case, a ball). You should find that in free fall, the ball experiences a constant acceleration close to 9.8 m/s 2, the average value of gravitational acceleration at the earth s surface. Set up the motion detector as displayed on the right. Go through the instructions above, this time reconnecting the motion sensor, and setting the device to Motion Sensor in the Lab Pro setup. DATA. Practice dropping the ball directly below the sensor and letting it bounce. Remember that the ball must remain within the cone of ultrasound, and not come closer to the detector than 40 cm on its way up from the rebound. If you drop from just in front of the sensor, this should not be hard. You are interested in the period of time from just after it has bounced until it is about to hit the ground again, or the period of free fall from going up to going down. 2. Before collecting data, make a prediction: draw graphs of what you think the position, velocity, and acceleration of the ball should look like during a single period of free fall (one single bounce). 0
3. Click the Collect button. Toss the ball once you hear the motion detector. Logger Pro will plot the position, velocity and acceleration of your ball as a function of time. 4. Your graphs will probably be rather complex. Rescale the axes so that you can see the numerical values well and so that only one bounce is shown. Print the graphs, all on one page and staple them to the back of this manual. Label this sheet as freefall plots. Identify the parts of your graph where the ball was in free fall and moving upward, and the parts where the ball was in free fall and moving downward. Label them on your graphs. In the analysis section, you will focus on the free fall portion of your graphs. (If there is no part of your graph that looks like what you expect from free fall motion, don t print them; go back and repeat the ball drop.) GRAPHS AND ANALYSIS. Use the Examine button to answer the following questions. a. On the velocity vs. time graph, decide where the ball had its maximum velocity, just after the ball bounced. Mark the spot and record the value on the graph. b. On the distance vs. time graph, locate the maxiumum height of the ball during free fall. Mark the spot and record the value on the graph. c. What was the velocity of the ball at the top of its motion? Why does this make sense? d. What was the acceleration of the ball at the top of its motion? Why does this make sense? 2. The graph of acceleration vs. time should be more or less constant. Do your results confirm that bodies near the Earth s surface fall with constant acceleration?
3. Click and drag the mouse across the free fall portion of the acceleration graph and click the Statistics button. The average is your experimental estimate of g, and the standard deviation your estimate of the uncertainty. Record these values below. Does your value match the published value of g=9.8 m/s 2 to within your uncertainty? 4. The graph of velocity vs. time should be linear, v=v0 gt. To fit a line to this data, click and drag the mouse across the free fall region and click the Analyze/Linear Fit menu item. Record the coefficient of the x term. How closely does the coefficient of the x term compare to the published value of g? 5. The position of an object in free fall is modelled y= y0+v0t /2gt 2, so your graph of position vs. time should be a parabola. To fit a quadratic equation to your data, select the free fall region of your graph and click the Curve Fit button. Select Quadratic from the list of models and click Try Fit. Examine the fit of the curve to your data, and click OK to return to the main graph. What do you expect the coefficient of the x 2 term to be? How does it compare to what you expect? 6. Determine the consistency of your acceleration values and compare your measurement of g to the published value. Do this by repeating the ball drop experiment four more times. Each time, fit a straight line to the free fall portion of the velocity graph and record the slope of that line. Calculate the average and standard deviation of your measurement of g. Does the variation in your five measurements explain any discrepancy between your average value and the published value? Put the results of your measurements in the table and show your calculations in the space below. 2
Drop Acceleration (m/s 2 ) 2 3 4 5 ANALYSIS. List some reasons why your values for the ball s acceleration may be different from the published value of g. 2. Would you expect to get a different value for g if you used a ball with a different mass? Under what circumstances? (Try dropping two objects of different mass, and see which one hits the ground first.) 3. A horizontal line on a position-time graph implies what about the corresponding segment on a velocity-time graph? 3
4. A horizontal line on a velocity-time graph implies what about the corresponding segment on a position-time graph? 5. A line segment on a position-time graph that is not horizontal implies what about the corresponding segment on a velocity-time graph? 6. A line segment on a velocity-time graph that is not horizontal and which crosses the time axis implies that something very specific has happened at the moment of crossing. What is it? 4