Measuring Impulse and Momentum Change in 1 Dimension

Transcription

1 Measuring Impulse and Momentum Change in Dimension PY205m Purpose In this lab you will: use a computer interface to collect and display data, simultaneously use a motion sensor and a force sensor, and use your measurements to relate impulse to change in momentum. Some useful equations and constants are given at the end of this document. 2 Experimental Setup Equipment: track, cart, fan, motion sensor with stand, force sensor with soft spring, Science Workshop interface, and balance. 2. Equipment setup A motion sensor should be placed at one end of the track, and connected to Digital Channels and 2 of the interface (colored plug in channel ). A force sensor, mounted on a bracket, should be firmly attached to the other end of the track, and connected to Analog Channel A of the interface. From the WebAssign, right click on the DataStudio file impulse.ds, and save it to My Documents. Double-click the file to open it in DataStudio. You should see a display that includes a single window containing two empty graphs, the top one with y-axis labeled Force and the lower one with y-axis labeled velocity. (Note that these labels should really be F x and v x!) 2.2 Force sensor setup Different objects (springs, hooks, rubber disks) can be attached to the force sensor. For the first part of this experiment, make sure the soft spring is attached to the sensor. (There are two springs, one stiff and the other soft.) If a different object is attached, take it off and put it into an empty slot in the mounting bracket, then attach the spring to the force sensor.

2 To see how the equipment works,. Click Start in DataStudio, then press gently on the spring on the force sensor. Try compressing the spring slowly and rapidly, to see what kind of graph is produced. 2. If the force does not read zero when the spring is not compressed, press the tare button by inserting the screwdriver through the hole in the mounting bracket: tare button. Roll the cart gently so it bounces off the spring on the force sensor; find a speed that produces a clearly visible signal, but does not flatten out at the top (this indicates a force too large to measure with these settings). 2. Ensure you understand the graphs of F x vs t and v x vs t that are produced simultaneously. Look at your graphs, and make sure you can answer the following questions. If necessary, do further experiments to make sure you understand. Write the answers in your notes.. What location in space is considered to be the origin? 2. In which direction (in your actual experimental setup) does the positive x-axis run? 3. When vx is positive, is the cart traveling toward the sensor or away from the sensor? 4. Does the graph of F x vs t show the force on the sensor due to the cart, or the force on the cart due to the sensor? (Think about the sign of F x ). 5. How can you expand the x-axis of the graph? 6. How can you move the graphs to center the section of interest in your field of view? When you understand how the setup works, choose Delete ALL data runs from the Experiment menu. 2.3 Note about reciprocity of forces The interaction between the cart and the force sensor is due to electric interactions between the protons and electrons in the cart and those in the sensor. It is a property of the electric interaction that the electric forces exerted by two objects on each other are equal and opposite. Therefore, the graph of F x on the cart versus t would be the same as the one you see, but the values of F x would be negative. 3 Simultaneous measurement of F x and p x Use your equipment to measure F x vs t and v x vs t while the cart rolls toward the force sensor, collides, and bounces off, rolling backward toward your hand. Do not let the cart hit the motion sensor. CHECKPOINT: Compare your graph with another group to make sure it is reasonable. Do the following analyses on a whiteboard:. From your graph (and any other necessary data), determine the change in the x-component of the cart s momentum from just before the collision starts to just after it ends. 2

3 2. Record the peak value of Fx during this interval. 3. From your graphs, estimate the average value of Fx during this interval. 4. From the graphs, estimate the impulse FxDt applied by the sensor on the cart during collision (with correct sign). 5. How does your calculated value of Dpx compare to your estimated value of FxDt? 6. How would you expect these values to compare? There is a more accurate way to determine the actual impulse on the cart. In the first short time interval t, the spring is only slightly compressed, and the force F x on the cart is small. The small impulse F x t makes a small change p x in the momentum. But F x t can be thought of as the area of a rectangle shown on the diagram, whose base is t and height is F x. So the area of the rectangle is equal to the impulse during t and also equal to the change in momentum p x during that short time interval. In the next time interval t 2, we can again represent the impulse (and the change in momentum) as the area of the next rectangle shown on the diagram. We can continue through the entire collision with the spring, and we see that the total area under the curve is equal to the total impulse (and the total change in the momentum, which is the sum of all the changes to the momentum). We re approximating the area under the curve by a bunch of rectangles, but if the little t s are small enough that the force isn t changing much during that short time interval, the total area of our rectangles is approximately equal to the area under the curve. This is an example of an integral, which can often be thought of as the area under a curve. More generally, an integral is the sum of a large (infinite) number of very small (infinitesimal) quantities. The integral of impulse is written F x dt, where the integral sign is a distorted S meaning sum and the dt stands for extremely small (infinitesimal) time interval. F x4 F x5 F x3 F x6 F x2 F x7 p x3 p x4 p x5 p x6 F x p x p x2 p x7 p x8 Fx8 t t 2 t 3 t 4 t 5 t 6 t 7 t 8 DataStudio can calculate the area under thef x vs t curve for you.. Select the force graph. 2. Using the cursor, draw a box around the entire peak on the F x vs t curve, to select it (only the selected points will change color). 3. Click on the S menu above the graph, and choose Area. 4. Save your graph including the area calculation so that you can submit it to WebAssign: (a) On the Display menu, choose Export Picture. (b) Save as a.bmp file to My Documents (which is your K drive). This is a very large file. (c) Right-click the.bmp file and open with Paint. (d) Save as a.jpg file. This is a manageably small file. 3

4 4 Important questions to answer: How does the x component of the net impulse found using F x dt compare to the x component of the change in the cart s momentum? (Remember that the actual value of the impulse applied to the cart is negative). When F x is the biggest it ever gets, what is p x? Is p x also at a maximum? Is p x proportional to F x? Relate your answers to the Momentum Principle. Save your graphs as a jpeg, as explained above CHECKPOINT: Compare your graph, data, and analysis with another group. Then have your instructor check both groups work. Enter your data and answer the follow-up questions for this section in WebAssign. August 5,

5 Things you must know: Definition of average velocity Definition of momentum Definitions of particle energy, kinetic energy, and work The Momentum Principle The Energy Principle Other fundamental principles: L dl A A = τ net,a t or = τ net,a dt L trans,a = r A p (point particle) τ A = r A F r cm = m r + m 2 r m + m Multiparticle systems: P tot M tot v cm (v << c) K tot = K trans + K rel K rel = K rot + K vib K trans 2 M totv 2 cm (v << c) I = m r 2 + m 2r K rot = L2 rot 2I = 2 Iω2 L trans,a = r cm,a P tot Lrot = I ω LA = L trans,a + L rot γ Other physical quantities: ( ) E 2 (pc) 2 = ( mc 2) 2 v 2 c F grav = G m m 2 r 2 ˆr U grav = G m m 2 r F grav mg near Earth s surface Ugrav mg y near Earth s surface F elec = q q 2 4πǫ 0 r 2 ˆr U elec = q q 2 4πǫ 0 r F spring = ks s opposite to the stretch U spring = 2 k ss 2 for ideal spring U i 2 k sis 2 E M approx. interatomic pot. energy E thermal = mc T E N = 3.6eV N 2 where N =,2,3... (Hydrogen atom energy levels) ksi E N = N ω 0 + E 0 where N = 0,,2... and ω 0 = (Quantized oscillator energy levels) m a ( ) d p dt = pv R mv2 (if v << c) where R = radius of kissing circle R ω = 2π T Y = F/A L/L (macro) x = Acos ωt ω = Y = k si d (micro) ks m speed of sound v = d ksi m a 5

6 ˆf = cos θ x,cos θ y,cos θ z unit vector from angles Moment of intertia for rotation about indicated axis L L R I = 2 5 MR2 I = 2 MR2 I = 2 ML2 I = 2 ML2 + 4 MR2 R Ω = (q + N )! q!(n )! S k ln Ω T S E S = Q T (small Q) prob(e) Ω (E) e E kt Constant Symbol Approximate Value Speed of light c m/s Gravitational constant G N m 2 /kg 2 Approx. grav field near Earth s surface g 9.8 N/kg Electron mass m e kg Proton mass m p kg Neutron mass m n kg Electric constant 4πǫ N m 2 /C 2 Proton charge e C Electron volt ev J Avogadro s number N A atoms/mol Plank s constant h joule second hbar = h joule second 2π specific heat capacity of water C 4.2 J/kg Boltzmann constant k J/K milli m 0 3 kilo K 0 3 micro µ 0 6 mega M 0 6 nano n 0 9 giga G 0 9 pico p 0 2 tera T 0 2 6