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PHYS 201 Version 3/3/10 Physics I: Mechanics, Wave Motion, & Sound pg. 1 of 5 Lab 6. Conservation of Energy Purpose: The purpose of this lab is to investigate conservation of total mechanical energy ( E 1 mv 2 mgy 1 kx2 ) as a mass moves under the influence of two conservative forces, 2 2 gravity and a spring force. Equipment: Large spiral spring Vertical table clamp 2 cross clamps (ring-stand clamps) 1 long, 1 short support rods Meter-stick clamp Two-meter stick Electronic mass balance Photogate electronic timer Ruler Tape Scissors 3 x 5 index card One 500 g and seven 100 g slotted weights Mass hanger Procedure: I. Determination of the Spring Constant 1. Measure and record the mass m s of your spring, as at the top of Table 1 below. (If you hook the spring’s two ends together, it will form a ring which fits conveniently on the balance pan.) As in the past, assume that the uncertainty in a mass value m as measured by the electronic balance is m 0.001m 0.01 grams . 2. Clamp the long rod vertically at the edge of the lab table, then clamp the two-meter stick to it in a vertical position. Arrange the cross clamps and the short rod so as to suspend the spiral spring (large end down) as near as possible to the two-meter stick without touching it. (You can hook the spring directly onto one of the screws of a cross-clamp.) Note: It is important to read through the entire Procedure at this point, and test your set-up to find where the lowest and highest positions of the mass hanging from the spring will be (see step 3, and Fig. 1). You need to mount the ruler so that you can complete the entire experiment without moving it! Also, the “gravitational reference position” defined in step 5 and Fig. 1(b) should be near the edge of the table, for convenience in positioning the photogate for the speed measurement of step 8. 3. Hang the 50 g mass hanger from the spring and record the ruler mark r r at the bottom of the hanger, as in Table 1 below. Add more mass to the hanger, in 100 g increments, each time recording the ruler mark at the bottom of the hanger. [Do not exceed a total of 750 g.] From this table of measurements, estimate the spring constant k . (You will make a more precise calculation of k in the Data Analysis section below.) II. Determination of Total System Energy at Various Positions 4. Now place only a single 500 g mass on the hanger (taping it in place so it won’t fall off). Also attach to the side of this mass a strip about 2 cm wide, cut from a 3 x 5 card, to serve as a “flag.” You will measure the time it takes this flag to pass through a photogate, to determine the speed of the oscillating mass (in step 8 below). Consequently, the flag’s width should be well-defined (measure it carefully): we want it to be not too wide, but wide enough that the time it takes to pass through a photogate will be significantly longer than the photogate’s timing resolution (which is 0.1 ms at best). Record the total mass M M hung from the spring, and the width l l of the flag, as at the top of Table 2 below. (As in previous labs, we will assume that our labeled weights have a relative uncertainty of 0.2%.) PHYS 201 Version 3/3/10 Lab 6 pg. 2 of 5 y y y y 0 r0 0 0 0 0 x3 y3 r3 x1 0 r1 0 0 x1 0 r1 (v = 0) x2 y2 r2 (v > 0) x x x x x (a) (b) (c) (d) (e) Zero-spring-force Gravitational reference Lowest point Highest point Gravitational reference position position of oscillation of oscillation position (at rest) (in motion) Lab 6 Fig. 1 5. With this mass hanging from the spring at rest, record (in Table 2) the ruler mark r1 r1 at the new equilibrium position of the bottom of the hanger, and call this the “gravitational reference” position: i.e., the position “ y 0 ” where gravitational potential energy U grav ( y ) m tot gy 0 [see Fig. 1(b)]. At this position, with the mass at rest, the total energy is then just equal to the spring potential energy, which is not zero because the spring is stretched. (The actual amount of spring stretch ( x1 ) at this gravitational reference position, and the corresponding amount of spring potential energy, will be calculated in the Data Analysis section below.) 6. Now pull the mass straight down an additional 30 cm or so. The precise amount of additional displacement is not important, but you must accurately record and be able to reproduce the ruler mark r2 r2 of the bottom of the mass hanger at this lowest point of oscillation [see Fig. 1(c) and Table 2]. 7. Release the mass from rest at its lowest point, and note the ruler mark at the highest point of oscillation, which the bottom of the mass hanger reaches on its first oscillation cycle. (In subsequent cycles, friction already begins to degrade the motion, making measurements inaccurate.) This measurement of a position “on the fly” is not easy, but repeat it until you can confidently record the result r3 r3 (in Table 2) to at least the nearest 0.5 cm! 8. Now situate the photogate electronic timer at the gravitational reference position. Using the timer in its Gate mode, on the 0.1 ms scale, again release the mass from rest at its lowest point, and time how long it takes the flag to make its first pass through the photogate. Make several trials to generate good statistics: record the time t for each trial as in Table 2. Data Analysis: 1. Complete Table 1 by computing, for each amount of mass hung from the spring, the total force F applied to the spring (i.e., the weight in N). Note: The total hanging mass ( m tot ) must include one-third the mass of the spring itself, added to the mass hung from the spring. PHYS 201 Version 3/3/10 Lab 6 pg. 3 of 5 Now use your TI-92 to fit a line to the graph of total force F (in N) vs. ruler mark r (in m): the slope of this line will be the spring constant k. Additionally, extrapolate this line to locate the ruler mark r0 at which the force on the spring would be zero. (For the uncertainty r0 , just use the same uncertainty that you assigned to each of your ruler mark measurements in Table 1.) By assigning this “zero-spring-force” position the coordinate “ x 0 ” [see Fig. 1(a)], we may write the magnitude of the spring force and the spring potential energy in their customary simple forms Fspr kx and U spr ( x) 1 kx2 . [Question: Why could you not find 2 this position simply by noting where the bottom end of the spring would hang if there were nothing hanging from it?] Record in your lab notebook the results of this linear fit. 2. Another way to estimate the spring constant k is simply to look at the ratio F / r as each successive mass is added to the hanger. Complete Table 1 by computing this ratio for each row after the first, and then find their statistical average value k k . Does this value of k agree with the one you found from the linear fit to the data, to within the statistical uncertainty k ? 3. Now note that we have three related coordinate systems for describing each position, as shown in Fig. 1: i. the ruler mark r which you recorded for each of the positions in Table 2; ii. a corresponding value x that tells the amount of spring stretch at that position (relative to the zero-spring-force position r r0 ); iii. a value y that tells the height of that position relative to the gravitational reference position r r1 . For each position recorded in Table 2, compute its corresponding x-value and y-value, and thus compute the spring and gravitational potential energies, U spr and U grav , with their uncertainties, according to the appropriate equations (which you should record in your notebook). [Note: For k k , use the statistical average value obtained in Step 2 above.] Reminder: The total hanging mass ( m tot ) must include one-third the mass of the spring itself, added to the mass hung from the spring. (Their uncertainties similarly add, to produce mtot .) 4. Compute the average t t of the times-of-flight of the flag through the photogate, which you tabulated in step 8 of the Procedure. (Keep in mind, however, that no matter how accurate your statistics may seem to be, t cannot be less than the a priori uncertainty of the photogate timer! Why not?) Using this average time of flight, determine the speed of the mass at the gravitational reference position [see Fig. 1(e)], and record the result in Table 2. From that, compute the kinetic energy at this position, and its uncertainty. (Again, record the equations you use.) 5. You can now complete Table 2 by writing down the total mechanical energy at each position. Check your calculations by entering all your data and calculated values into the spreadsheet “Lab6Data” found on the “shares” drive in the folder \szpilka\P201Labs. If all your calculations have been performed correctly, each of them should receive an “OK” from the spreadsheet! Print a copy of your completed spreadsheet and attach it in your lab notebook. PHYS 201 Version 3/3/10 Lab 6 pg. 4 of 5 Discussion and Conclusions: Is the total mechanical energy in each of the four columns of Table 2 the same? Should it be? Did your measurements and calculations demonstrate conservation of energy to within the experimental uncertainties, or did they point to the presence of some other significant effect or systematic error which we did not address here? If the latter, can you suggest what might have been overlooked? How do the individual energies (spring potential, gravitational potential, and kinetic) change as the mass moves from one position to another? How do these changes conserve total mechanical energy? Tables of Measured and Computed Data Table 1: Spring mass m s m s (in grams) = Mass hung from Total hanging Total force on Estimated spring spring, Ruler mark, mass, spring, constant, m (in grams) r r (in cm) m tot (in grams) F (in N) F / r (in N/m) 50 150 250 350 450 550 650 750 PHYS 201 Version 3/3/10 Lab 6 pg. 5 of 5 Table 2: Total mass hung from spring in Part II, M M (in grams) = (This value does not yet include any contribution from the mass of the spring itself.) Width of flag, l l (in cm) = Gravitational Lowest point Highest point Gravitational ref. position of oscillation of oscillation ref. position (position 1) (position 2) (position 3) (position 1) (at rest) (in motion) Ruler mark r (in cm) r x-value x (in cm) x y-value y (in cm) y 1 Time-of-flight for flag 2 to pass through photogate: 3 individual trials 4 t (in sec) 5 Average time of flight t (in sec) t Speed v (in m/s) v Spring potential U spr energy (in J) U spr Gravitational U grav potential energy (in J) U grav Kinetic energy K (in J) K Total mechanical E energy E (in J)

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