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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

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.)
       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).
           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

   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.]

         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
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

                              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)
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)
       x-value               x
       (in cm)              x
       y-value               y
       (in cm)              y
Time-of-flight for flag
  to pass through
     photogate:              3
   individual trials
     t (in sec)
Average time of flight      t 
      (in sec)
                            t 
        Speed                v
       (in m/s)
  Spring potential         U spr
       (in J)
                            U spr
    Gravitational          U grav
  potential energy
       (in J)
                          U grav
   Kinetic energy            K
       (in J)
  Total mechanical           E
        (in J)