Two-Dimensional Motion and Conservation of Energy

As an object slides down an incline, it loses gravitational potential energy
DU = m g Dy,
where Dy is the change in the y coordinate as the object moves. That energy accompanies an increase in kinetic energy
DK = m ( vB2 - vA2) / 2,
where vA is the object's velocity at the greater height and vB is its velocity at the lower height. If there is no energy lost to friction, we say that energy is conserved, and
DU = DK,
which can also be expressed as
g = ( vB2 - vA2) / ( 2 Dy )
giving us a new method for measuring g.

Additionally, if the angle of the incline is q, the acceleration of the object is

a = g sin q.
Note that the title of this lab is actually a cheat. Motion down an incline, while described in terms of x and y coordinates, is actually constrained to one dimension: we could describe it simply in terms of a distance along the incline.

We will use an inclined air track, which reduces friction to a minimum, two photogates, a counter/timer, a meter stick, a micrometer and a balance. The results you obtain from this experiment are highly dependent on the accuracy of your measurements, so be careful!

Name:

Lab Partners:

When entering numeric data, use exponentials: ie., 1.6 * 10-19 = 1.6E-19.

Procedure

  1. Before elevating the track, make sure it is level. Do this by placing a cart on the track in the middle, turning the air supply on, and adjusting the feet (evenly on each side of the track) so that the cart does not move toward either end.

  2. Use the micrometer to measure the thickness of the block used to elevate the track:
    th = cm

  3. Use the meter stick to measure the distance between the feet on the air track:
    xfeet = cm

  4. Place a photogate over the level track and measure the effective length of the cart. This is the length the cart moves while the photogate is active. To do this, plug the photogate into the counter/timer, turn the counter/timer on, turn the air supply on, and slowly move the "floating" cart past the photogate until the light turns on. Note the position of the edge of the cart:
    x1 = cm

    Then move the cart until it turns off, and note the new position of the same edge:
    x2 = cm

    The difference of these positions is the effective length:
    leff = x2 - x1 = cm

  5. Measure the mass of the cart:
    m = g

  6. Elevate the air track at the end with the single foot, and place the photogates EXACTLY 50 cm apart and at the appropriate heights to be triggered by the cart as it rides down the track. Set the counter/timer to "split timers", "gate mode" and turn off "input hold" and "memory". Plug the timer at the high end of the track into the jack marked "A", and the timer for the low end into jack "B".

    Record the times for gates A and B for each of ten trials. Be sure to reset the timer before each trial, and be sure to release the cart from the very end of the track on each trial. To do this reliably, hold the cart against the end stop with the eraser end of a pencil, and release the cart by quickly moving the eraser away from and above the cart. Practice for a few trials before recording data will improve your consistency. Record all digits in the timer display.

    tA,1 = s tB,1 = s
    tA,2 = s tB,2 = s
    tA,3 = s tB,3 = s
    tA,4 = s tB,4 = s
    tA,5 = s tB,5 = s
    tA,6 = s tB,6 = s
    tA,7 = s tB,7 = s
    tA,8 = s tB,8 = s
    tA,9 = s tB,9 = s
    tA,10 = s tB,10 = s

  7. Now set the counter/timer to "timer", "pulse mode", and perform ten more trials as above. The time measured is now the elapsed time as the cart travels the 50 cm from one photogate to the other. You may round the timer display to four digits to the right of the decimal point.

    t1 = s t6 = s
    t2 = s t7 = s
    t3 = s t8 = s
    t4 = s t9 = s
    t5 = s t10 = s

Analysis

  1. Compute the angle of elevation using the fact that when elevated, the distance between the track feet is the hypotenuse of the triangle whose side opposite the angle of elevation is the thickness of the elevating block. Do not round this number!
    q = sin-1 ( th / xfeet )

    = degrees

  2. Compute Dy = 50 sin q
    = cm
    and DU = m g Dy
    = ergs
    using 981 cm / s2 for g. An erg is a unit of energy, equal to one g cm2 / s2.

  3. For each of the ten trials compute vX,i = leff / tX,i:
    vA,1 = cm / s vB,1 = cm / s
    vA,2 = cm / s vB,2 = cm / s
    vA,3 = cm / s vB,3 = cm / s
    vA,4 = cm / s vB,4 = cm / s
    vA,5 = cm / s vB,5 = cm / s
    vA,6 = cm / s vB,6 = cm / s
    vA,7 = cm / s vB,7 = cm / s
    vA,8 = cm / s vB,8 = cm / s
    vA,9 = cm / s vB,9 = cm / s
    vA,10 = cm / s vB,10 = cm / s

  4. For each of the ten trials compute KX,i = m vX,i2 / 2:
    KA,1 = ergs KB,1 = ergs
    KA,2 = ergs KB,2 = ergs
    KA,3 = ergs KB,3 = ergs
    KA,4 = ergs KB,4 = ergs
    KA,5 = ergs KB,5 = ergs
    KA,6 = ergs KB,6 = ergs
    KA,7 = ergs KB,7 = ergs
    KA,8 = ergs KB,8 = ergs
    KA,9 = ergs KB,9 = ergs
    KA,10 = ergs KB,10 = ergs

  5. Compute DKi = KB,i - KA,i for each trial:
    DK1 = ergs DK6 = ergs
    DK2 = ergs DK7 = ergs
    DK3 = ergs DK8 = ergs
    DK4 = ergs DK9 = ergs
    DK5 = ergs DK10 = ergs

  6. Compute the average of the DKi:
    DKavg = ergs
    and the absolute error in the DKi:
    DKabs = ergs
    Compare the average change in kinetic energy with the change in potential energy; was energy conserved? Why or why not?

  7. Compute the average speed through each photogate from the first ten trials:
    vA,avg = cm / s

    vB,avg = cm / s

    and the average elapsed time from the second ten trials:
    tavg = s

  8. Compute the average acceleration during this experiment:
    aavg = ( vB,avg - vA,avg ) / tavg

    = cm / s2

    and compute gavg = aavg / sin q
    = cm / s2

  9. Using conservation of energy, compute g
    = cm / s2

  10. In order to compare g with gavg, we need to compute the experimental error in gavg. The relative error in gavg is equal to the RMS (Root Mean Square) error
    DgRMS = ( Dvrel2 + trel2 )1 / 2
    where Dvrel is the relative error in the Dvi and trel is the relative error in the ti.

    Compute the change in velocity Dvi = vB,i - vA,i: for the first ten trials:

    Dv1 = cm / s Dv6 = cm / s
    Dv2 = cm / s Dv7 = cm / s
    Dv3 = cm / s Dv8 = cm / s
    Dv4 = cm / s Dv9 = cm / s
    Dv5 = cm / s Dv10 = cm / s

  11. Compute the average of the Dvi:
    Dvavg = cm / s
    their absolute error:
    Dvabs = cm / s
    Dvrel = Dvabs / Dvavg * 100 %
    = %
    the absolute error in the ti:
    tabs = s
    and trel = tabs / tavg * 100 %
    = %

  12. Compute the absolute RMS error in gavg:
    DgRMS * gavg / 100 = cm / s2
    Are the two values g and gavg consistent? Why or why not?

©2004, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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