Jason Fong
702847140
date of experiment: 10-17-00
partner: Rosanne Hong
Physics 4AL Lab 5
Experiment 2
Abstract / Intro
The purpose of this experiment was to measure the value of g, which is the acceleration
due to gravity. The acceleration was measured by dropping a ball through two photogates onto
an impact pad. Measurements were taken for the time to travel between the two consecutive
intervals of the ball's fall: the first interval being between the photogates, and the second being
from the lower photogate to the impact pad. The distance of the intervals were measured so that
the velocity at the end of each interval could be determined. The two velocities could then be
used to calculate the value of the acceleration g. Five runs using five different distances were
used in the experiment. The values obtained for g were smaller than the expected 9.85 m/s2 for
all of the runs, with the longer distances giving more accurate answers.
Formulas
In order to make the calculation for the value of g, the equation for g needed to be
derived from the kinematics equations of the ball's fall through each of the two intervals. The
coordinate system was set up so that the y axis is along the path of the ball's drop. The bottom of
the ball determines the ball's y-position since the bottom is the first part to cross the beams of the
photogates and it is also the first part to strike the impact pad. The position of the top
photogate's beam is taken to be y=0 since timing will begin after the ball crosses that beam. The
lower photogate's beam marks the end of the first interval, and the distance between the top and
lower photogates' beams is denoted by d. The distance from the lower photogate's beam to the
impact pad is denoted D. The time to travel across the first interval (between the photogates) is
denoted by t1 and the time to travel across the second interval (from the lower photogate to the
impact pad) is denoted by t2. The formula for the value of g is given by the formula:
An attached page contains the derivation of the above formula from the kinematics equations for
the ball's fall through the two intervals.
Procedure
The experiment was setup by placing two photogates directly above the transducer of an
impact pad. The photogates were separated by a distance of 0.0755 ± 0.0005 m. For each of the
runs, the pair of photogates were a different distance away from the impact pad: 0.0505 ±
0.0005 for the first, 0.1015 ± 0.0005 m for the second, 0.2000 ± 0.0005 m for the third, 0.3015 ±
0.0005 m for the fourth, and 0.4005 ± 0.0005 m for the last run. The uncertainty is based on the
fact that the ruler was marked down to thousandths of a meter (tenths of a centimeter), so
measurements could be taken to an accuracy of half a tick mark, or ±0.0005 m. A ball was then
dropped through the two photogates and directly onto the transducer of the impact pad. It was
important for the ball to strike the transducer directly since there would be a slight delay for the
impact to propagate through the pad to the transducer if the ball struck somewhere other than
directly on top of the transducer. The two photogates and the impact pad were plugged into a
Pasco Science Workshop (PSW) box so that it could take readings to pass along to a computer.
The computer was setup to record the time taken for the ball to travel between the photogates
and the time taken for the ball to travel from the lower photogate to the impact pad. The value of
gravity was then calculated using those times and the measured distances of the intervals.
Data Analysis / Error Analysis
For each of the different drop distances, the average over the ten drops was taken for the
value of g and its error. The values thus calculated are summarized in the table below:
Distance D (m)
Gravity g (m/s2)
0.0505 ± 0.0005 m 6.4933 ± 0.1087 m/s2
0.1015 ± 0.0005 m 8.3380 ± 0.0222 m/s2
0.2000 ± 0.0005 m 9.0398 ± 0.0126 m/s2
0.3015 ± 0.0005 m 9.3076 ± 0.0201 m/s2
0.4005 ± 0.0005 m 9.4280 ± 0.0221 m/s2
The average value of g for each run was plotted against the distance of the drop, and the resulting
chart is included on an attached page. The error bars on each point is based on the average value
of the error calculated for each value of g. It would not make sense to calculate an average value
for g based on all of the values for all of the drops since the longer distances gave more accurate
results. It would make more sense to use only the longer distances for finding a value for g.
The first distance gave a result that was very far off from the correct value of 9.85 m/s2,
but the next distance gave a much better result. Each of the longer distances gave a closer result,
but the difference in accuracy was to a lesser degree than the differences in the shorter distances.
Based on this observation, and on the graph of the results, it appears that the accuracy of the
measurement of g and the drop distance have a logarithmic relationship. This is because the
accuracy improves very quickly for small distances, and improves more and more slowly as the
distance increases. The source of these errors could have come from many different sources, but
it seems that the most likely sources will be the ones that have a greater effect on the shorter
distances.
One such source of error is in the timer of the PSW box. The timer has a resolution of
about 0.1 milliseconds, so it cannot measure to any greater accuracy. In the equation for g, it can
be seen that errors in time will affect the value of g quadratically since the times are multiplied in
the equation. The following table1 summarizes the effects that a 0.1 millisecond (0.0001 s)
difference in the time can have on the calculated value of g:
Gravity g (m/s2)
Distance D
% Change in g
(m)
original
t1 + .0001 s
t2 + .0001 s
t1 + .0001s
t2 + .0001 s
0.0505
6.5909
6.6007
6.5356
0.1487%
0.8390%
0.1015
8.3928
8.3991
8.3598
0.0751%
0.3932%
0.2000
9.0202
9.0252
9.0004
0.0554%
0.2195%
0.3015
9.3000
9.3038
9.2856
0.0409%
0.1548%
0.4005
9.4289
9.4327
9.4169
0.0403%
0.1273%
The chart shows that for increasing distances, the changed times cause a smaller percent
difference in the calculated values of g. But the longer the distance, the slower the percent
difference approaches zero. This is also reflected in the scatter plot graph because the calculated
1
The values in the table are based on the first drop for each distance used. Thus, the numbers do not match up with
the first table giving the average value of g over all of the drops for each distance. Errors are not given since this
table is only intended to be a rough estimate of the effects of changes in time.
value of g is more accurate for larger distances, but the increased accuracy slows as the distance
is increased.
Another source of systematic error in the timer is that if it runs slow or fast, it will affect
the calculated value of g. If the timer is systematically wrong so that it ran fast by 0.1%, it
would introduce the following errors summarized in this table2:
Gravity g (m/s2)
Distance D
% Change in g
(m)
original
t1 + 0.1%
t2 + 0.1%
t1 + 0.1%
t2 + 0.1%
0.0505
6.5909
6.5996
6.5689
0.1320%
0.3338%
0.1015
8.3928
8.3985
8.3704
0.0679%
0.2669%
0.2000
9.0202
9.0246
8.9978
0.0488%
0.2483%
0.3015
9.3000
9.3034
9.2779
0.0366%
0.2376%
0.4005
9.4289
9.4322
9.4067
0.0350%
0.2354%
This table also shows the same pattern seen in the previous table and in the graph. The accuracy
increases with increasing distances, but the rate of improved accuracy slows with increasing
distances. This seems to indicate that the timer was not properly calibrated since the pattern of
rapid to slow increasing of accuracy with increasing distance is shown in both the graph and in
the above charts. The percent differences shown in this chart for a 0.1% difference in the times
is not very large. In order to account for the larger errors in the calculated values for g, either the
timer has a very large error, or there are other factors compounding the error in the timer. Most
likely it is the latter case that there are other factors involved.
Another source of error is in the measurement of the distances of the intervals, d and D.
There is a possibility of a systematic error if the bottom of the meter stick was not truly at 0.
This was resolved by using the meter stick to measure a 5 cm gauge block. The result was that
there was no error in the 0 mark of the meter stick. Another source of systematic error was in the
slight tilt in the arms suspending the photogates. This resulted in the measurements taken along
the sides of the photogate to be not exactly the same as the height at the center of the photogate
where the ball would pass. However, obtaining accurate measurements in the center of the
photogate would be difficult since there is no solid material to measure against. The angle that
the meter stick was at when taking measurements would also introduce an error since it was
2
The values in the table are based on the first drop for each distance used. Thus, the numbers do not match up with
the first table giving the average value of g over all of the drops for each distance. Errors are not given since this
table is only intended to be a rough estimate of the effects of changes in time.
difficult to hold the meter stick in an exactly vertical position. There also was an uncertainty of
±0.0005 m in the measurements since the ruler was marked down to 0.001 tick marks.
Higher values of D would cause g to be higher since D appears in the numerator of the
formula for g. Higher values of d would cause g to be lower since d is subtracted in the formula
for g. The errors in g from D and d were combined and the resulting error in g is given in the
attached data spreadsheet under the "delta g" column. The formula used in the calculation was
the following:
Another source of error was in the air resistance on the ball as it falls. The air resistance
will slow the ball as it falls, so the measured value of gravity will be smaller. The effects of air
resistance increases with increased drop distance since the force of air resistance increases with
the increasing velocity that the ball will acquire while falling a greater distance.
If the ball does not fall directly on top of the transducer in the impact pad, it can also
introduce an error. This is because there will be a delay for the impact of the ball to propagate
through the pad to the transducer. This will lengthen the time for the second interval. The effect
of this was tested by shifting the impact pad 0.1 meters off center for a 0.0505 ± 0.0005 m drop.
The original value of g was 6.4933 ± 0.1087 m/s2, and the shifted pad gave a result of 6.3303 ±
0.1075 m/s2. This is a change of 0.1630 ± 0.2162 m/s2, or 2.5103%. The delay introduced by
the shifted pad was estimated by taking 0.0885 s as t1 (the time to travel the first interval in the
first drop for the 0.0505 m distance; an arbitrarily picked drop since this is only a rough
estimate), then using the two different values of g to calculate t2 and finding the differences in
the two t2 times. The result thus obtained is a 0.0003 s delay in the time for the interval from the
lower photogate to the impact pad.
After the pad was shifted, it was recentered and the drop for 0.0505 ± 0.0005 m was
performed again to check the reproducibility of the experiment. The resulting value for g was
6.4866 ± 0.1083 m/s2. This is a difference of 0.0067 m/s2, or 0.1032%. The small change shows
that the experiment is reasonably reproducible.
Conclusion
In this experiment, the calculated values for g came up short for all of the runs. However,
the accuracy of the values for g appear to increase logarithmically with the drop distance. The
appearance of the graph of gravity versus drop distance suggests this logarithmic relationship.
The effects of systematically incorrect time measurements also show this same logarithmic
relationship in the accuracy of the values of g with increasing distances. This suggests that the
timer for this experiment was miscalibrated. Nonetheless, the calculated values of g approach
the correct value of 9.85 m/s2 as the drop distance is increased. Thus, the experiment will
probably show more accurate measurements for g if the drop distance is increased, but not to the
extent that air resistance plays too large of a role.