YEARS 7–12 IDEAS
ARTICLES
FOR THE CLASSROOM
Measuring Gravitational Acceleration using Timing Gates (continued)
• using more heights and greater heights, to check the linearity
of the result,
Analysis: The equation s = ut + ½at 2 describes uniformly
accelerated motion. If the interval between the light gates is
considered, then h 2 = u t 2 + ½ g t 22
We multiply both sides by 2/t 2 to find
2h 2 /t 2 = 2u + g t 2 which
Analysis: The equation s = ut + ½ at 2 describes uniformly accelerated motion. If the interval
is in the
y = b + mx,
between
the form
light gates
is considered, the
then equation
h 2 = u t 2 + ½ for
g t 2 a 2 straight line, familiar
from
maths
be is in
found
graphically
We
multiply
both classes.
sides by 2/t 2 Then
to find g,
2h 2 h /t 1 2 = and
2u + t g 1 t 2 can
which
the form
y = b + mx, the
equation
for a slope
straight and
line, familiar
from maths classes. Then g, h 1 and t 1 can be found
from the
intercept.
graphically from the slope and intercept.
5
4.5
3.5
3
2.5
• rolling the ball at different speeds effectively varies h 1 ,
• the round bottom of the ball breaks gate 2 differently and
effectively varies h 2 , thus t 2 .
Conclusion: The result was almost accurate (to 6 sig. figs). The
expected value at our school, allowing for latitude and altitude, is
9.79565 m.s –2 . This authoritative value used the World Geodetic
System 1984 ellipsoidal gravity formula. Latitude is ϕ. Height
above sea level is h.
2
1.5
1
0.5
0
Some recognised sources of error are:
• The ball should be rolled as straight as possible. This also
means the ball is less likely to hit, rather than pass through,
the lower timing gate. Repeated impacts might eventually
damage the timing gate.
y = 9.8039728x + 0.9013015
R² = 0.9998298
4
• using more points at each height, to reduce the standard
error, …
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
t2 (s)
0.45
While it is usual to plot the independent variable on the x-axis and the dependent variable on
the y-axis, this guideline can be ignored if there is good reason. The slope and intercept
While it the is information
usual to we plot
the so independent
variable
on
x-axis
contained
needed,
we plotted a calculated
value, 2h
, on the
y-axis,
2 /t 2 the
and
our
dependent
variable
on
the
x-axis.
This
simplified
the
data
analysis.
and the dependent variable on the y-axis, this guideline can be
Plot
on the y-scale
2h 2 is
/t 2 good
and on reason.
the x-scale The
t 2 . Use slope
Excel’s and
built-in
fitting for contained
a straight line.
ignored
if there
intercept
Print the equation with the slope and intercept values on your graph. R 2 tells you how well
the line information
we
needed,
plotted
calculated
the
fits. The value of
R 2 lies
between 0 so
(no we
predictive
value) a and
1 (perfect). value,
gEGF(ϕ) = {9.7803267714 [1 + 0.00193185138639 sin 2 ϕ/√(1 –
0.00669437999013 sin 2 ϕ)] – (3.083293357 × 10 –6 + 4.397732 ×
10 –9 cos 2 ϕ) h + 7.2125 × 10 –13 h 2 }m∙s –2
The theoretical background is found in Hinze et al (2005) or Li
and Götze (2001)
Launching the ball by rolling it slowly over the edge gave an
effectively uniform vertical speed u, which is required for the data
analysis to be valid.
The
had the
the value
9.804 and
m.s -2 = our
g
2h 2 slope
/t 2 , on
y-axis,
dependent variable on the x-axis.
-1
The
intercept
was 0.90130
m.s = analysis.
2g t 1 = 2u
This
simplified
the data
So t 1 was u/g = 0.04597 second
Plot on the y-scale 2h /t and on the x-scale t . Use Excel’s built-
2
2
So h 1 = ½ g t 12 was 10.36 mm, 2 which
was plausible. The light gates
are 25.4 mm thick. The
in through
fitting 12.7
for mm
a straight
line.
Print
the equation
slope
and
fall
was slowed
slightly
by upward
and outward with
forces the
as the
ball teetered
2
over
the
edge.
intercept values on your graph. R tells you how well the line
The timing data were precise to microseconds, but only reliable
to three or four decimal places. With more practice rolling the ball
slowly and repeatably over the edge, we hope to improve the
reliability. The timing was vastly better than using stopwatches
or mobile phones. These record only two or three decimal
places, and starting and stopping the timing often use slightly
different stimuli, leading to systematic timing errors of about 0.1
second. We have observed such timing errors across a range of
experiments, all with similar systematic errors.
fits. The value of R 2 lies between 0 (no predictive value) and 1
(perfect).
The slope had the value 9.804 m.s –2 = g
The intercept was 0.90130 m.s –1 = 2g t 1 = 2u
So t 1 was u/g = 0.04597 second
So h 1 = ½ g t 12 was 10.36 mm, which was plausible. The light
gates are 25.4 mm thick. The fall through 12.7 mm was slowed
slightly by upward and outward forces as the ball teetered over
the edge.
The height data were precise to millimetres. To measure g to three
decimal places will require distance measurements to about 0.1
millimetre. This will be challenging.
Discussion: The fit to a straight line was remarkably good (ie
R 2 =0.9998298). We extrapolated outside our data to find the
intercept, which usually gives worse estimates. The large value
of R 2 , al