8.4 Measuring g in lots of different ways
Activity 130E: Experiment
Often, the only way to improve a measurement is to find a better way of doing it. Here you can try a
variety of different ways of measuring the acceleration g under gravity.
The methods suggested here are:
1.
g by tower using automatic release and electronic timing over a known distance (with one
drop height measured many times or several drop heights)
2.
g by free fall using two light gates for speeds at known separation
3.
g by trolley accelerating down a slope (measuring the acceleration down slopes of several
different gradients)
4.
g by dropping a steel ball bearing from as large a height as possible and timing with a stop
watch
5.
g by videoing free fall
Getting as close to g as possible by reducing uncertainty and eliminating systematic
error
In this experiment you will attempt to measure the local acceleration under gravity g. The point is to
do this as well as you can, to know how well you have been able to do, and to compare your results
with those of other groups that have tried a different method.
You need to consider the percentage uncertainties in each measurement; these will come from
repeated readings and cross-checking. You will need to assess the overall uncertainty in the value of
g. At the end you should be able to state your best estimate for g and within what range you believe
the true value to lie.
In this experiment systematic error (bias) is often important. You need to think about reasons why
your result may contain systematic error, in the measuring methods or in the assumptions that you
are making. Any final graphs may help you to see if systematic errors seem to be present.
1: g by tower and electronic timer
You will need
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an electronic timer / scaler capable of measuring to milliseconds or better using break to start,
break to stop automatic switching
a tower about 1 m tall fitted with shorting release mechanism to start (three metal pins, two
with circuit connections) and a trapdoor, breaking the circuit to stop the clock (or use firmly
clamped stands)
four long connecting leads
a steel metre rule
a large-diameter ball bearing to fit the start mechanism
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break-to-start contacts
steel ball
about 1 m
break-to-stop contact
trap door
This method uses an electronic timer to time the fall of a ball. The timer is started by breaking
contacts between the ball and a release device, and stopped when the ball hits a trapdoor, again
breaking a contact.
Here are detailed instructions:
1.
The ball bearing is to be released from the shorting pins (breaking the circuit starts timer). It
falls freely under gravity until it hits the trapdoor (and breaks a circuit to stop the timer). Practice with
your partner dropping the ball bearing cleanly, by withdrawing your fingers rapidly.
2.
Measure the height s dropped with an accurate ruler from the trapdoor to the bottom of the
ball bearing.
Repeat the drop several times (many times if you cannot vary the height of drop with your tower), and
record the times. Decide if any are too different from the others and should be considered as
unreliable outliers. If you have a reason why a result is wrong (e.g. you know you fumbled the release
of the ball) then discard it and repeat.
3.
If you can adjust the height dropped on your apparatus, repeat the drops for several different
heights. Calculate the mean values of the times of drop and plot them on the x-axis against heights
dropped on the y-axis.
The raw data s plotted against t should fit a power law of order 2 since
1
𝑠 = 𝑔𝑡 2
2
(with start velocity u = 0).
4.
s/t2.
In another column of your data table you can estimate g for each of your times by tabulating 2
Take the mean of these as an estimate of g
2
5.
To get your data into straight-line format make a new data column in your spreadsheet for
values of t2 and plot s on the y-axis against t2 on the x-axis. Fit a best straight line. Twice the gradient
gives you another estimate for g.
Comment on any possible systematic error if your best-fit line does not pass through the origin.
6.
Think about whether there could be bias in your experiment, giving a value which is
systematically too high or too low.
7.
Decide your final value for g together with your estimate of the uncertainty in the measurement
and compare it with the mean UK value of 9.8 m s-2.
2: g by speeds measured at two light gate pairs of known separation
You will need
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a data logger with light gates and PC set to measure velocities at A and B
a metal cylinder to drop through the light gates (spheres tend to interrupt gates by a variable
length – it is hard to keep the diameter aligned with gates). The cylinder needs to be a few
centimetres long and of sufficient diameter to interrupt both beams reliably. Find out if the
cylinder drops more stably with its axis vertical or horizontal
a clamp stand with a G-clamp and bosses to hold light gates securely
a steel rule to measure the distance between light gates
a bucket and sand to catch the cylinder safely
a plumb line (pendulum bob and thread)
cylinder dropped
through gates
light gate
light gate
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This method measures the velocity of fall of a cylinder at two places a known distance apart, using
light gates.
Here are detailed instructions:
1.
Clamp the light gates to the stand over the edge of the bench. Place the bucket just below the
lower gate to catch the cylinder. Practice with your partner dropping the cylinder so that it interrupts
both light gates reliably. You can release the cylinder from any height above the top light gate, but it is
instructive to see the spread of initial speeds if you keep the height above the top gate fixed, for all
separations of the gates.
2.
The gates can be separated by vertical distances up to about 1 m. Use a plumb line to ensure
they are in vertical alignment.
Repeat the drop several times for each separation of the gates and record the initial speed u and the
final speed v for each separation that you use. Decide if any values are too different from the others
and should be considered unreliable outliers. If you have a reason to suspect that a result is wrong
(e.g. you saw the cylinder tip over during the fall) then discard it and repeat the reading.
3.
In another column of your spreadsheet table calculate the value of (v2 – u2). Plot values of (v2
2
– u ) on the y-axis against s on the x-axis. Fit a best straight line. The gradient divided by two gives
you an estimate for g because
(𝑣 2 − 𝑢2 ) = 2𝑔𝑠
Comment on any offset (possible systematic error) if your best-fit line does not pass through the
origin.
4.
In another column of your data table you can estimate g for each of your runs by tabulating
(𝑣 2 − 𝑢2 )
2𝑠
=𝑔
Take the mean of these as your measurement of g and look at their range, taking half the range as an
estimate of the uncertainty.
5.
Think about whether there could be bias in your experiment, giving a value which is
systematically too high or too low.
6.
Decide your final value for g together with your estimate of the uncertainty in the
measurement and compare it with the mean UK value of 9.8 m s-2.
4
3: g by a trolley accelerating down a ramp
You will need
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a data logger with a pair of light gates and PC set to measure velocity
a runway with a trolley (as friction free as possible)
a card set vertically
a clamp stand, G-clamped to the bench to raise the runway at one end by small heights up to
about 0.3 m for a 2 m runway
spirit level to check that lab bench is horizontal when measuring the slope of the runway
This method measures the acceleration of a trolley down a ramp of known slope, with a data logger
recording two velocities and the time between which they are measured, so giving the acceleration .
Here are detailed instructions:
1.
Clamp the runway with a small angled slope so that trolley accelerates slowly but reliably
down the ramp.
2.
Measure the heights above the horizontal bench for each end of the runway and record them
in a table, with an estimate of their uncertainty. Release the trolley and measure the velocity at the
first gate and then at the second gate and the time between the gates. Use v = u + at to calculate the
acceleration.
Repeat the run several times and compare the accelerations. Decide if any are too different from the
others and should be considered unreliable. If you have a reason why a result is wrong (e.g. you saw
the wheels were not properly engaged in the track) then discard it and repeat. Take the mean of these
accelerations and look at their range, taking half the range as an estimate of the uncertainty.
3.
In your data table construct a column to calculate:
sin(angle of slope to horizontal) = difference in heights / length along runway
How uncertain is your slope estimate?
In the absence of friction the component of the acceleration of gravity down the slope should be given
by the equation a = g sinθ.
Repeat for other angles of θ.
4.
Plot values of a on the y-axis against sin θ on the x-axis. Fit a best straight line to your points.
The gradient of the line gives your best estimate for g. Comment on any offset (possible systematic
error) if your best-fit line does not pass through the origin.
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5.
Think about whether there could be bias in your experiment, giving a value which is
systematically too high to too low. Can you reduce frictional drag any further?
6.
Decide your final value for g together with your estimate of the uncertainty in the
measurement and compare it with the mean UK value of 9.8 m s-2.
4.g by dropping a steel ball bearing large distances
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a steel ball bearing
a stop watch
a tape measure
a long drop
Here are detailed instructions:
1.
The ball bearing is to be released from a given height s. Time how long it takes to reach the
ground using a stopwatch. You will need to estimate the uncertainty in this measurement and what
causes the uncertainty.
2.
Measure the height s dropped with a tape measure.
Repeat the drop several times and record the times. Decide if any are too different from the others
and should be considered as unreliable outliers. If you have a reason why a result is wrong (e.g. you
know you fumbled the release of the ball) then discard it and repeat.
3.
Adjust the height dropped, repeat the drops for several different heights. Calculate the mean
values of the times of drop and plot them on the x-axis against heights dropped on the y-axis.
The raw data s plotted against t should fit a power law of order 2 since
1
𝑠 = 𝑔𝑡 2
2
(with start velocity u = 0).
4.
s/t2.
In another column of your data table you can estimate g for each of your times by tabulating 2
Take the mean of these as an estimate of g
5.
To get your data into straight-line format make a new data column in your spreadsheet for
values of t2 and plot s on the y-axis against t2 on the x-axis. Fit a best straight line. Twice the gradient
gives you another estimate for g.
Comment on any possible systematic error if your best-fit line does not pass through the origin.
6.
Think about whether there could be bias in your experiment, giving a value which is
systematically too high or too low.
7.
Decide your final value for g together with your estimate of the uncertainty in the measurement
and compare it with the mean UK value of 9.8 m s-2.
6
5. g by videoing free fall (see page 120 for the basic method)
1. Drop a light ball against a dark background with a ruler vertically placed. Video the event with your
ipad on the ‘vernier video physics’ software.
2. Use the ‘vernier video physics’ software to:
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click on ‘calibrate’, set the scale by placing the markers from the top to the bottom of the ruler
and setting the distance this corresponds to.
set the origin and scale – align the (x,y) intercept with the release point of the ball and orient
this carefully so it is in line with the vertical drop of the ball.
Now step through the video until you get to the release point of the ball and click on the ball’s
position. Move forward a frame, move the circle and click on the point where the ball is now.
Continue until all the points are marked.
Now set the circle to the size of the ball and place it at the release point. Click on track and
the software should follow the actual path of the ball.
Now click on top right hand corner graph icon and scroll through to the Ydistance and Yvelocity
against time curves.
Ydistance should be a curve as the distance travelled will increase every second as the ball is
accelerating.
Yvelocity should be a straight line through the origin as u = 0 ms -1.
v = at
What does the gradient give you?
What are the uncertainties in this method?
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