Measurement of the acceleration due to gravity
Aim
The aim of this experiment is to measure the local value of the acceleration due to
gravity, g, by studying the free-fall motion of steel balls. Some general concepts of
error analysis will also be introduced.
Background
In this experiment, g will be determined by applying the well-known kinematic
equations:
s = ut +
1 2
gt
2
v = u + gt
[1]
[2]
to the motion of a free-falling object, and measuring the variables s, t and v as
appropriate in order to determine g.
Apparatus
Retort Stand
Metre rule
Magnetic ball release system
Forked light gates (x2)
Steel balls
Plumb line
Counter-timer
Release switch
Set up and familiarisation
•
Make sure that the red and green wires from the release switch are connected
to the upper and lower 4mm sockets respectively on Terminal E of the
counter/timer.
•
Plug the upper light gate into Terminal F, leaving the lower light gate
unconnected for the moment. Move the lower light gate to the lowest possible
position on the retort stand.
•
Align the retort stand by eye until it appears perpendicular to the bench top.
•
Move the magnetic release to a convenient height, and position the upper light
gate about 40cm below. With the release switch in the “up” position, connect
the plumb line to the magnetic release (held by the steel washer). Increase the
holding force on the washer by rotating the knob on the release system if
necessary. Align the light gate with the magnetic release, ensuring that the
cord of the plumb line just intercepts the beam of the light gate (when this
happens, the red light on top of the gate will go out).
•
Remove the plumb line, and place one of the steel balls in the magnetic
release. Adjust the knob until the magnetic force just supports the ball,
allowing it to be released “cleanly” when the field is switched off.
•
Press the “MODE” button on the counter/timer until the “tE→F” light is
illuminated. This means that the timer is set to measure the time between
“event at E” (release of the ball) and “event at F” (ball passing through the
beam of the light gate).
•
Press the “START” button on the counter/timer, which should be showing
0.000 ms on the display.
•
Place the plastic box in a suitable position on the bench to catch the ball.
•
Move the hold switch to the down position to release the ball. The “STOP”
light should now be lit, with the display showing the time taken in
milliseconds for the ball to fall from the holder to the light gate. If “STOP” is
not lit, and the time is continuing to increase, the gate is probably misaligned.
Check the alignment again, and get help from a demonstrator if necessary.
Experiment 1
In this experiment you will vary the distance s between the magnet holder and upper
light gate, and measure the associated times of fall for the steel ball, extracting the
value of g from Equation [1] by plotting the data in a suitable form.
Questions 1
What are the dependent and independent variables in this experiment? How would
you normally plot the data in order to obtain a straight line graph from which g can
be determined? Which variables would conventionally be plotted on the x and y axes
respectively?
If you think about the experimental setup, you might realise that it is rather difficult to
measure the actual distance fallen by the ball without some unknown offset ∆s arising.
Question 2
How can you plot your data so that the unknown offset ∆s does not affect your
measurement of g? (Hint: think about an unconventional choice of axes). Discuss this
point with a demonstrator before continuing.
Set up the experiment as described in the previous section, with a distance between
magnetic holder and light gate of about 90cm. Measure the distance between
convenient points on the magnetic holder and light gate as carefully as you can, and
write it down, including the appropriate uncertainty. Set up the timer, release the ball,
and measure the time of fall, t. Repeat the measurement 5 times and calculate the
mean and standard deviation for your values of t (see “Basic Notes on Errors”) if you
are unfamiliar with the idea of standard deviation). Repeat the whole procedure for 6
different values of s (you will find it more convenient to move the light gate, keeping
the magnetic holder position fixed), and plot your data to determine g and its
associated experimental error. You may either plot the data by hand or use Excel with
the LINEST function to plot/analyse your data. How does your measurement compare
with the accepted value of g? Comment on any discrepancy.
Experiment 2
Unplug the release switch from the counter/timer, and plug the lower light gate into
terminal F. Select tE,F using the MODE switch. Line up the magnetic release and the 2
light gates using the plumb line as described above, with the light gates and the
release equally spaced by about 45cm. Place the ball in the magnetic release with the
release switch in the “up” position as before, press START on the counter timer and
release the ball. The following readings should now show on the display (cycle
through these by pressing the tE,F button, below the START button):
•
Lights E, F and t1→2 brightly lit: time taken for the ball to pass from gate E
(upper) to gate F (lower). Should be around 100ms.
•
Light E on, light F off: time taken for the ball to fall through the gate E beam
(t1, should be a few ms).
•
Light F on, light E off: time taken for the ball to fall through the gate F beam
(t2, should be < t1).
Measure the diameter of the ball with a micrometer or vernier callipers (including
appropriate uncertainty) and hence calculate the velocity of the ball when it travels
through each of the 2 light gates.
Repeat the measurement 5 times and obtain the mean values and standard deviations
for t1→2, v1 and v2.
Repeat the whole procedure for about 5 different light gate separations, and plot an
appropriate straight line graph (include error bars if necessary) in order to obtain a
value for g and its experimental uncertainty.
Analysis of Experiments
In your lab diary, write a brief summary comparing the relative accuracy and
precision of the two experiments, and discuss the contribution of systematic and
random errors in both cases. Can you suggest any ways in which any of these errors
can be reduced? How could these suggestions be tested?