A Determination of g, the Acceleration Due to Gravity,
from Newton's Laws of Motion
Objective
In the experiment you will determine the cart acceleration, a, and the friction force, f,
experimentally for different known masses m1 and m2. You will then use these results to
obtain g, the acceleration due to gravity. You will also learn how to deal with random errors
and systematic errors in experiments.
Equipment
Cart, pulley with mounting clamps, string, spark timer, timing tape, wood stop, table clamp,
weight hanger, weights, sand bag, masking tape, ruler and triple beam balance.
Introduction
In beginning physics, students test their ability to understand and apply Newton's second
law
by doing homework problems involving the constant acceleration of masses
subject to constant net forces. The masses may be moving vertically or on surfaces that are
horizontal or inclined, with or without friction. Often there are two or more masses
involved connected by ropes with possibly one or more of the masses hanging from pulleys.
Generally the mass of each object is given but the gravity force of the earth on it, i.e., its
weight is not. To obtain the weight it is necessary to use the relation
where g is
called the acceleration due to gravity and must be given. Usually, the student is told that,
although g varies with location, a good average value of g is 9.80 m/s2 for motion near the
earth's surface.
In this experiment you will use an arrangement of the type described above to actually
determine g at your laboratory location given some known masses, an understanding of
Newton's second law and a record of the motion.
Mass m2 is a cart, able to roll (with some friction) on your lab table. m2 is connected to the
hanging mass m1 by a light (negligible mass) string passing over a pulley. When the system
is released from rest with the spark timer activated, the position of m2 (and m1) is recorded
as dots on the tape every 1/60th of a second. It is clear that the acceleration of m2 (and m1)
and the resulting record will depend on the value of g. (If you could do this experiment on
the moon for instance, would you expect the acceleration of m2 to be greater or less than that
on earth? Why?)
Theory
Let T be the tension in the string while m1 is moving downward. Then the string exerts an
upward force T on m1 and a force T to the right on m2. Let f be the kinetic friction force
opposing the motion of m2.
is the weight of m1. These individual forces are
shown in Figure 2a.2 below. (Note that the vertical forces on m2 have been omitted as they
will not be needed in this particular analysis.)
The net force on m1 is downward and equals m1g - T. The net force on m2 is to the right and
equals T - f. Since the accelerations of m1 and m2 must have the same magnitude, we can
designate this magnitude by the letter a. Newton's second law when applied to ml and m2
then gives:
for m1
and
for m2.
On adding these two equations, the result is:
.
On solving for g,
(2a.1)
Brief Comments on the Treatment of Experimental Errors
Experimental error or uncertainty is inherent in any experimental result at some level,
however small. It is set by a combination of the design of the experiment, the quality of the
apparatus and the skill of the experimenter. Generally it is possible to separate the sources
of experimental errors into two categories: random and systematic.
A. Random Errors
The existence of random errors in a measurement can be inferred if repetition of the
measurement does not give the same result each time. By definition, random errors
are those that tend to average out upon repetitions of the measurement. For random
errors then, the more repetitions of a measurement the less the uncertainty in the
resulting average. From the mathematics of statistics it can be shown that the
uncertainty due to random error in the average of N measurements decreases as
when N is "large". Thus 400 measurements should give an average with half the
uncertainty due to random error as compared to 100 measurements.
The first step in treating the random error in a large number N of repeated
measurements is to calculate the average. The average is the desired result and it is
the uncertainty of the average due to random errors which must then be determined.
If the N measurements are labeled y1, y2, y3, ... yN the average as denoted by
defined as:
is
(The average is also sometimes called the mean in statistics.)
To establish the uncertainty in the average, the usual procedure is to first calculate
what is called σ (sigma), the standard deviation of measurements. If it is believed
that no one measurement is more accurate that any of the others (as we will assume)
then σ is defined by:
The significance of σ is that any one additional y measurement has about a 2 in 3
chance of falling between ± σ of the average,
. Statistical analysis further shows
that the average, has about a 2 in 3 chance of falling within ± σ of the true value.
Thus, after specifying the average of N repeated measurements, it is common to
append ± σ as a measure of the uncertainty due to the randomness of the
measurements.
Example
Suppose you have the following 5 measurements of the same quantity.
Note: One or two more decimal points must be kept in the calculations than are in
the data until the calculations are complete.
The standard deviation of these measurements is:
The uncertainty in the average is
= 0.0104. The result
is
then 1.0434 ± 0.0104. At this point some rounding off is appropriate and the result
would be reported as 1.04 ± 0.01 or possibly as 1.043 ± 0.010.
It should be pointed out that the number 5 is stretching the concept of large N
beyond reasonable bounds but that this is common practice when a large number of
measurements are not available.
B. Systematic Errors
A systematic error is one which tends to repeat and thus create a shift in the average
from the true value. Systematic errors may be provided by the experimenter, the
apparatus or by poor design of the experiment. Because they are not revealed by
repeated measurements, care must be exercised to investigate and account for all
possible sources of systematic errors. This can be very difficult to do. Such errors
sometimes remain unknown until other experimenters with other apparatus obtain
convincing evidence that a previous result is off by more than the originally reported
uncertainty.
Error Analysis in 2048C Laboratory Experiments
A. Treatment of Random Errors
Generally in your lab experiments in 2048C you will not measure the same quantity
over and over. Often, however, you will have a moderate number of data points
which are used to determine a single experimental quantity. Under such
circumstances the data can be treated in two ways. The first way is the graphical
method such as you used in the g experiment where the data are used to create a
straight line plot the slope of which is the desired experimental quantity. The second
way is to convert the data to a form equivalent to a series of repeated measurements.
In the case of the g experiment, this would correspond to determining the accelerator
between every pair of data points on the tape and creating a whole set of values for
acceleration a, which you would then average. In general the graphical method is
preferable and it is the only one you will use. It has the advantage that it gives a
useful visual picture of the data. It allows you to see whether, in fact, the data
exhibit the expected linear behavior and it also makes it easy to see the scatter or
random errors in the data.
To treat the random errors in your data using the second (non-graphical) approach
you would use the results presented earlier for repeated measurements to calculate
the uncertainty in the average. In the case of the graphical approach, the random
errors in the data points produce a statistical error or uncertainty in the slope. The
statistical treatment of random data errors in the graphical approach involves the
same principles, but somewhat different mathematics, as the treatment for the effect
of random errors in repeated measurements. These mathematics will not be
discussed in this course in any detail but have been embodied in a computer program
which you will use inside Excel. This program will analyze and plot your data and
give you the uncertainty in the slope produced by random errors in your data.
B. Treatment of Systematic Errors
You will be expected to discuss for each experiment possible sources of systematic
errors. In general, this means giving plausible reasons as to why your results might
differ from the expected result by more than the uncertainty due to random error.
Procedure
Set up the equipment as shown in Figure 2a.1. Set the length of the string so that the
hanging mass m1 will not hit the floor when the cart reaches the edge of the table. Use about
30 cm of timer tape and adjust for at least 40 cm travel of ml and the cart. Fasten the tape to
the bottom of the cart with a masking tape.
The amount of data which you will take in this experiment and the time required to obtain it
are both small. Whether you obtain good data or not hinges on your experimental skills and
attention to instructions.
You must observe the following precautions while setting up.
1. Be sure that the pulley height is adjusted so that the string from the pulley to the cart is
horizontal.
2. Be sure that the pulley, the cart, and the timer are all in one line with the cart aimed at the
pulley.
To take data, hold the cart in a steady position with m1, just below the pulley. Have your
partner take any slack out of the timing tape and then have him or her start the timer. Wait
an instant for the timer to come up to speed and then abruptly release the cart.
Practice the above once or twice before taking any data.
Take at least three motion records with the following values of m1 and m2 :
1) m1 = 0.750 kg, m2 = cart
2) ml = 1.000 kg, m2 = cart
3) m1 = 1.000 kg, m2 = cart plus sandbag.
As you obtain each tape, write on it the values for ml and m2 so you do not get the tapes
mixed up later.
Inspect your tapes to see if the timer marks appear consistent with motion that starts from
rest and increases in speed as time goes on. When you are convinced that you have
reasonable looking tapes, ask your instructor to look at them to see if he or she concurs.
Determine the force of friction using the following method:
Since friction is not negligible in this experiment, the kinetic friction force, f, opposing the
motion must be measured. To do this we use Newton's first law. That is, if you hang a
weight on the string that produces a force equal and opposite to the kinetic friction, then the
cart once set into motion ought to move at constant velocity.
With m2 = cart, start by using a mass m = 25 g initially. Give the cart a nudge and observe
the motion. Increase or decrease m as needed to produce a constant velocity. You can
determine the needed value of m to produce a constant velocity by direct observation. Be
sure to determine the friction force for all values of m2 that you use.
Because the friction force is now given by mg and you are to determine g in this experiment,
you must replace f by mg in Equation (2a.1) and solve for g. Show that the result is:
(2a.2)
Data Analysis
The assumption in this experiment (which your graphs should verify) is that the friction
force is constant. If so, it follows from Equation 2a.1 or 2a.2 that the acceleration a is
constant. In this case the distance, D, that the cart travels in time, t, is given by:
(2a.3),
and the instantaneous velocity, v, of the cart after time, t, is:
(2a.4),
Since the cart is released from rest, it is true that vo = 0 at the instant of release. However,
your timer may not have made a mark at just that instant or you may not have released the
cart cleanly. In any case, the likely uncertainty as to which tape mark represents t = 0 makes
Equation 2a.4 rather than 2a.3 the best choice for analyzing the data.
You should recognize that Equation 2a.4,
is of the form
so that a
graph of v versus t on linear paper should be a straight line whose slope is the acceleration a.
Figure 2a.3 illustrates this.
Note that if you were to change the instant of time that you label t = 0, the graph would be
shifted horizontally. vo, the intercept on the v axis, would change but the slope representing
the acceleration a would remain the same. Hence, such a graph will give a value for a
unaffected by any initial timing uncertainty.
Since the marks on the tape represent distances traveled, you will have to use these distances
to determine instantaneous velocities in order to plot Equation 2a.4. To do this, measure
and record the distances ΔD in cm between the centers of successive timing marks.
Pick a convenient point to start measuring and try to measure as many distances as you can.
Carefully measure to at least the nearest 0.05 cm (half a millimeter). Use your clear plastic
ruler to do this. Omit measuring values of ΔD that are less than 3 or 4 millimeters. Also do
not include the last two or three points on the tape as they may have been recorded when the
cart was stopped by the wood block.
Run Excel and enter in separate columns the measured distances ΔD and the
corresponding time t in seconds for each of your tapes. Identify each run and record the
corresponding cart mass m2, mass of the pulling weight m1, and the mass of the hanging
weight m used to determine friction force. As discussed, the time you choose to call t = 0
doesn't matter in determining the acceleration from the graph of Equation 2a.4. It is
convenient to let t = 1/60 s = 0.0167 s correspond to your first value of t.
m1 =
m2 =
m=
Time in
seconds
Measured ΔD in
cm
v = ΔD/Δt in cm/s
0.0167
0.0333
0.050
...
...
In another column, calculate the velocity v (by using equations in Excel, do not use your
calculator to do this) for each time interval by dividing ΔD by 1/60 sec which is the length
of each time interval corresponding to ΔD. Graph v versus t using the “Regression” feature
of Excel which can be obtained from “Tools” – “Data Analysis” menu. The time t should be
plotted on the x-axis and the measured distances ΔD on the y-axis. Make sure to click on
the boxes for Residuals, Residual Plots, Line Fit Plots and New Worksheet Ply to produce
the information and graphs that you need for analysis.
Go to your “Line Fit Plot”. Add a trend line and display the equation of the best fit line for
v versus t. Get the slope of this line (what does this slope represent?). Also determine the
uncertainty in the slope by inspecting the resulting tables.
Repeat the calculation of velocity v and the graphing instructions on this paragraph
for the other 2 runs.
Because of the linear relation between v and t in Equation 2a.4, the average velocities you
have determined are also instantaneous velocities midway through the time interval between
marks. Hence a plot of your versus t on linear graph paper will give you a plot whose
slope is the acceleration a.
Calculate the value for the acceleration due to gravity g for each of your runs using Equation
2a.2 and your values for a, m1, m2 and m.
Tabulate the 3 values of a and their uncertainties, and also the 3 values of g that you got.
Calculate the standard deviation σ of the three measurements of g as a measure of its
uncertainty due to the randomness of the measurements.
Calculate the percentage difference between each of your results and the value of g = 980
cm/s2 (9.80 m/s2) expected at this location.
NOTE: Include printouts of your data tables and graphs from Excel in your report.
Also include the tabulation of the 3 values of a and the corresponding uncertainties,
and also the 3 values of g, σ and the calculated percentage differences.
Questions:
a. Inspect the “Line Fit Plot” and check how well the measured values of the velocity v
plotted on the y axis coincide with the “Predicted Y” on the computer fitted line. Inspect
the “Residuals” and the “Residual Plot”. Are your residuals reasonable or are there
obvious outliers or data points that are possibly entered in error?
b. By how much do each of your values for g differ from 980 cm/s2? How does each
difference compare with the corresponding uncertainty Δg due to the random errors in the
data? Does it appear that there is a systematic error in either or both of your results?
c. If you worked carefully and your equipment worked properly, you may see the effect of a
small systematic error inherent in the design of this experiment. In order to get the cart
wheels to spin faster and faster as the cart accelerates, the table must exert a static friction
force on the wheel rim in a direction opposite to the cart motion. This is not the friction
force you measured when the cart was moving at constant velocity and so it has not been
accounted for. A calculation of this retarding force on the cart based on the mass and
geometry of the wheels and the physics of circular motion shows that it should cause the
measured value of g to come out about 2% low for the values of m1, and m2 used here.
What result do you obtain if your g values are each increased by 2% (about 20 cm/s2) to
account for the systematic errors above? Are your data accurate enough to see this
effect? Do you still have systematic errors larger than the above?
d. List a few other possible sources of systematic errors that should be checked if time
allowed. (Hint: Think about some of the assumptions you likely have made about the
equipment that may or may not be valid.)
Because each run was made under different conditions, i.e., with different masses, they
do not represent an exact repetition of a single measurement. Each run may be subject
therefore, to different systematic errors. Until such errors are accounted for, averaging of
your results is not strictly valid.
© 1996 Dr. H. K. Ng.
All Rights Reserved.
Revised Fall 2013 B. R. Reyes