Download Lab for October 14: acceleration due to gravity and Newton`s second

EXPERIMENT 4
DETERMINATION OF ACCELERATION DUE TO GRAVITY AND
NEWTON’S SECOND LAW
I. Introduction.
There are two objectives in this laboratory exercise. The first objective, (A), is the study
of the behavior of a body in the gravitational field. The second objective, (B), is to verify
Newton’s second law.
II. Theory
A.
Determination of the value of the acceleration arising from the gravitational field.
The force between two bodies is attractive and depends on the masses of the bodies and
the distance between them. The magnitude of the force on a body of mass m, at a distance R
from the center of the earth and at, or above, the surface of the Earth, is given by,
Fg  G
mM e
Re2
(1)
where G is the universal gravitational constant, Me is the mass of the earth and Re is the distance
from the center of the earth. If the body is on the earth, the radius of the earth. The force Fg is
called the gravitational force.
Newton’s second law gives us the acceleration, a, of a body of mass, under a net external force F
as:

 F
a
m
(2)
Thus, if a body has no other force on it than the gravitational force, the magnitude of the
acceleration of a body of mass m in free-fall at the surface of the earth is given by:
ga
GM e
Re2
(3)
that is, the acceleration of a body at the surface of the Earth depends only on the mass of the
earth, Me and it radius Re - it is independent of the mass of the body. This acceleration is termed
"the acceleration of gravity", denoted by g. It is taken as a constant but actually depends slightly
on the location (because of inhomogeneity in the earth's crust, such as denser rock formations
and the different elevations of the surface, etc.).
III.
Equipment and Procedure (acceleration due to gravity)
Equipment
Photogate, picket fence, PASCO 750 Interface and computer.
Procedure
Find the value of the acceleration of gravity, g, at Athens, OH
1.
2.
3.
Connect the photogate to the first digital input in the interface and turn on the Interface.
Select Data Studio from the desktop. Follow the "Computer Instructions" to set up the
photogate to collect position and velocity data. A photogate is set up at the end of the
laboratory table so that a picket fence may be dropped through it.
One student should drop the picket fence through the photogate and one should start the
collection of data before the picked fence starts to cross the photogate. Make sure the
picket fence does not hit the photogate.
Enter the velocity data into your notebook. Fit the position data with a quadratic and the
velocity (really speed) data with a linear fit as indicated in the computer instructions. The
coefficient of t2 yields one-half the value of the acceleration in the gravitational field. The
slope of the fit of the velocity data may be taken to be the average acceleration and the
associated standard deviation in the slope to be the standard deviation in the acceleration.
a
dv
dt
 a   slope
(6)
In this case these are, respectively, the average value of the acceleration due to gravity
and its standard deviation.
Enter the average value of the acceleration and the standard deviation of the mean of the
acceleration in your notebook and print the graphs showing the curve fits to the position
and velocity data.
III. Data
Print out and enter into your notebook the position and velocity data for the picket fence
dropped vertically through the photogate and analyze it to obtain the value of the acceleration of
gravity it according to the procedures outlined in section II and the Computer Instructions.
IV. Results
Determine the value of g from fitting to the position and velocity data. For the linear fits,
determine the standard deviation in the mean of from the standard deviation in the mean of and
write the result as g  g   g . Show your calculation for the standard deviations in the mean of
on your summary sheet.
Compare each value of g to the accepted value in Athens, OH, 9.800±0.003ms-2 (from the
National Geodetic Survey 06-22-05) and determine the percent error in your measurement with
respect to the accepted value.
V. Conclusion and Discussion (acceleration due to gravity)
Does the value of g fall within the standard deviation of the accepted value? Discuss any
actual sources of error. What are the sources of error (if any) in your procedure?
VI.
Equipment and Procedure (Newton’s Second Law)
Equipment
Airtrack with glider, photogate, PASCO 750 Interface and computer.
Procedure
1.
2.
3.
4.
Make sure the airtrack is level. The instructor will show you how to determine if it is
level and how to level it if necessary. The glider must move smoothly (it should not
appear to slow down or "catch" along the track). If it doesn't move smoothly ask the
instructor to look at it with you.
Follow the computer instructions for Newton’s Second Law at the end of this lab.
Attach a mass holder to a string about 3.0 m long and pass it over the pulley, tying the
other end to the hook on the glider. Place the assigned number of grams on the glider and
use the scale provided to determine the mass of the assembly of glider plus masses,
including the holder. Enter this data into your laboratory notebook.
Transfer masses of 2-g each, one at a time, from the glider to the hanger so that the total
mass of the system remains constant. Begin with a run with only the holder to provide the
accelerating force ( =mholderg ). Thereafter transfer a 2-g mass to the holder and obtain the
average acceleration and standard deviation again. Repeat the procedure until all the 2-g
masses have been transferred to the holder. The magnitude of the net force is the weight
of the hanger and masses on it so that the acceleration of the glider is given by
a
5.
dv mholder g

dt
mtotal
(9)
To hold the glider at rest, place a pencil eraser or finger on the side of the glider. Release
the glider. Let the mass move some distance and take position and speed data (which will
appear under “Run#1” in the left window). But stop the glider so that the masses don’t hit
the floor. Make sure the string remains on the pulley. The computer will provide a set of
positions speeds for each mass placed on the glider from which the acceleration is
obtained by taking the slope of the velocity versus time graph, Eq. (9).
VII. Data
After the first run (with all the masses on the glider), transfer a fixed amount of mass
each time to the mass holder and release the glider as indicated in the procedure. Enter into your
laboratory notebook the total mass (includes the holder), the acceleration, and the standard
deviation of the acceleration for each additional mass added to the holder.
Data: Part B Newton’s Second Law
Mass of hanger, mh =
___________kg
Added masses to glider ma ___________kg
Mass of glider from balance ___________kg
Total Mass of the System ms = (holder + mass on glider +glider)
_____________kg
Table 1. Masses, force and linear accelerations for a mass ms.
Run
Mass of Holder +
Added Mass
Force
Acceleration
(Fit to Position)
(=(mh+ma)g)
Acceleration
(Fit to
Speed).
Std.
Deviation
1
2
3
4
Mass of the system from the inverse slope of a vs F, ms 
F
, and the standard deviation of
a
the mass.
Mass of glider = mgleider = ms- (mh + ma)
VIII. Results
Plot, on separate graphs, ai vs Fi for the acceleration obtained from fitting the position
versus time and the slope of the velocity versus time data and attach the standard deviation of the
mean for the each acceleration determined from the slope. Obtain the best fit to the data keeping
in mind the standard deviations associated with the each acceleration. The inverse slope of the
acceleration versus the force is the mass of the system (total mass). The standard error in the
mass is the determined by standard error in the fit. Determine the mass of the system on the scale
provided.
IX. Conclusion (Newton’s Second Law)
Does the data verify a linear relationship between the force and the acceleration of the
glider. Does the mass determined from the graph of acceleration vs force fall within the standard
deviation of the mass measured on the scale? What are the possible sources of error?
N.B. The mass as measured by the accelerations is called the inertial mass because it is a
measure of the resistance to change of motion while that measured on the scale is called the
gravitational mass which is a measure of the strength of a body as a source of gravitational force.
In principle the properties of these two masses are different but very careful experiments yield a
ratio of 1 to about fourteen decimal places for the two masses.
Computer Instructions
Acceleration due to Gravity
Physically plug the photogate into digital input port #1 on the 750 interface.
1.
In the Experiment Setup window (click Setup on the toolbar if it isn’t shown), you’ll see
an image of the front of the Pasco 750 Interface. Click input port #1 on that image, and
select the photogate sensor from the list.
2.
Choose the position and velocity data sources. In the Constants tab, enter the bar spacing
for the picket fence so that the interface will report distances correctly. For the large
picket fence it is 0.05 m and for the smaller picket fence it is 0.03 m. You may close the
Experiment Setup window.
3.
Now to collect some data and display it on a graph: Left Click on Run#, wait a short
while and left click again until a rectangle appears. Type your initial and those of your
partner into the box. Then answer “yes” when “Rename” appears in the main window.
This will place the initials on the graph. Drag and drop the Position data source (from the
top left window) onto the graph icon in the Displays window (bottom left). A graph will
appear in the Main window. You'll want to show Position and Velocity on the same
graph, so select Velocity from the data source window and drag & drop it onto the Graph
that was just created. If you drag & drop it onto the Graph item in the Display window, it
will open a new graph in the main window. This will also work, but you'll have to print
two pages instead of one.
4.
To start the run, select one student to drop the picket fence and one student to start the
data acquisition. It is important that the picket fence not hit the sides of the photogate as
it travels through it. The data run is started with the Start Button on the toolbar. Once
data acquisition is in progress, the Start button becomes a Stop button.
5.
Your first data set will be named Run #1. There's an autoscale button on the top left of
the graph display. You may have to further correct the x and y axes by clicking and
dragging them.
6.
To analyze your data, a) select “Run #1” on the position graph, and then go to the G
button and select Quadratic fit from the Fit pulldown menu. The coefficient of t2 will
yield half the value of g. b) Select the velocity graph, and do a linear fit. The program
will return a slope which is the average value of g and a quantity after the  sign which
the is the standard deviation in the value of g.
Newton’s Second Law
1.
Click on the first digital input, and select the sensor labeled “Motion Sensor” from the
menu that appears.
2.
An icon showing the Motion Sensor will appear attached to the interface. Below the
interface image are the sensor configuration options. Configure the sensor in the
following way: in the “Measurement” tab, select “position” and “velocity” and de-select
“acceleration”. Click on the “Motion Sensor” tab see if the sensor is working, it should
at this point read the distance of the glider on the track, if not see the TA, set the
sampling rate to 20 Hz. You are now ready to make the measurements. You can
minimize the experiment setup screen.
3.
Drag the “Position, Ch 1” data set from the Data panel (top left) and drop it onto the
Graph icon in the Displays panel (bottom left). Repeat for velocity.
4.
You'll want to show Position and Velocity graphs. Set up the display as before.
5.
To start the run, select one student to hold the air car in place and release it when a
second student clicks the start button. You'll want to stop the run before the air car hits
the end of the track and before the masses hit the ground.
6.
Again, rescale the graphs so that your data is most clearly displayed.
7.
Analyze your data as before, and print a sample of the position and velocity graphs.
8.
To fit a curve: In the graph window, click the “Fit” button and select the fit type from the
pull-down menu. In calculating g from the position vs time data choose “quadratic”.
This will fit the data with a quadratic and a box will appear with the value of the
coefficients for each power. A is the coefficient for the quadratic term. Proceed
similarly for the velocity but call for a “linear fit”. The program will return the slope and
standard deviation (the quantity following the slope) for the average acceleration and an
RME (Root Mean Square Error). Disregard the RME. The “slope” can be taken to be
the average value of the acceleration to be used in Eq. (1).
9.
Record the value of the slope and the associated standard deviation (which often is taken
as the standard error).
10.
If you are interested in the drag of the system, curve fit the first 6 points on the position
graph to a quadratic and the last 6 points in the position graph to measure the change in
fit to the graph.