Download gravitational acceleration

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Gravitational Acceleration
GRAVITATIONAL ACCELERATION
Equipment: Pasco Track, Pasco Cart, Vernier Motion Sensor, Logger Pro 3.1, Lab Pro
Interface, Meter Stick, Cart Masses, Wood Blocks
About this lab:
Neglecting friction,
> The assumption that gravitational force is a vector is tested by examining the agreement
(or disagreement) with the assumed sin(θ) variation of acceleration with incline angle θ,
for constant mass.
> The assumption that inertial and gravitational masses are proportional (equal, in our
units) is tested by examining the constancy (or non-constancy) of acceleration with
gravitational mass, for constant angle.
> The assumption that the gravitational force does not change as the cart moves closer to
the center of the earth is tested (within large uncertainties) by the quality of the fits to
1
constant acceleration expressions. (In reality the force increases as
2 , but the %
r
variation with height in r 2 is very small in the lab.)
Reference:
Cutnell & Johnson: Physics, 6th: Figure 4.34, 4.35
Study the Mechanics applet at:
http://www.ngsir.netfirms.com/englishhtm/Motion.htm M4: Inclined plane
(Scroll down to the bottom. From the mechanics menu, select M4 (inclined plane).
Start with zero friction and rebound at A. "Back" resets and permits new input.
You can input negative initial velocities, but the graph only shows the return
(downhill) motion, not the initial uphill motion. The velocity at top on downhill
return is the same in magnitude as initial, only reversed. Rebound at A is also
elastic, i.e. (scaler) energy conserving and (vector) velocity reversing.
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Gravitational Acceleration
Figure 1 Kinematic quantities in frictionless inclined plane motion
Note that you can alter the plane by dragging on the large red dots at top or bottom
of the incline (reset first with Back). And, clicking anywhere on the graph window
switches the display of vector quantities between the kinematic quantities (velocity,
acceleration) and the force components.
Observe the changes in the kinematic quantities. Pay particular attention to the
spacing between adjacent points (0.2 second intervals). Pause for examination.
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Gravitational Acceleration
Figure 2 Forces acting in frictionless inclined plane motion
In terms of the block position, study the forces acting. Pay particular attention to
the "Normal" force exerted by the plane on the block. Observe the kinetic and
gravitational potential energies. Try to interpret the changes in these quantities. (If
you have not yet discussed energy in your lecture class, just observe them for future
reference.)
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Gravitational Acceleration
Figure 3 Ballistic Projectile Motion
Study the horizontal spacing between successive projectile positions in this figure.
Is there any horizontal acceleration? What about the frictionless, inclined plane
motion? Why is there horizontal acceleration in the inclined plane motion, but not
in this ballistic motion?
(Consider all the forces acting on the block/projectile in the two cases)
Early in the seventeenth century, Galileo experimentally examined the concept of
acceleration to learn more about freely falling objects. Lacking precise timing devices (he
used his pulse sometimes) he limited acceleration by using fluids, inclined planes, and
pendulums. Similarly, you will study the ramp angle dependence of gravitational
acceleration, and then extrapolate to the acceleration on a “vertical ramp” (free fall).
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Gravitational Acceleration
x
g
h
gcos 
Figure 2
The gravitational force component perpendicular to the track is mg
cos(θ), and the component along the track is mg sin(θ) (whether the cart moves up
or down the track). The net acceleration perpendicular to the track is zero ( the
normal force cancels one component of the gravitational force). Assume no track
friction, for simplicity.
sin(θ) = h/x (x measured along track)
To the extent that the track is frictionless, changing the angle “tunes” the
effective earth's gravitational force of attraction: mg sin(θ) .
The accelerating force is gravitational. i.e. proportional to the gravitational
mass of the accelerated object. The acceleration is inversely proportional to
the inertial mass of the accelerated object, so
To the extent that the track is frictionless, changing the accelerated mass tests
equivalence (equality, in our units) of gravitational and inertial mass.
Gravitational mass enters on the force side of Newton's second law ; inertial mass enters
on the response side of Newton's second law:
Force = m grav *g sin(θ)
Response = m
inertial*acceleration
.
Sonic ranger d vs. t data enables calculation of cart velocity and acceleration, by
successive numerical differentiation with respect to time.
Study
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Gravitational Acceleration
> Acceleration as a function of incline angle (a vs. sin(θ)), at fixed mass
> Acceleration as a function of mass (a vs. m), at fixed angle.
GENERAL PROCEDURE:
Open Logger Pro: Gravitational Acceleration.cmbl. Aim the sonic ranger on narrow
beam.
1) Keep the cart load fixed (larger may be better, to reduce relative importance of
friction) and vary the angle. Record and analyze several runs at various angles. For each
good run, measure the height difference between the two track ends (h) and calculate
sin(θ) = h/x (x along track, hypotenuse of triangle). Obtain a value of cart acceleration in
three ways (from distance, velocity and acceleration graphs.)
(For displacement, Analyze: Curve Fit: Quadratic; use twice the fit coefficient of the t2
term for acceleration. For velocity, Analyze: Linear Fit; use the slope. For acceleration,
Analyze: Statistics; use the mean. In each analysis, select data points that look reliable.
See the graph example below.)
Make a neat data table showing your three a values, h. x, and sin(θ) = h/x.
For each good run, enter your best estimate of acceleration a and of sin(θ) in the data
table on Page 3 of the Logger Pro program. (You may average all three values of a, or
reject some if you think them less reliable.) Graph a vs sin(θ). Analyze: Linear Fit.
Compare the slope to the accepted value of g (9.80 m/s 2). Record the ratio: R 1 =
slope/9.80 = g exp/g accepted .
Print the graph of cart acceleration a vs. sin(θ) and submit. Write on it (front or back):
your ratio R1 . (Wheel friction may reduce g
exp
.)
An alternative determination of g exp from this Logger Pro graph (Page 3) is from the
value of the extrapolated linear curve fit curve at 90 o (sin(θ) = 1, corresponding to free
fall - a long extrapolation). Place the cursor on the extrapolated curve at sin(θ) = 1 and
read the value of a. Record the ratio R2 = a(900) /9.80 and enter it also on your graph.
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Gravitational Acceleration
2) Now keep the angle fixed and vary the cart load. (Larger θ may reduce the relative
importance of friction.) Take data as before (a from distance curve, from velocity curve
and from acceleration data).
Make a table of a average vs. mass. (Don't forget to include the mass of the cart.)
Examine it for any evidence of systematic dependence of acceleration on mass. Plot a
average vs. mass m if you wish. Write your conclusions on a graph.
*If you launch uphill the cart will slow down, stop, then speed up while returning.
(DON’T LET IT HIT THE SONIC RANGER!) v will change sign. Will a?
Figure 4 Experimental arrangement
Procedural details:
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Gravitational Acceleration
Click the Lab Pro button and verify that the motion sensor is connected to the correct
Dig/Sonic input. On Page 2 of the Logger Pro file, double click the Velocity and
Acceleration headers and check the definitions of these quantities. Look in the
Experiment menu: Data Collection and check that it is set for 15 samples per second for 2
seconds. (You may experiment with these initially to find settings that you like. Uphill
launches may take longer times. Acceleration may look pretty choppy at higher sampling
rates.)
Spin each cart wheel before beginning work. If one wheel shows excessive friction,
consult your instructor.
Remember that the motion sensors have a minimum range, ~ 25 cm.
Don't let the carts fall on the floor, or hit a sensor!
In analyzing a run, autoscale graphs. If you don't want to display a portion of your data,
select and magnify (or use Edit: Strike Through Data Cells (reversible)). (Select rows by
dragging down the row number column, or select the first row, hold down the Shift key
and select the last.
(For a spherically symmetric earth, the gravitational acceleration varies as 1/r2 (r
from center). But, for vertical drops small compared with earth radius R, the
assumption of constant acceleration is very good, though not exactly correct. The
equations above follow the constant acceleration assumption.
The constant acceleration assumption would be very poor, however for a satellite
with a very eccentric orbit.)
Observe the quality of the fits (to valid portion of graphs).
Make a neat data table: Run number, mass, three acceleration values for each run (a d, a V,
a acc), the average acceleration (a av) (you may reject one if you think it is poorly
determined, but not just because you don't like the number), and the corresponding value
of sin(θ). When you are finished, copy this data table onto your report form. Enter your a
av and sin(θ) values in the Page 3 table of the Logger Pro program. Analyze this: Linear
Fit.
Report
Your report should contain: your data table (neat and legible), your plot of
a av vs. sin(θ) and a plot of a average vs. mass (if your instructor specifies), one data graph
of d, v, and a vs. time showing fits, names of all partners, section/instructor, date, with
your ratios R1 and R2 and your conclusions regarding acceleration dependence on mass.
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Gravitational Acceleration
Figure 5
Kinematic curves. All curves are fit over the same time region. Note
that acceleration does not seem constant over the entire run. This may be due to
friction. The most constant region of the acceleration curve may provide a fittingregion guide.