AH-04 ONE DIMENSIONAL MOTION-1 (1) (1)
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AH-04
ONE DIMENSIONAL MOTION
Rev 3/28/2021
OBJECTIVE
The purpose of this Lab is to verify the equations of one dimensional motion. This will be done by
measuring the distance, time and velocity of an object that moves with constant acceleration,
and hence calculating the value of acceleration due to gravity by using these equations.
MATERIALS
1.
ME-6960 PasTrack
2.
ME-1240 Smart Cart
3.
ME-9495A Angle Indicator
4.
ME-8971 End Stops
5.
Capstone software
THEORY
An object in one-dimensional motion under constant acceleration satisfies the following
equations of motion:
v
x
f
=
v
xi
+
a
x
(
t
f
−
t
i
)
(1)
x
f
=
x
i
+
v
x
i
(
t
f
−
t
i
)
+
1
2
a
x
(
t
f
−
t
i
)
2
(2)
v
x
f
2
=
v
x
i
2
+
2
a
x
(
x
f
−
x
i
)
(3)
Where
x
i
= initial position on the track (at time =
t
i
)
x
f
=
final position on the track (at time
=
t
f
)
v
xi
= Initial velocity in the x-direction (at time =
t
i
)
v
x
f
= Final velocity in the x-direction (at time =
t
f
)
a
x
= Acceleration in the x-direction (which is constant, not a function of time)
In the absence of air resistance, objects falling under
the influence of gravity have a constant downwards
acceleration. On and near the surface of the Earth this
acceleration has a value of approximately 9.81 m/s
2
which is denoted by the symbol ‘g’. We will study the
distance, time and velocity of an object as it slides
without friction on an inclined plane, and hence
determine its acceleration for the two cases. As seen
in the figure, for an inclined plane, the value of
acceleration along the plane will be:
a
x
=
g
sin
(
θ
)
(4)
g
g Cosθ
θ
g Sinθ
Equation (1):
v
x
f
=
v
xi
+
a
x
t
f
Here
v
xf
is the velocity at time
t
f
and
v
xi
is the velocity at time
t
i
=
0
. The equation
of a straight line is:
y
=
b
+
mx
Equation (1) is an equation of a straight line, if we take
v
xf
as the y-axis and
t
f
as the x-
axis. The slope of the line “
m
” will be the acceleration
“
a
x
”,
and the y-intercept “
b
” is
“
v
xi
”.
From the smart cart we can get the velocities at different times, and plot a graph between
v
xf
on the y-axis and
t
f
on the x-axis. The slope of the line will give us the value of
a
x
.
For the motion on the inclined plane, it should come out to
a
x
=
9.8sin
θm
/
s
2
(with
x
along the
inclined plane).
Equation (2):
x
f
=
x
i
+
v
x
i
(
t
f
−
t
i
)
+
1
2
a
x
(
t
f
−
t
i
)
2
We will measure the position of the cart. The best fit line for the graph of
x
f
on the y-axis and
time on the x-axis will yield a parabola. This parabola equation will have form:
x
f
=
At
2
+
Bt
+
C
And acceleration of the cart is seen to be equal to
2
A
.
Equation (3):
v
x
f
2
=
v
x
i
2
+
2
a
x
(
x
f
−
x
i
)
A plot of
v
x
f
2
on the y-axis Vs position (i.e.
x
f
¿
on the x-axis should come as a straight line
with the slope being
2
a
x
and y-intercept being
v
x
i
2
.
Equation (4):
a
x
=
g
sin
(
θ
)
Once the value of acceleration for the cart (i.e. “
a
x
”
) is obtained, Equation 4 can be used to
find the value of “
g
”. If “g” comes out close to the correct value, that means that “
a
x
“, and
hence equations 1, 2, and 3 are correct.
The procedure will involve finding the values of acceleration of the cart i.e. ”
a
x
by the three
equations, and from them, finding the value of “g”. If the value of “g” comes out good, then we
can assume that
a
x
is good, which means the equations are good.
PROCEDURE:
1.
Set up the track so that it is tilted to an angle
θ
about 3° to 5
⁰
. You can place a book at one
end. Better to put something under the middle legs to that all six legs carry the weight of the
track.
2.
Attach an end-stop at the lower end to prevent the cart from rolling off to the ground, and
place some soft item on the track before the end stop so that the cart does not hit it. Keep its
magnets pointing away from the track.
3.
Click on the PASCO Capstone icon on the computer to open the software to use in this
experiment. In the Tool Palette (on left side of screen), click “Hardware Setup”. This will open
the Hardware Panel. Then press the power switch on the cart to turn it on (the red led should
start blinking). In “Searching for Wireless Devices”, click the Bluetooth icon. Capstone should
detect all Bluetooth devices that are nearby. Select the Smart Cart with the serial number of
your Smart Cart. Click on it. Your instrument is now connected to the Software.
4.
The Options in the Smart Cart will appear. Select “Smart Cart Position Sensor” and turn off
the rest. Click Hardware Setup once again. This will close the Hardware Panel, and you can
now select the type of display (graphs and/or table) from the Display Panel. Select “Sensor
Data”.
Equation (1):
5.
Select Velocity on the Y-axis, and Time on the X-axis. If you see a graph and a table, minimize
the table. You can adjust the size of the graph, and the scale of the X- and Y-axes.
6.
At the lower left of the screen is the ‘RECORD’ button. Start ‘RECORD’ and release the Smart
Cart from near the top of the track. Press the same button again to stop the data recording
after the cart reaches the bottom of the track. The data for velocity of the cart should show
on the graph. (
note: if the Velocity is negative, rotate the Cart 180° and retry
).
7.
Click on the “Highlight Range…” icon to get a colored square on the screen. On the graph,
move and adjust its size so that a portion of the data that is “good” is inside the box. Then
click the icon for Curve Fits.
8.
Observe the data on the V-T curve. Equation (1) indicates that it should be a straight line. The
slope on the V-T curve is the value of acceleration of the cart ‘
a
x
’. The value of the
regression coefficient i.e. R, indicates how good the data fits the equation chosen. R should
be greater than 0.95 for the fit to be acceptable. Note the value of the slope in the data
table.
9.
Repeat steps 6 to 8 a few times till you are adept at doing this. Then clear all data from the
software, and repeat the experiment at least four times (with different angles and masses on
the cart – your choice of values). Take average of the values of acceleration as found, and
use it to find value of acceleration due to gravity ‘g’. Find the percent error in ‘g’.
(Trick: You can press RECORD and roll the cart four times and then stop recording. You will get all
four data sets on the same screen).
Equation (2)
10. Now change the variable on the Y-axis to ‘Position’. The data that you have already taken will
show on the X-T curve (no need to roll the cart again). Equation (2) indicates that it should be
a parabola. On this curve, the quadratic fit will give the equation
X = At
2
+ Bt + C
. The value
of acceleration
a
x
of the cart will be equal to
2A
. Note the value of acceleration in the
data table.
11. Get the acceleration for all four runs, take their average, and find the value of ‘g’ from the
average acceleration
a
x
. Find the percent error in ‘g’.
Equation (3)
12. Change the Y-axis back to velocity. Click the icon for Velocity. This will open a box. Select
“QuickCalc”. In QuickCalc, select V
2
. Change the X-axis from Time to Position. You now have
the V
2
versus Position graph. Fit a straight line to this and obtain the value of “
2
a
x
” from
the slope. Hence find the acceleration
a
x
.
13. Do this for all four data sets, and get the average value of
a
x
,
g and percent error.
Note
:
1.
1n the software, instead of manually doing start and stop, you can set up automatic start and
stop conditions. Click “Recording Conditions”, and set the start condition to when the cart has
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moved, say 10 cm, and a stop condition when it has moved, say, 80 cm. Then data recording
will begin when the cart has moved 10 cm from where you release it, and stop when it reaches
80 cm.
2.
Attach a) graphs showing the curve fits, b) Pictures of your setup.
3.
Uploading a video of your experiment being performed will get you 5 points extra credit.
AH-04
One-Dimensional Motion
REPORT
FORM
Date:
_______________
Inclined Plane
T r I a l
N u m b e r
Average
value of ‘g’
Percent
Error in
‘g’
1
2
3
4
Angle of incline
X
X
Mass of the cart
X
X
‘
a
x
’ from velocity versus
time graph (m/s
2
)
X
X
Value of “g” from a
x
(m/s
2
)
‘
a
x
’ from the position
versus time graph
(m/s
2
)
X
X
Value of “g” from a
x
(m/s
2
)
‘
a
x
’ from the V
2
versus
Position graph
(m/s
2
)
X
X
Value of “g” from a
x
(m/s
2
)
RESULTS
Found by Inclined Plane in this Lab with
Found by
Picket
Fence in
earlier Lab
Found by
Pendulum
in earlier
Lab
X-T Graph
V-T Graph
V
2
-X Graph
Value of ‘g’
(m/s
2
)
Percent
Error in ‘g’
SAMPLE DATA
Sample data and corresponding graphs for one data is shown. The angle and mass are not correct
T r I a l
N u m b e r
Average
value of ‘g’
Percent
Error in
‘g’
1
2
3
4
Angle of incline
3°
X
X
Mass of the cart
280 g
X
X
‘
a
x
’ from velocity versus
time graph (m/s
2
)
0.434
X
X
Value of “g” from a
x
(m/s
2
)
8.29
‘
a
x
’ from the position
versus time graph (m/s
2
)
0.217*
2 =
0.434
X
X
Value of “g” from a
x
(m/s
2
)
8.29
‘
a
x
’ from the V
2
versus
Position graph (m/s
2
)
0.869/
2 =
0.435
X
X
Value of “g” from a
x
(m/s
2
)
8.31
AVERAGE VALUE OF “g”
Figure 1: Velocity – Time Graph
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Figure 2: Position – Time Graph
Figure 3: V
2
– Position Graph.
ADDITIONAL INFORMATION
See these videos for additional information on this experiment.
Using time of fall
2.35 min
https://www.youtube.com/watch?v=wBIydqBHFes
POINTS TO THINK ABOUT
1.
Your results and their errors.
2.
Are your results within ‘acceptable’ range of error (
what is an ‘acceptable’ range
?)
3.
Do your results verify the equations of motion within acceptable errors?
4.
What are the most likely sources of error, and how can the errors be reduced?
5.
If so, why are the values of ‘g’ different in the two graphs (X-T and V-T).
6.
What was the effect of angle and mass on the acceleration found from the inclined plane,
and how does this compare with what is expected from the equations of motion?
7.
Which of the two methods did you find to give better results.
8.
How can the accuracy of this experiment be improved?
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Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Engineering Mechanics: Statics
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ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY