Centripetal Force and Acceleration
Date: 13/06/2014
Group members:
1. May
2. Patty
3. Milk
4. Nuey
5. Mook
Section: 10-9
Objective
In this experiment, you will
• Collect force, velocity, and radius data for a mass undergoing uniform circular
motion.
• Analyze the force vs. velocity graphs.
• Determine the relationship between force, mass, velocity, and radius for an
object undergoing uniform circular motion.
• Use this relationship and Newton’s second law to determine an expression for
centripetal acceleration.
Materials
- LabQuest
- Vernier Photogate
- Vernier Dual-Range Force Sensor
- Vernier Centripetal Force Apparatus
- Masses
Figure 1
PART 1 – FORCE VS. VELOCITY
Procedure
1. Attach a Dual-Range Force Sensor and a Vernier Photogate to the Vernier
Centripetal Force Apparatus (CFA), as shown in Figure 1.
2. Connect the force sensor and the photogate to LabQuest.
3. Set up data collection.
a. Tap Mode. The default Photogate Mode►Motion works for this
experiment. Select
b. To control when data collection stops, tap the option labeled “with the
Stop button” in
c. From the Graph menu select Graph 1. Change the axes of the graph so
that it shows
d. Tap the Meter tab.
4. Determine the mass of the sliding mass carriage. Add mass to both the sliding
and fixed mass carriages as directed by your instructor. The mass of the sliding
and fixed carriages should be the same so that the beam is balanced. Record the
total mass of the sliding carriage and extra mass.
5. Position the fixed carriage so that its center is 10 cm from the axis of rotation.
Adjust the position of the force sensor on the rail so that, when the line is tensed,
the center of the sliding mass carriage is also at 10 cm.
6. Zero the force sensor.
7. Spin the beam by twisting the knurled spindle of the CFA with your fingers.
When the force reaches 3–4 N, begin collecting data. Let data collection continue
for 20‑40 s. When you stop data collection, use the friction between your hand
and the knurled spindle to stop the beam.
8. If your graph of force vs. velocity is not linear, create a “calculated column”
that will enable you to obtain a linear graph (linearized graph F vs. v2 step how
to do this if you cannot do it by yourself.
PART 2 – INVESTIGATING THE EFFECT OF MASS AND RADIUS
When a quantity (in this case, force) is a function of more than one
variable, it is usually the case that the slope of the graph is related to the
parameters held constant during the experiment. Examine the units of the slope
of your graph of F vs. v and radius that has the same units as that of your slope.
Substitute the known values of these parameters; how closely does the value of
this expression agree with that of your slope? Predict the effect of doubling the
mass on the value of the slope. What effect would doubling the radius have on
the slope? You can test your conclusions by varying first the mass and then the
radius as follows.
Procedure
1. Change the mass on both the fixed and sliding carriages and record the value
of the total mass of the sliding carriage and any extra masses.
2. Re-zero the force sensor, then spin the beam as you did before. Once the force
reaches 5–8 N, begin collecting data. When you stop data collection, stop the
beam as you did in
3. Change the system mass again and record the value of the total mass of the
sliding carriage and any extra masses, then repeat Step 2.
4. Return the mass on both the fixed and sliding carriages to the original value
used in Part 1. Decrease the radius of both the sliding and fixed mass carriage by
2-3 cm. Record the value of the radius. Adjust the value of the User defined
distance in the Mode window so that it is 1/10 of the circular path of the mass
carriage. A warning will appear. Choose to discard the data.
5. Re-zero the force sensor, then spin the beam as you did before. Once the force
reaches 5–8 N, begin collecting data. When you stop data collection, stop the
beam.
6. Now, change the radius so that it is 2–3 cm greater than your initial value. Be
sure to adjust the User defined distance as you did in Step 5. Record the value of
the radius, then repeat Step 6 and 7.
7. Repeat step 8 of last section to linearize your last plots. Again, create a
calculated column for v2. Use this calculated column to plot F vs. v2
Results
The run 3 graph (radius=0.10m, mass=100g)
The run 4 graph (radius=0.10m, mass=200g)
The run 7 graph (radius=0.10m, mass=300g)
The increased radius graph (radius=0.13m, mass=200g)
EVALUATION OF DATA
1. Open Logger Lite in your computer and connect the LabQuest interface
to your computer. Import the data from your experiments.
2. Choose one set of data and plot F. v. V (not linearized). Write a statement
that describes the relationship between the force acting on the carriage
and its tangential velocity.
The run 4 graph (radius=0.10m, mass=200g)
According to the graph, the x-axis of graph is its tangential velocity while
the Y-axis of the graph is the force acting on the carriage. Base on the shape of
graph, the slope of the graph is positive and the graph is a curve line, which is
parabola, not a linear. From the graph, it shows that when the force increases,
the velocity also increases too so that makes the line moves up likes this.
Therefore, we can conclude the relationship between the force and the velocity
that while the velocity is increasing, the centripetal force of that object will be
increased and the relationship of the graph is a kind of quadratic.
3. Set the graph to show the linearized plot (use the calculated column v2
the line that best fits your linearized graph. Determine the units of the
slope; simplify these as much as possible.
The run 4 graph (radius=0.10m, mass=200g)
The equation of linear line is y = mx+b, while m is the slope of the graph, b
is the y-intercept, y is equal to the force and x is equal to 𝑣 2 so the equation of
this graph is y = 2.477x+0.01195. The slope of the graph is calculated from
dividing delta Y by delta X so we will divide N by
𝑠2
the units of the slope is 𝑁 × 𝑚2 .
𝑚2
𝑠2
(the unit of 𝑣 2 ). Therefore,
4. Examine the value of the slope of each of the first three runs (linearized
plots). What relationship appears to exist between the value of the slope
and the total mass of the sliding carriage?
The comparing three slopes (run3,4,7) graph
According to the graphs, the value of slope of the third run (mass = 100g)
is equal to 1.492 while the value of slope of the fourth run (mass = 200g) is equal
to 2.477 and the value of slope of the seventh run (mass = 300g) is equal to
3.523. You will see that if the total mass of the sliding carriage is increased, the
value of slope of the graph will increase too. This is because when we increase
the mass of the object, the centripetal force will be increased so it affects to the
graph in the Y-axis and makes the value of slope increases. Therefore, we can
conclude the relationship between the value of the slope and the mass of the
object that adding more masses will affect in increasing the value of slope of the
graph and the relationship of the graph is a kind of linear.
5. To study the effect of changing the radius, select one of the runs in which
you changed the radius from its original value. Compare the value of the
slope for this run to that for your first run (r = 0.10 m).
The first run (radius=0.10m, mass=200g)
The second run (radius=0.13m, mass=200g)
𝑣2
Based on the formula 𝐹 = 𝑚 × 𝑟 , if we increase the radius of the object
from its rotational center point, the velocity of the object will be decreased that
will affect in decreasing the value of the slope of the graph between the force and
the calculate column (𝑣 2 ). But according to these two graphs with its mass is
constant (both 200g), the value of slope of the first run (radius = 0.10m) is equal
to 2.477 while the value of slope of the second run (radius=0.13m) is equal to
3.503. That means from the result of the graph, when we increased the radius of
the graph, the value of slope of that graph--in contrast--increased. Therefore, we
concluded that it must have some mistakes when we were doing the experiment.
We might add too much force on the carriage or might measure something
wrong.
6. Repeat Step 5 for another run in which you changed the radius. Does the
change in the radius have the expected effect on the value of the slope?
Compare your findings with those of other groups in class.
After I have compared my findings with the other groups, my changed
radius graph doesn't have the expected effect on the value of the slope likes the
other groups' graphs due to some mistakes were happened. While the values of
slope of the other groups are decreased when increasing radius based on the
formula and their graphs, my group's value isn't.
7. Write an equation relating the net force, mass, radius and velocity of a
system undergoing circular motion.
𝑣2
𝐹 =𝑚× 𝑟
While F = centripetal force
m = mass of the object
v = velocity
r = radius
8. Use what you have learned and Newton’s second law to write an
equation for the acceleration of the object undergoing circular motion.
According to the Newton's second law, the formula of linear motion is F =
ma which ‘F’ is the force, ‘m’ is the mass and ‘a’ is the acceleration of the object.
In the rotational motion we can also apply the same formula as the linear motion
but ‘F’ will be converted to the centripetal force while ‘a’ will be converted to the
𝑣2
centripetal acceleration and ‘a’ is also equal to 𝑟 . Then if we want to find the
acceleration of the object undergoing circular motion, we just find it by dividing
𝐹
the value of centripetal force by the mass of that object or 𝑎 = 𝑚. Moreover, we
𝑣2
can also find the centripetal acceleration by using 𝑟 formula too. From
vocabulary.com website, they define the 'centripetal' as "Centripetal is an
adjective describing a force that brings things toward the center, not unlike the
force of a black hole."(Vocabulary.com, 2014)
Extra points
According to the Newton’s first law of motion, when the object is moving,
it wants to keep going continually likes that, unless there is the force acting on
the object. This law also applies to the rotational motion too, but we call it the
centrifugal force that is not the real force because it is just an illusion force of our
brains. An example of the centrifugal force that I have experienced from my real
life is I used to play on the chair by sitting on it and spinning the chair. When I
was spinning, although I spin myself to the left side, my body tended to bend to
the right side. This is because the centrifugal force pushed me to the opposite
side. However, this force isn’t real so we might better explain it in terms of
centripetal force. This interaction occurs because when I was rotating, there was
the force that pushed me to the rotational center point called ‘centripetal force’
but my body wanted to maintain my self to the former direction that I used to
stay, according to the law of inertia, so my body tried to bend to the opposite side
to balance the force.
Conclusions
𝑣2
According to the experiment and also the formula 𝐹 = 𝑚 × 𝑟 , we can
conclude that the mass of the object and the radius of that mass from its
rotational center point both affect to the velocity of rotational motion. Increasing
the mass of the object will affect in increasing the velocity that makes the
centripetal force be increased so the value of slope of the graph between force
and 𝑣 2 increases, obviously seen from the comparing three slopes (run3,4,7)
graph. In the other hand, increasing the radius of the mass from its rotational
center point will affect in decreasing the velocity that makes the value of slope of
the graph between the force and 𝑣 2 decreases but due to my graph, there are
some errors which make the result changes because the value of slope of the
increased radius graph comparing with the first one increases. This is because
my group has made some mistakes during the experiment. However, we promise
that next time we will improve our experiment to get better result and minimize
errors by doing the experiment more circumspectly and trying to maintain the
time when doing the experiment because when we hurriedly do the experiment,
we can make the mistakes easily. Moreover, based on the result and the
knowledge we have learned from this experiment, I think we can do some more
other experiments in the future such as change from the horizontal rotation likes
what we have done in this experiment to the vertical rotation by trying to
increase the mass and the radius as same as this experiment, then trying to
compare the result of these two experiments and also trying to find that what
makes the result changes.
Reference
Vocabulary.com. (2014). Centripetal. Retrieved June 17, 2014 from
http://www.vocabulary.com/dictionary/centripetal