Centripetal Force and Acceleration

ObjectiveIn 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

PART 1 – FORCE VS. VELOCITY

Procedure1. 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.

Date: 13/06/2014Group members:

1. May2. Patty3. Milk4. Nuey5. Mook

Section: 10-9

Figure 1

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 RADIUSWhen 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.

Procedure1. 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 in3. 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. v2Results

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 v2 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 m2

s2 (the unit ofv2). Therefore, the units of

the slope is N × s2

m2.

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)

Based on the formulaF=m× v2

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 (v2). 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.

F=m× v2

rWhile F = centripetal force

m = mass of the objectv = velocityr = 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

centripetal acceleration and ‘a’ is also equal to v2

r. Then if we want to find the

acceleration of the object undergoing circular motion, we just find it by dividing

the value of centripetal force by the mass of that object or a=Fm . Moreover, we

can also find the centripetal acceleration by using v2

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

According to the experiment and also the formula F=m× v2

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 andv2increases, 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 v2 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 fromhttp://www.vocabulary.com/dictionary/centripetal