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LPC Physics
Centripetal Force
Centripetal Force
Purpose:
The purpose of this lab is to measure centripetal force and compare it with predicted
values, and to demonstrate that a centripetal force acts to hold an object in a circular
orbit.
Equipment:
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Centripetal Force Apparatus
Hooked Mass Set
Stopwatch
Balance
Carpenter’s Level
Meter Stick
Photogate, LabPro
Theory:
The change of velocity of an object is, by definition, an acceleration. If an object is to
accelerate there must be a force acting on the object which points in the direction of the
acceleration (in the direction of the velocity change). Velocity is a vector quantity,
meaning it has both a magnitude and a direction. A change in velocity (that is, an
acceleration) may be due to a change in the magnitude (faster or slower), a change in
direction, or both.
If an object is moving in a circle, it necessarily undergoes an acceleration as can
be seen by its constant change of direction.
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vi
∆v = vf - vi
vf
Figure 1 Subtraction of vi from vf gives ∆v, or acceleration
In an exaggerated example, an object traveling counterclockwise in a circle will change
the direction of its velocity as it moves from its starting point at 12 o’clock to its final
point at 9 o’clock. The change in velocity, ∆v = vf - vi is shown pictorially above as
pointing directly toward the center of the circle. This means that the object is
accelerating toward the center of the circle, which means there must be a force pointing
toward the center of the circle. This force is known as a centripetal force. Centripetal
means “center seeking”, and should not be confused with the more commonly known socalled centrifugal (“center fleeing”) force which is actually not a force at all.
A note on vector subtraction:
Vectors are added “tip to tail”, meaning that the end (with no arrow) of the second vector is placed at the tip
(with the arrow) of the first vector and the resultant vector is drawn from the end of the first vector to the tip of
the second. Just like with “regular” numbers, it doesn’t matter which is the first and which is the second vector.
v1 + v2 :
v1
v2 + v1:
v2
v2
vf
v1
v1
v2
vf
Vectors are subtracted by placing the ends (no arrow) of the two vectors together, and drawing the resultant
vector from the tip of the second to the tip of the first. Just like with “regular” numbers, it does matter which is
the first vector and which is the second vector. Transposing the two results in a negative answer.
v1 - v2:
v2 - v1:
v1
vf
v1
vf
v2
v2
If an object is to travel in a circle, the centripetal force must be equal to
Fcentripetal
mv 2
=
r
where
m = the mass of the object
v = the velocity of the object
r = the radius of the circle
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Eq. 1
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Centripetal Force
From this equation you can see that to travel in a circle of smaller radius requires a larger
centripetal force, as does a larger mass, and a higher velocity. So what happens if an
object is attempting to trace a circle of radius r1 a constant velocity, and the centripetal
force that exists is not large enough to meet the condition in Eq. 1?
velocity
At any point around a
circle, the instantaneous
velocity is pointed tangent
to the circle. Thus, an
object’s instantaneous
momentum points tangent
to the circle, and upon the
removal of a force, the
object will continue to
move in the direction of
its momentum.
r1
r2
If the centripetal force is
not great enough to keep
the object in a circle of
radius r1, the object’s
tangential momentum will
Path of object carry it outward until it
reaches the radius that will
drift
make Eq. 1 true. In
essence, the object drifts
outward to “find” the
r1
appropriate radius.
If the force were removed entirely, the object would travel in a straight line. Because
in this case there still exists a force, the path of the object is curved, and eventually (at
the right radius) becomes a circle again.
Figure 2 An object will drift to a circle of larger radius if the
centripetal force acting on it is too small
Conversely, if the force applied is greater than that required to push an object in a circle
of radius r1, the object will be pushed inward to a smaller radius that meets the condition
of Eq. 1.
But wait, you might say, Eq. 1 includes velocity as well as radius. And we all
know that a force may change an object’s velocity. So why mightn’t, in the first case
above, the object just decrease its velocity rather than move to a larger radius? The
velocity won’t change for the same reason that gravity will never change the horizontal
velocity of a projectile…a force must be at least partially in the object’s direction of
motion to change its velocity. If a force acts perpendicularly to an object’s direction of
motion, the velocity will neither increase nor decrease, but the force will change the
direction of motion.
It is very important to remember (even more to realize for the first time) that
Centripetal Force is not a force in the same way that gravity, rope tension, the normal
force, etc. are forces. The centripetal force is simply a label for any of these other forces
that happen to be causing an object to move in a circle. In the case of the Earth orbiting
the Sun (in a mostly circular fashion), we don’t label the force of gravity and the
centripetal force acting on the earth to push it in a circle. We say that the force of gravity
acts on the Earth, and acts as a centripetal force to push the Earth in a circle.
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The Earth-Sun System
Wrong!
Fcentripetal
Fgravity
Sun
Earth
Ftotal = Fcentripetal + Fgravity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Right!
Fcentripetal = Fgravity
Sun
Earth
Ftotal = Fgravity which is acting as Fcentripetal
Figure 3 The exerted on the Earth by the Sun: Right and Wrong interpretations
The same could be said about a ball on a string being swung in a circle. Because the ball
is traveling in a circle, there is a centripetal force acting on it. The force that is causing
the centripetal acceleration is the force of rope tension.
A Ball on a String
Right
Wrong
ΣF = Frope tension which is
acting as Fcentripetal
ΣF = Fcentripetal + Frope tension
Ftotal
Ftotal
Frope tension
Fcentripetal
Frope tension
is
Fcentripetal
Figure 4 Centripetal forces on a tennis ball: Right and Wrong interpretations
And this is the way it will go every time you have an object that travels in a circle. The
first step is recognizing the circular motion, and realizing that a force (centripetal force)
is necessary to push the object in a circle. The next step is determining what type of
force is acting as the centripetal force. Any force you can think of can act as a centripetal
force.
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Centripetal Force
So what is this lab all about? You’re going to determine the centripetal force necessary
for an object of known mass to travel in a circle of a certain radius at a calculated
velocity. If you’ve been to a carnival (or to Camp Snoopy™ at the Mall of America),
you may have seen the giant swings. About 50 people are lifted 20 feet off the ground
and swung in a large loop around a central axis. As the angular velocity increases, the
rider feels “pushed”outward until the chains are no longer vertical but at some small
angle. At this point, it is the horizontal component of the rope tension that acts as the
centripetal force.
ω
Frope tension
Frope tension, vertical
Frope tension, horizontal = Fcentripetal
Figure 5 A close-up view of forces on a rotating swing
Now imagine that a spring were attached to the swing chair such that when stationary the
chair is pulled in toward the central axis. Now when the chairs are spun, the spring will
stretch until the chair hangs vertically. In this instance the spring force acts as the
centripetal force.
ω
ω=0
Fcentripetal = Fspring
Figure 6
Because the spring force is proportional only to the distance the spring is stretched, once
we’ve stopped spinning we can determine what was the centripetal force on the swing by
applying enough force to stretch the spring the same distance.
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Centripetal Force
ω=0
Fappliedl = Fspring
Figure 7
You will be performing this same experiment on a smaller scale. You are provided with
a rotating arm that has a hanging mass on one end and a counter-weight on the other.
The index on the platform (thin rod) provides a means of measuring the radius of
rotation. A photogate or stopwatch may be used to determine the velocity of the rotating
mass. From this you will calculate a “theoretical” centripetal force. Once the spinning
has been completed, you will stretch the spring by means of hooked masses strung over
the provided pulley. This will determine the “experimental” centripetal force. You shall
then compare the two.1
Experiment:
1. Level the base of the centripetal force apparatus.
2. Determine the mass, m1, of the hanging mass.
3. Connect the AC adapter to the LabPro by inserting the round plug on the 6-volt
power supply into the side of the interface. Shortly after plugging the power supply
into the outlet, the interface will run through a self-test. You will hear a series of
beeps and blinking lights (red, yellow, then green) indicating a successful startup.
4. Attach the LabPro to the computer using the USB cable that is Velcro-ed to the side
of the computer box (do not unplug the USB cable from the computer!). The LabPro
computer connection is located on the right side of the interface. Slide the door on
the computer connection to the right and plug the square end of the USB cable into
the LabPro USB connection.
5. Connect the Photogate to the DIG/SONIC1 port of the LabPro. If you are using a
one-piece Photogate, a PASCO or very old Vernier Photogate, you will need to use
the digital adapter. If you are using a newer Vernier (with removable cable), simply
remove the cable with the Phono plug, and connect the Photogate Cable with a
British-Telecom plug on one end.
6. Open the file centripetal_force.cmbl (or .xmbl) in the Experiments folder on the
desktop. This will start the program Logger Pro3.3 and bring up the appropriate data
file. If you do not have an auto-ID sensor (which is the likely case), a dialog box will
1
The theory section of this experiment was written by Jennifer LK Whalen
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pop up asking you to confirm the sensors being used. If you have the suggested
sensor attached to the LabPro in the suggested port, click “OK”. If the “OK” button
is not active, ask your instructor for help.
7. Click on ExperimentÆSet up sensorsÆLab Pro, then click on the photogate icon.
Change the setting to “pulse timing.”
8. Place the index in an intermediate location and measure R, the radius of circular
motion.
9. Reposition m1 so that it is directly over the index. Rotate the apparatus and reposition
the counter weight so that it rotates smoothly.
10. Attach the spring to m1.
11. Rotate the shaft so that m1, while rotating, stretches the spring and lines up with the
index. You will need to concentrate on keeping the weight directly over the shaft.
Don’t worry if you are off slightly—but avoid systematic errors (consistently too
short or too long). By repeating the experiment, you can eliminate random errors.
12. Place the photogate so that the beam will be interrupted when the hanging mass
rotates at the proper radius. To avoid damaging the photogate, start with the gate
too low and raise it into position once the mass is revolving at the proper radius.
13. Determine the period of rotation by timing 30 - 50 revolutions (T = t/Nrev). Repeat
at least 3 times.
14. Using the set of weights, determine the force required to stretch the spring the same
amount as when m1 was rotating. While you are at it, try adding and subtracting a
small amount of weight (how much weight can you add or subtract so that you don't
see any measurable change in the spring) to find Fmax and Fmin . These will allow you
to determine your experimental uncertainty.
15. Move the index closer to the shaft (for a smaller radius) and repeat steps 8 - 14.
16. Move the index farther from the shaft, repeat steps 8 – 14.
17. Copy your data from Logger Pro to an Excel spreadsheet or Graphical Analysis to
perform data analysis.
NOTE: Because the pendulum timer expects its beam to be broken twice per period,
your ∆t values are twice what they should be.
Analysis:
1. Starting from Newton’s Second Law, and the equation for centripetal motion:
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F = ma, a =
v2
R
derive the following relationship:
Fc =
4π 2 mR
T2
Eq. 2
where T is the period of one revolution.
2. Using your data, and Excel or Graphical Analysis spreadsheets, determine average
values of the centripetal force for each of the three radii in the experiment. For
convenience’s sake, these will be referred to as the “theoretical values” of the
centripetal force.
3. Determine the standard deviation for each of the values of Fc you calculated in Step
2. This will be your theoretical uncertainty in the experiment.
4. Calculate the force (m2g) necessary to stretch the spring the same amount as provided
by the centripetal force. This will be referred to as the “experimental value” of the
centripetal force.
5. Determine the uncertainty in the experimental value, given by (Fmax - Fmin)/2.
6. Determine the percent difference between the experimental and theoretical values in
this experiment. Determine whether the percent difference between the theoretical
and experimental values is less then the percent uncertainty in each case.
7. If the values are all within the total uncertainty, discuss methods of improving the
experiment so that the uncertainty is reduced. If the values are not within the
uncertainty, suggest possible reasons why. As in all experiments, discuss and
interpret trends or anomalies suggested by the data.
Results:
Write at least one paragraph describing the following:
• what you expected to learn about the lab (i.e. what was the reason for conducting
the experiment?)
• your results, and what you learned from them
• Think of at least one other experiment might you perform to verify these results
• Think of at least one new question or problem that could be answered with the
physics you have learned in this laboratory, or be extrapolated from the ideas in
this laboratory.
Clean-Up:
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Before you can leave the classroom, you must clean up your equipment, and have your
instructor sign below. How you divide clean-up duties between lab members is up to you.
Clean-up involves:
• Completely dismantling the experimental setup
• Removing tape from anything you put tape on
• Drying-off any wet equipment
• Putting away equipment in proper boxes (if applicable)
• Returning equipment to proper cabinets, or to the cart at the front of the room
• Throwing away pieces of string, paper, and other detritus (i.e. your water bottles)
• Shutting down the computer
• Anything else that needs to be done to return the room to its pristine, pre lab form.
I certify that the equipment used by ________________________ has been cleaned up.
(student’s name)
______________________________ , _______________.
(instructor’s name)
(date)
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