Download Centripetal Force

Bob Somers
1/13/05
Per. 4
Daniel Lau, Patrick Noland
Centripetal Force
Purpose
In this lab we set out to demonstrate how centripetal force and it’s “center-seeking”
properties were attained using existing forces that we could measure. This proved to
make the actual formulation of the lab procedure very easy because we didn’t have to set
out to use a special new force. All we had to do was use the forces that we already knew
about to create the effect of centripetal force. This also allowed us to do the lab
calculations without much effort.
Theory
The most important thing that we had to remember in this experiment is that centripetal
force is not a separate force in and of itself. Centripetal force is merely a sum of the
forces that cause an object to move in a circular path. This allowed us to account for the
centripetal force by using other forces which we already knew about and knew how to
calculate.
The underlying principle of this lab is that we used the property of tension in a rope to
create a situation of equilibrium, or at least as close to equilibrium as humanly possible.
The apparatus was designed to allow us to hold the system in the air while swinging the
stopper around without influencing (or at least negligibly influencing) the freedom of the
string to slide back and forth.
The great thing for us is that tension pulls toward the center of the rope. This means that
if we can swing the rope and stopper fast enough to counteract the force of gravity, the
tension in the rope (our supplier of centripetal force) and force of gravity acting on our
counterweight mass will cancel each other out and the rope will be in a state of
equilibrium.
If the system is in equilibrium, we can then assume that the forces opposing each other
are equal and opposite, leading to our chief assumption that Fg should be equal to Fc, or
the tension in the string. We have enough information to calculate both values without
relying on the other, allowing us to check ourselves by comparing percent difference
between the numbers.
Because the stopper was revolving so quickly, it would have been impossible to get
acceptable data by trying to calculate its velocity using only one revolution. Therefore, it
was necessary to time multiple revolutions and then multiply our distance traveled by a
factor of that amount.
Procedure
1. Choose a mass of 50g or more to hang from the clip on the end of the apparatus.
Attach this and hold the apparatus near your face to avoid facial damage.
2. Wait for your lab partners ready themselves. They should be instructed to count a
set number of revolutions (remember to start with ZERO, not one!) and record
the time with a stopwatch.
3. Spin up the stopper so until the mass is hovering and a consistent distance from
the edge of the glass tube. Your job as the stopper-spinner is to ensure that during
the timing of the trials the counterweight mass maintains this distance without
varying too much.
4. Have the time keepers begin counting and keeping time. Continue spinning the
stopper consistently as they do this.
5. Once they’ve yelled “Time!” you may slow and catch the stopper. Remember to
keep it near your face to prevent the need for facial reconstruction.
6. Repeat steps 1 – 5 with different masses and record the mass, time, and number
of revolutions for each trial.
Data
Mass of stopper:
Mass of hanger (including alligator clip):
Radius of stopper:
Trial
1
2
3
Mass
0.1039 kg
0.2039 kg
0.2539 kg
23.2 g
3.90 g
54.99 cm
Time
10.00 s
7.34 s
6.13 s
= 0.0232 kg
= 0.00390 kg
= 0.5499 m
Revolutions
14
14
14
Observations
• The string drooped as it swung through the air. It was not actually a “horizontal”
circle.
• The string sometimes got off kilter and made a somewhat diagonal circle.
• The masses would spin in circles below the glass tube while the stopper was
spinning.
• The heavier the mass, the faster and harder you had to swing the string to put the
system in equilibrium.
• The faster you swung the string, the further out it spread and the more horizontal
it became.
Calculations
First we find the distance that the stopper actually traveled during the course of fourteen
revolutions.
x = 2 π r rev
x = 2 π (0.5499) 14
x = 48.4 m
The following equations will be calculated using the data from trial one. By following a
similar process you can obtain the stated results for trials two and three.
We find the force of gravity to find the tension in the rope by multiplying the mass times
the acceleration of gravity. This serves as our comparison number.
Fg = mg
Fg = (0.1039)(9.80)
Fg = 1.02 N
By similar processes, we find that the second trial yields a 2.00 N force of gravity and the
third trial yields a 2.49 N force of gravity
Next we find the velocity that the stopper was moving at by dividing the distance found
above by the recorded time.
v=d
t
v = 48.4 m
10.0 s
v = 4.84 m/s
By similar processes we find that the second trial yields a velocity of 6.59 m/s and the
third trial yields a velocity of 7.89 m/s.
We now shove the velocity, measured mass of the stopper, and measured radius into the
centripetal force formula to find the tension in the rope.
Fc = mv2
r
Fc = (0.0232)(4.84)2
0.5499
Fc = 0.987 N
By similar processes the second trial yields a centripetal force value of 1.83 N and the
third trial yields a centripetal force value of 2.63 N.
Finally we calculate the percent difference, using the calculated force of gravity as our
comparison number.
(0.987 – 1.02) x 100 = -3.24 % difference
1.02
By similar processes we find that the second trial yields a percent difference of -8.30 %
and the third trial yields a +5.58 % difference.
Error Analysis
There were a ton of possible sources of error in this lab, all of which were rather
significant, lending towards the somewhat high percent difference numbers in the
calculations. Just for these trials alone we can see a span of -8.30 % to +5.58 %, which is
almost a 14 % span of difference, just in between similar trials.
The most obvious source of error is uncontrollable to some extent, and that is the
introduction of a variable amount of human error caused by a human having to try and
keep the counterweight mass hovering at the same height during the duration of the
experiment. This can be used to explain our negative error, which can be interpreted to
mean that we had more centripetal force than force of gravity. Because the potential for
human error (and for that matter, magnitude of error) is so great, it is the most likely
offender when it comes to generating negative error.
Another possible source of error is the actual circle made by the swinging stopper. We
assumed that this circle was perfectly horizontal, however this was a false, but still
reasonable assumption. Since we cannot stop gravity from acting on the stopper, there
will always be a component of gravity no matter how fast we swing the stopper. The
stopper can never swing perfectly horizontal, and therefore the true centripetal force was
merely a horizontal component of the true tension in the rope. This explains our +5.58 %
error, because the tension in the rope, which was equal to the force of gravity, was only
the hypotenuse of a right triangle of which the true centripetal force was the horizontal
leg.
Conclusion
As we witnessed in the experiment, centripetal force is not a force in and of itself. It is
merely the sum of other forces that act upon an object that cause that object to move in a
circular path. Even though this particular lab had a very high margin of error, it was an
excellent way to demonstrate circular motion and how forces can cooperate to make an
object move in a circular path. Simply the fact that such large masses are able to “defy
gravity” (when in all truth, they are not defying gravity at all) is a visual signal that
something unseen is acting upon the event we’re watching unfold.
The results show a significant difference in forces, but it was all accounted for by error.
The closeness of the figures we calculated was able to effectively explain that the tension
of the rope was pulling everything inwards and holding it in an equilibrium of sorts,
keeping the counterweight mass from plummeting to the ground.
I personally learned in this lab that while human error can certainly get in the way of
discernable results, if you know what you’re looking for you can still pick out specific
pieces of information that support your case. I also learned the valuable skill of counting
from zero rather than one. Correcting this mistake brought two of our trials back into the
acceptable margins of error.