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May 18, 2002
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Physics of Alpine Skiing
Skiing is enjoyed by millions of people every year. Skiers enjoy moving fast down steep slopes but
seldom stop to think about what is making them move. Skiing is an example of an everyday activity
affected by elements of physics. Many aspects of skiing depend fully on physics and surely without
physics, there would be no skiing.
The basic concept of skiing is simple. Attached to each foot is a long narrow board with tips in front of
you that flip up. The bottom of the ski is smooth so you slide so one can slide over the slippery, cold
snow with greater ease. With your skies you go to the top of an incline and slide down, turning left
and right because, as any skier knows, for some reason you go faster and faster and too fast if you
were to go straight down the slope. As you go down, you wear goggles because the air flies past you
so fast it begins to blind you. From this basic concept of sliding down an incline many types of skiing
have developed. One type is ski jumping, where you fly off a jump and try to suspend yourself in midair as long as possible. Another is ski racing where you fly down a course of poles as fast as you
possible can. Or cross-country skiing, where instead of going down an incline you use your own
human force to push yourself along the snow, still wearing skies. There are many types of skiing,
different ways to ski, conditions which suite different people, and different types of ski equipment but
in general many different variables but everything comes down do to one sure thing: without physics,
non of this would be possible.
Out of all the things that effect the skier’s movement as they go down a hill, gravity is perhaps the
simplest to understand. Gravity is the force that holds "us" down, the force that keeps everything from
flying away, keeps objects grounded. Gravity is the force that earth exerts on an object, pulling the
object toward earth’s center. If you through something in the air, it comes back down to the ground
because of gravitational pull. Acceleration due to gravity equal 9.81 m/s, this number can often be
useful when used in equations to find values for a skier’s speed (velocity) or mass. In the case of
skiing, gravity pulls the skier down the hill. The gravity tries to pull the skier straight toward the center
of the earth and meanwhile a normal force is being exerted on the skier, which acts as the opposing
force to gravity. The normal force acts perpendicular the surface the object is on, in this case
perpendicular to the mountain the skier is on. If the skier was on level ground she would not move
because the force of gravity would be pointing down and normal force would point straight up and
they would cancel each other out and there would be no change in velocity, however this is not the
case on the mountain. The combination the perpendicular normal force and the gravity will result in
the skier being pulled down the mountain at the same angle the mountain’s slope is at.
As the skier skies down the mountain, she goes threw an acceleration. 2 That is, her speed changes
as her velocity increases. Acceleration is defined as a change in velocity and that is what happens to
the skier. She accelerates because gravity is pulling her down. There is a positive acceleration as she
gets started down the mountain and picks up speed. As she comes to a stop she goes threw a
negative acceleration. People get hurt skiing not because of their acceleration or speed but because
they decelerate to quickly. For instance, if a skier were skiing at a ridiculously fast speed of 50mph
they would be very safe until they hit an object such as a tree. Then momentum must be conserved
and the tree won’t move as the skier flies into it.
Friction often affects a skier. As the skier moves down the mountain, the snow can resist the motion
of the skier, slowing her down and providing friction. A frictionless surface is one that does not have
such resistance to an object moving across it, snow however does have friction. As the skier skies
down the mountain, friction is the force that keeps her from flying down with an acceleration of
9.81m/s; it pulls her in the opposite direction. The friction acting on the skier is acting in directed
opposition to the displacement of the skier. The largest amount of friction is used when stopping. You
great a huge amount of friction by digging the edge of the ski facing down the mountain into the snow,
facing sideways to the slope. This creates so much friction that you are immediately stopped. The
coefficient of friction in the case of skiing depends on conditions. If it is a colder day (below about 25F
degrees) than the snow is dryer and therefore slicker meaning the ski will slide over it with greater
ease making the coefficient of friction in this case smaller. On a warmer day (about freezing, 32F
degrees) the snow is wetter and will clump making the surface not a smooth, here the coefficient of
friction is higher.
Reducing friction is the object of ski racers. The more the skier is able to reduce friction against their
skies, the faster they will go. Wax is applied to the bottom of skies in order to reduce the amount of
friction between the ski and the snow. Applying wax to skies helps to prevent them from drying out,
when the ski is dry the bottom is not as smooth thus making a greater coefficient of friction between
the ski and the snow.
As there is friction working against the skier, work is also done. The amount of work is found by using
the equation:
Force x displacement x cosine 180 = Work
Work can only be done unto an object if there is a force working against it, this is another way we
know that the skier is accelerating and working against a force. We know there is friction working
against the skier, therefore work is done in order for her to act against the friction. Work is only done if
there is a force to act against, and there is no force if the object, in this case the skier, is not
accelerating.
Friction is not the only thing that slows skiers down. When skiing down a mountain standing straight
up your body is almost acting as a parachute to air causing air resistance. In order to reduce air
resistance, ski racers where very tight suits called gs suits which fit snug against their bodies and
provide the skier with the least amount of area for air to get trapped in. Ski racers also ski in what is
known as the "tuck" position making them very aerodynamic. In the tuck, the skier makes herself as
small as possible by crouching down with her arms stretched out in front of her giving her body a
pointy shape as it goes down the mountain. When ski racers do larger courses such as a downhill or
super G course they are in the tuck position more because the gates are more spread out and they
don’t have to turn as quickly, therefore they are in the tuck position more. These skiers sometimes
have special curved poles that fit around their bodies while they are in the tuck position; the curved
poles allow the skier to be as small as possible.
The conservation of energy is one of the most important concepts in understanding the physics of
skiing. Potential energy is energy associated with a falling object, an object that has an initial
elevation and with a position such that relative a certain frame of reference it could fall. The formula
for finding potential energy is
Mass x gravity x height = Potential energy
A skier gains potential energy as she goes up the lift. This is because she is gaining height. Potential
energy is always gained when an object is working against the force of gravity because the farther the
object goes up the higher it is and therefore will have a greater potential energy. When the skier is at
the bottom of the mountain, height is equal to zero and therefore potential energy is equal to zero.
As the skier heads down the mountain, kinetic energy is gained. Kinetic energy equals:
1/2mass x velocity2
So the kinetic energy of the skier will depend on how much she weighs and how fast she is going. As
she accelerates down the hill, kinetic energy is gained. This is also in relation to potential energy, the
skier gains kinetic energy as she goes down the mountain but looses potential energy at the same
time. Therefore, potential energy lost equals kinetic energy gained. Right before the skier hits height
equals zero, the kinetic energy is at its maximum and as the skier hits height equals zero she will
continue to move on the flat surface because she still has energy. As the energy runs out, she will
decelerate and slow down, eventually stopping all together. The total energy is found by adding
potential energy to mechanical energy.
Mechanical energy is another form of energy that affects the skier. The mechanical energy of a skier
is directly related to work and friction. The more friction affects the skier over a longer distance, the
greater the amount of work done will be and the more friction and work increase, the more
mechanical energy decreases. Therefore:
Amount of work done = loss of ME
The formula for mechanical energy is:
Mechanical energy = Kinetic energy + potential energy
Or
Mechanical energy = ½ mass x velocity 2 + mass x gravity x height
If the forces of friction and work were insignificant, than the mechanical energy and therefore the
energy of the skier would be conserved. This would mean that the forces of gravity and normal force
and the only two forces acting on the skier. That is, her energy when she first began to ski would be
the same as when she was at her maximum speed. At the top of the mountain, mechanical energy
equals the potential energy because there is a large height but no velocity and therefore no kinetic
energy. And likewise, as the bottom of the mountain mechanical energy equals kinetic energy
because there is no longer any height and so potential energy must equal zero. So if kinetic energy at
the bottom equals potential energy at the top, energy is conserved.
As one can see, there are many aspects of physics that go into alpine skiing. From the seemingly
simple concept of sliding down a mountain with each foot attached to a long smooth board you find
that many things attribute to what most skiers never take notice of. Without gravity, you would not fly
down the mountain but without friction you would go to fast and never be able to stop, making skiing
to dangerous. This what makes physics fun; the different concepts apply in so many days to our
everyday life, not just in the heart of winter on top of a mountain.
Proposal for Experiment
I have now collected a large amount of research concerning the physics of skiing and discovered just
what makes skiers go down the steep mountain. I have learned that all different types of potential,
kinetic, and mechanical energy contribute to a skier’s motion as well as gravity, acceleration, forces
and friction. Now that I know what makes them move I am going to study what makes skiers go fast.
What contributes to the speed of a skier? What makes them go faster and slower?
There are obviously many things that contribute to a skier’s speed. From skiing myself I have
discovered a few things about what makes me go faster but now I will go into why. When I ski down a
steeper trail I go faster than I would when skiing down a beginner trail, therefore I know slope must
have an effect on skiers. When one is skiing and they ski over grass they slow down so friction
appears to have some sort of effect on the skier. When I ski with people who are heavier than I am,
they go faster so my last variable it weights. I will use slope, friction, and mass as my variables for my
experiment.
When I say I will measure what makes the skier go fast, I mean I will measure their acceleration. All
the carts in the experiments will start from rest and therefore I will be measuring what the change in
their velocity is to find out if they are indeed going faster or slower. By definition acceleration is a
change in velocity whether the object is getting faster or slower so by measuring acceleration I should
be able to tell which variable affect a skier’s speed.
For my experiment I will use a dynamic tract with the motion sensors in the physics classroom. My
other materials will include a wood and felt 100g block, the ring stand to hold up the track, a protractor
to measure the angle of the track, two 100g weights that I will use for excess weight, and the
computer connected to the two photo gates to record data.
PROCEDERS
Experiment 1: How does a skier’s slope affect how fast the skier skis? (This lab was based largely
after Lab #3 in October)
1. Run Data Studio on computer.
2. Put the track at a 15 degrees angle with the table.
3. Measure the height of one of the points on the track, preferably closer to the top of the track
and record the height. This will be the starting height for the cart for this step.
4. Place the felt side of the block down on the track at the recorded height with the eye facing
forward, and the card the photo gates read on the block, and let the block be pulled down the
track by gravity while recording a time vs. velocity graph on Data Studio using the photo gates.
Time should be on the X-axis and velocity is on the Y-axis. The slope of this graph will give the
velocity
5. Repeat steps 3-4 two more times for 15 degrees. Make a new graph for each run, each time
finding the slope and at the end average the slopes.
6. Repeat steps 3-5 with theta = 20, 25, 30, 35, 45, and 50 degrees. Do each degree three times
to try to eliminate human error.
7. Record all data in an appropriate chart (see end of proposal).
Experiment 2: Does a skier’s weight effect how fast they go?
1. Set up the track at a 30degree angle.
2. Run data studio.
3. For all runs of this experiment, start the cart from the same set point at the top of the track.
4. Running data studio let the block be pulled down the track by gravity at the 30-degree angle,
felt side down. Repeat 2 more times, each time making a graph of time vs. velocity on data
studio and finding the slope (which equals the acceleration) and averaging each group of data.
5. Repeat step 4 but tape a 100g weight on the block, which does not interfere with the card,
which the photo-gate reads.
6. Repeat step 4 but with two 100g weights taped onto the block.
7. Average the slope of the graphs of each weight interval and record on appropriate chart (in
back of proposal).
Experiment 3: Does the amount of friction on the bottom of a skier’s skis effect how fast they go?
1. Set up track at 30 degrees.
2. Run data studio.
3. For all runs of this experiment, start the cart from the same set point at the top of the track.
4. Running data studio let the cart slide down the ramp starting from the given spot with the felt
side down. Repeat two more times and find the slope of all the lines and average them.
5. Repeat step 4 with the wooden side of the cart down.
6. Repeat step 4 with a piece of paper taped to the bottom of the block.
7. Record all information in chart in back of proposal.
8. (For the Analysis) For each of the three surfaces, run the experiment with 5-6 different slopes.
That is, for instance for the paper surface, run the block down the track at 5-6 different
MEASURED AND RECORDED angles (such as 20, 30, 40, 45, 50, 55). Do each angle 3 per
surface to eliminate human error. Make graphs for all runs of the experiment then average the
slopes of the graphs with the like angles. That is, for every run of 40 degrees you do (you do it
three times), average the three slopes to come up with an average slope for 30 degrees, this is
the acceleration. So, for each of the three different surfaces, you should have average
accelerations for each angle, recorded for the analysis.
Hypothesis:
Experiment 1: When one thinks about a skier going down a steep trail, naturally one would think they
are going faster than someone on a beginner trail even when they start from the same height.
However, both skiers are starting from rest with the same coefficient of friction and because
acceleration due to gravity is the same they should have the same acceleration. Because their
accelerations are similar, the different graphs made in Experiment 1 should all look the same
because in a time vs. velocity graph, the slope of the line produced equals acceleration. So if the
accelerations are equal, so are the slopes of the line in the time vs. velocity graph.
Experiment 2: Velocity and energy are different. When a skier has a greater mass, they will have
greater energy but maybe not greater acceleration. I think this experiment will prove again that
acceleration due to gravity is constant and therefore the acceleration of the skier will be unaffected by
the mass.
Experiment 3: The acceleration of the skier will depend of the coefficient of friction. That is, the higher
the coefficient of friction, the more friction will be present below the skier and their acceleration will go
down because the friction is holding the skier back. I think my experiment will prove that as more
friction is added, as the surface of the block goes from felt to wood and paper, the slower the block
will go because the friction has increased and therefore the acceleration has decreased.
Also, knowing the different surfaces of paper, felt and wood, I think they will all have different
coefficient of frictions and therefore affect the acceleration differently. For example, the wood side of
the block will probably have a higher coefficient of friction than the paper or felt. Then the felt will most
likely have the lowest coefficient of friction because it will create the least amount of friction.
Procedure for Analysis
Experiment 1: When analyzing the change in acceleration with different slopes for the track, I will look
at the different graphs of theta vs. acceleration for the different experiments. I will primarily look at the
slopes of the line to determine if my hypothesis holds true, that because acceleration due to gravity is
constant the slope of the track will not change the acceleration. This line is made with time on the xaxis and velocity on the y-axis where the slope stands for a change in velocity or the acceleration.
Experiment 2: When analyzing the change in acceleration with different masses for the block I will
again look to the different graphs made in the different runs of the experiment. I will have found the
average slope for each set of runs, these average slopes should, if my hypothesis holds true, be very
similar. The slopes equal a time vs. velocity graph where the line made represents a change in
velocity, or an acceleration. I will then look at a different graph where theta is on the x-axis and the
average of these accelerations is on the y-axis. I think the line of this graph will show no change in
average velocity. I will also make and study a mass vs. acceleration graph that should not have a lot
of variation in the y-axis if my hypothesis holds true.
Experiment 3: Analysis for friction is a little more difficult. Through the analysis it must be proved that
friction and acceleration are inversely proportional, that is as one goes up the other goes down. Take
the average accelerations and their thetas in the procedure of experiment 3. The first thing to do is
make a graph, a graph of theta on the x-axis and the slope (i.e. the acceleration) on the y-axis. Graph
the theta of the already found points vs. their average accelerations. There should be three graphs for
the three different surfaces. You will need this graph after the next part of the analysis.
We then must find the coefficient of friction or m (standing for the Greek letter Mu). This requires the
equation
M(skier) x A (skier) = m(skier)x gravity (sin theta –m cos theta)
This equation transforms into the equation
m = (a –gravity sin theta)/ (gravity cos theta)
Here, m= Mu, A=Acceleration, Gravity = 9.81 meters per second squared
And M= mass
This is to solve for m. Theta equals the angle the track (or ski slope). Using the average acceleration
found in the procedure for experiment 3 (#8), find the Mu for all the different accelerations and record
them. You should have a different set of coefficient of friction for all three different surfaces.
Once you have all the different coefficient of frictions, transform the equation for finding Mu into the
equation
A = gravity (sin theta – m cos theta)
Plug in the different Mus and the angle of the track for theta for the different surfaces to come up with
a new set (which hopefully matches the old set) of accelerations.
Take the graph already made of theta vs. acceleration and in a different color, plot the theta on the xaxis and the accelerations you just found for the y-axis. There should be three different graphs.
All the points should lie on the same line. If this happens (which it should) than it proves that
acceleration and friction are disproportional and as one gets bigger, the other gets smaller.
Through these procedures and their analysis’ I should be able to determine what effects the skier’s
acceleration. This should help me through the experiment of come up with a statement about the
acceleration of a skier.
Analysis of Experiments
Analysis of Experiment 1:
Question: How does a skier’s slope affect how fast the skier skies (i.e., how does the slope the skier
is skiing at affect their acceleration)?
In this experiment, I originally wanted to take 100g block of wood with a felt surface and slide the
block down a track at various angles. The speed of the cart would be measured by two photo gates
set up on the track and would measure and make a graph of change in time verses change in
position. The slope of this graph would equal the acceleration of the cart. The different slopes would
range from 15 to 50 degrees in intervals of five. By looking at the how the acceleration changed as
the track became stepper, I would be able to see how the slope of the track effected the track’s
acceleration.
When I began the testing of this experiment I encountered a number of problems, which I overcame. I
first was that since the block of wood was sliding down the track, it tended to shift while in motion
because the bottom of the block had no grooves to keep the block following the same line. This would
alter the results. The time verses position graph was not a straight line like it should be and the
correlation coefficient of the linear least line was between .3 and.7. To fix this problem I added more
weight (going from the original 100g to 300g) making the experiment more constant to where the new
correlation coefficient equaled .95 to .999. This weight would remain the weight for this experiment
and experiment three (and the analysis section of experiment 3).
The chosen slopes were also not appropriate. If the track was at a slope below 20 degrees, it was not
steep enough for the cart to slide down. And if the track was above 35 or 40 degrees the results were
not constant and were very unreliable. So I changed the slope of the experiment to ranging from 20 to
40 degrees in intervals of five. This worked a lot better.
After completing this experiment it was very evident that I was wrong in my hypothesis. In my
hypothesis I said that the slope of the track would not effect acceleration but my experiment proved
that I was incorrect. I discovered a positive association between the slope and the acceleration. To
analyze the data I made a graph of the different thetas used, (20, 25, 30,35, and 40) and put this on
my x-axis. On my y-axis was the average of the slope (or acceleration) for that particular theta. As
theta increased, so did the acceleration.
Later in the experiment, I calculated the coefficient of friction of this particular surface to be .249 using the equation:
A = gravity (sin theta –m cos theta)
Where gravity = 9.81 meters per second squared
And Theta and Acceleration come from the results from the experiment in the analysis of experiment 3
This equation backs up what was proved in this experiment, that as the theta of a ski mountain becomes larger, the greater the acceleration there will
be. With the constant coefficient of friction, if you plug in the point 20 degrees the acceleration comes out as 1.05 meters per second squared. Then if
you plug in the point 40 degrees the acceleration comes out to be 4.44 meters per second squared. Here the relationship is shown, as the theta
becomes larger so does the acceleration.
As I stated earlier, the data points become less and less constant as the slope becomes larger than
35 or 40 degrees. The graph reflects this (as seen below), those two points still follow the same
relationship as the other points but they are not perfectly in the same line as the other points. If I were
to do this experiment again in the future, I would only use points between 20 and 35. Instead of just
having three points like this, I would change the intervals of slope to intervals of 3 or 4. This way there
would be less variation in the results and the line in the graph would be smoother.
Chart of Experiment one (slope is actually measured in meters per second squared)
Run#
Theta
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Slope (acceleration)
20 .4476m/s
20 .5654m/s
20 .609m/s
25 1.0186m/s
25 .8288m/s
25 .85m/s
30 1.275m/s
30 1.3m/s
30 1.26m/s
35 3.2m/s
35 3.442m/s
35 3.8m/s
40 3.978m/s
40 3.91m/s
40 4.2m/s
Height of Track
39cm
39cm
39cm
46cm
46cm
46cm
54cm
54cm
54cm
61cm
61cm
61cm
69cm
69cm
69cm
Mass
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
Graph of Theta vs. Average Slope.
Ave. Acceleration
Slope Experiment
5
4
3
2
Series1
1
0
0
20
40
60
Theta
Analysis for Experiment 2:
Question: Does a skier’s weight effect how fast they are skiing?
In this experiment, I originally planned to take the 100g block of wood with a felt surface, the same
block used it the previous experiment, and slide it down the same track. The variable would the
weight of the block; the three different weights I planned to use were 100g, 200g, and 300g. The track
would be set at 30 degrees. Again, the two photo gates would be used to find the change in velocity,
or acceleration. The photo gates would record data and display it in the form of a graph where time
was on the x-axis and position was on the y-axis. Again, the slope of the line on this graph is the
acceleration. If the average acceleration changed when more weight was added, then it would be
determined that weight affects the speed of a skier.
Unlike the first experiment, I kept my original mass for this experiment. The only change I made was
the constant slope that was used. I used 20 degrees instead of 30 degrees because when the track is
at 20 degrees it gives more constant graphs with higher and more reliable correlation coefficients.
Unlike the first experiment, when I had completed this experiment and I examined my hypothesis I
concluded that my hypothesis had been correct, weight does not change the acceleration of skier
because acceleration due to gravity is constant. As I added more and more weight, the acceleration
stayed the same. There was a very slight difference in accelerations, as is seen in the graph, but if
one looks at the scale of the graph one would notice how truly small the difference is. The variation
also does not follow a pattern and therefore it can be concluded that the accelerations were
essentially equal.
Chart for Experiment 2 (slope is actually measured in meters per second squared):
Run #
Theta
1
2
3
4
5
6
7
8
9
Height of Track
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
Mass
100g
100g
100g
200g
200g
200g
300g
300g
300g
Slope (acceleration)
.4985m/s
.54m/s
.53m/s
.569m/s
.54m/s
.562m/s
.52m/s
.55m/s
.51m/s
Graph of Mass vs. Average Acceleration:
Average Acceleration
Mass vs Average Acceleration
0.56
0.555
0.55
0.545
Series1
0.54
0.535
0.53
0.525
0
100
200
300
400
Mass
When looking at the points, the greatest variation in slope is two points that have a five one
hundredths of a second difference from each other, all accelerations are between .4985 meters per
second squared and .569meters per second squared. It is very hard on a track that is not always
perfect because of the lack in consistency in the data runs to get perfect data, so this data is very
close to being perfect but it still proves my theory.
This was a successful experiment, which in my mind needs few improvements. If I were to do this
experiment again in the future, however, I might add different weights to the three. I might possibly
add a significant amount more, such as 1000g more to see if truly my experiment would hold true. I
would also find a cart that has groves on the bottom that would help the cart to slide down the track in
a straight line.
Analysis for Experiment 3:
Question: Does the amount of friction on the bottom of a skier’s skis effect how fast the skier goes?
Originally in this experiment, I planned to slide the same block of wood with a felt surface down a
track at 30 degrees. The block would weigh 100g. I would do the experiment 9 times, every three
times with a different surface. Three runs had a felt surface, three had a wood surface, and three had
a paper surface. Again, I would use the two photo gates to measure the acceleration. The photo
gates would produce a graph on the computer where time in on the x-axis and position is on the yaxis. My hypothesis was that friction and acceleration are inversely proportional and I hoped the
graph would portray this.
I changed this experiment like I had changed the previous two experiments, I changed the slope of
the track to 20 degrees instead of 30 degrees in order to get more accurate results and I changed the
weight to 300g instead of 100g for the same reason.
The results came out as followed; I really had no idea what they meant at first except I do believe
they look accurate because the acceleration for the different forms of friction are very similar.
Chart for Experiment 3 (acceleration is measured in meters per second squared):
Run #
Theta
1
2
3
4
5
6
7
8
9
Height of
Track
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
20 37cm
Friction Slope
(acceleration)
Felt
.554m/s
Felt
.57m/s
Felt
.57m/s
Wood .39m/s
Wood .32m/s
Wood .31m/s
Paper .47m/s
Paper .48m/s
Paper .49m/s
I now need to analysis the data in order better interpret it using the transformed equation:
A = gravity (sin theta – m cos theta)
Here, A= acceleration,
m = Mu
gravity = 9.81 meters per second squared.
Using this equation for my analysis, I now need to find Mu, or the coefficient of friction. I will then plug
this number into the original equation (which I will have transformed so it is in terms of acceleration) in
order to interpret the data. This is explained in more in-depth in the procedure section.
Like in experiment one, I changed the theta of the track five times while using felt as the surface of
the cart and recorded the acceleration of the cart sliding down the track at that angle. The results are
as followed:
Analysis Experiments for Experiment 3:
Felt Analysis (acceleration is measured in meters per second squared):
Run #
Theta
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Slope
(acceleration)
20 1.097m/s
20 1.08m/s
20 1.03m/s
25 1.92m/s
25 1.94m/s
25 1.93m/s
30 3.10m/s
30 3.15m/s
30 3.0m/s
35 3.7m/s
35 3.4m/s
35 3.8m/s
40 3.96m/s
40 3.90m/s
40 3.77m/s
Friction Mass
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
Felt
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
Graph of Theta vs. average slope:
Using this information, I now need to find Mu, or the coefficient of friction. I used 25 degrees as the
theta and the corresponding acceleration for that value into the following equation in order to
calculate mu.
M = (gravity sin theta - a)/ (gravity cos theta)
I calculated Mu to be .249. This answer makes sense because Mu must be positive.
After finding the Mu I plugged it into the equation:
A = gravity (sin theta – m cos theta)
Where g = 9.81 meters per second squared
Mu = .249
Theta equals the theta of the track
I found that by plugging in the five different slopes used and the coefficient of friction into the
equation, I came out with very similar accelerations as I had calculated before. The new
accelerations are as followed (acceleration is measured in meters per second squared):
Theta
Acceleration
20
1.057
25
1.93
30
2.8
35
3.62
40
4.3
When plotted, these are the results:
Acceleration
Theta vs. Acceleration for Felt analysis
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Series1
0
10
20
30
40
50
Theta
This graph very closely resembles the graph of theta vs. acceleration for the measured values of the felt
experiment. These numbers will be important later to prove my theory.
Wood Experiment:
Just like in the first part of the analysis of experiment 3, when analyzing the friction of wood I collected
data the same way I had for the friction of felt. The data is as followed:
Chart for Analysis of Wood (acceleration is measured in meters per second squared):
Run #
Theta
1
2
3
4
5
6
7
8
Slope (acceleration)
20 .91m/s
20 .92m/s
20 .89m/s
25 2.1m/s
25 1.9m/s
25 2.05m/s
30 2.87m/s
30 2.84m/s
Mass
300g
300g
300g
300g
300g
300g
300g
300g
9
10
11
12
13
14
15
30 2.88m/s
35 3.43m/s
35 3.44m/s
35 3.19m/s
40 3.7m/s
40 3.68m/s
40 3.661m/s
300g
300g
300g
300g
300g
300g
300g
Graph of Theta vs. Average Acceleration:
Just like in the experiment for the friction of felt, I again need to calculate the coefficient of friction (Mu) using
the equation:
M = (gravity sin (theta) – acceleration) / (gravity cos (theta))
Where gravity = 9.81 meters per second squared
Through this equation, I calculated Mu to be .23
So, again I will plug the Mu and the five different thetas into the equation:
A = gravity (sin (theta) – m cos (theta))
The results were as followed (acceleration is measured in meters per second squared):
Theta
Acceleration
20
1.1
25
2.016
30
2.869
35
3.7
40
4.5
This acceleration closely resembles the previous accelerations calculated. When these points are plotted this is
the result:
Acceleration
Theta vs. wood analysis
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Series1
0
10
20
30
40
50
Theta
This graph very closely resembles the previous graph made that calculates theta vs. acceleration of the cart
when wood is the surface. This proves my hypothesis that acceleration and friction and inversely proportional.
Paper Experiment:
In the experiment for testing the friction of paper, I followed similar steps as I previously had, finding the
acceleration of the cart with 5 different slopes but for this experiment the surface of the cart was paper. The
results for the experiment are as followed:
Chart for Analysis of Paper experiment (acceleration is measured in meters per second squared):
Run #
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Theta
Slope
(acceleration)
20 .51m/s
20 .51m/s
20 .47m/s
25 1.8m/s
25 1.85m/s
25 1.75m/s
30 2.9m/s
30 2.839m/s
30 2.85m/s
35 3.72m/s
35 3.6m/s
35 3.5m/s
40 4.0m/s
40 4.8m/s
40 4.4m/s
Mass
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
300g
Once I had found the average acceleration of each slope, I made a theta vs. slope graph that is as followed:
Analysis of Paper Experiment
Ave. Acceleration
5
4
3
Series1
2
1
0
0
10
20
30
40
50
Theta
Using this information, I then calculated the coefficient of friction or Mu, using the equation:
m = (gravity sin (theta) – acceleration) / (gravity cos (theta))
Where gravity = 9.81 meters per second squared
And theta = one of the measured thetas in the found measurements and the acceleration is the corresponding
acceleration.
Mu was calculated to be .26. I then plugged this mu into the equation:
A = gravity (sin (theta) – m cos (theta))
Here I plugged in not only the mu but the five different theta’s used to come up with new accelerations which
are as followed (acceleration is measured in meters per second squared)
Theta
Average Acceleration
20
0.958
25
1.8
30
2.66
35
3.5
40
4.3
These calculated results are almost exactly the same as the measured results recorded before. The graph for
these points is as followed:
Acceleration
2nd Analysis of Paper Graph
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Series1
0
10
20
30
40
50
Theta
This linear line shows what experiment one did, that as theta becomes larger, so does the acceleration.
Taking the three Mus and their accelerations, I then made a graph of Mu vs. acceleration. To do this, I
formed an equation much like the linear equation:
Y = mx + b
Where b is the y intercept
And m is the slope
But to get an equation where I have acceleration in terms of mu I transformed the equation:
A = gravity (sin theta – m cos theta)
Into
A = 4.2 –9.06 m
Where 4.2 is the y intercept and – 9.06 is the slope
This was done by finding the sin and cos of angle 25 and substuting sin 25 for sin theta and cos 25 for cos theta.
I used 25 because it always had very constant results. Once I had this new equation, I plugged in the three
different coefficient of frictions as the x value in the equation and came out with an acceleration as my y value.
These were the results:
Theta
Mu
Acceleration
25
0.249
1.944
25
0.23
2.1162
25
0.26
1.8444
These were the results of the graph:
Mu vs. Acceleration
2.15
Acceleration
2.1
2.05
2
Series1
1.95
1.9
1.85
1.8
0.22
0.23
0.24
0.25
0.26
0.27
Mu
This proves my theory that as the coefficient of friction becomes larger, the acceleration gets smaller and as the
coefficient of friction becomes smaller the acceleration increases. As one can see from the graph, this is the
trend.
If I were to do this experiment again in the future, I would use more dramatically different forms of
friction. The three forms I used here had coefficient of frictions that were very close to each other, I could
possibly in future experiments use sand paper or another material that would have a very different Mu and
therefore truly test my theory