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20191027001707lab_report_example_.pdf
Physics 1730
FRICTION AND THE INCLINED PLANE
Abstract
Our goal was to determine whether the mass, surface area and the angle of incline have an effect on the coefficient of friction. When an object is in contact with another surface, there is a resistance force that acts towards it, this force is called friction.
We performed three separate sets of experiments on an adjustable inclined plane. In the first experiment, we investigated the effect of surface area on the coefficient of friction on a horizontally placed adjustable inclined plane using a narrow and wide block. To investigate the effect of the angle of incline on the coefficient of friction, we used an adjustable incline plane at varying degrees of incline and made the use of a block with constant mass. We also determined the angle of incline at which an object slides down at a constant motion. To investigate the effect of mass on the angle of incline we used blocks of varying mass in all the procedures. From our results, we concluded that mass, surface area and the angle of incline do not affect the coefficient of friction. We found that the percentage difference between the coefficient of friction in all procedures was 5.9%. To estimate our uncertainty, we calculated the standard deviation which was 0.05 for all coefficients of friction recorded. Experimental errors came mainly from friction between the string and the pulley system and by tapping on the block to assist in motion. We can improve the experiment by lubricating the string and ensuring that the only source of friction is from the block and the inclined plane’s surface. Introduction
As an object surface interacts with another surface, there is a resistance to motion. The purpose of this experiment is to determine the various factors that influence the coefficient of friction. The factors that are investigated are mass, surface area and the angle of the inclined plane. Frictional force is an example of contact force in which there is direct contact between two surfaces. It is the opposing and parallel force to the direction of motion when two surfaces that are in contact slide across each other. Its magnitude is usually dependent on the materials in contact with each other provided that the surfaces are not deformed beyond the elastic limit. The two forces involved are kinetic friction and static friction. Static friction occurs when there is no relative motion, it exerts an equal and opposite frictional force on the object. This force has to be overcome in order to experience motion between the two surfaces. Kinetic friction is the resistant force to the movement between two sliding surfaces as the objects are moving relative to each other. The force that overcomes frictional resistance can be measured to determine the magnitude of friction. As the object is in motion, it experiences kinetic friction. The factors that affect the coefficient of friction will be determined using a horizontal and inclined plane. The objective is to prove that the mass, area of contact between the surface and the angle of the inclined plane do not have an effect on the coefficient of friction.
Theory
Considering the two forces, kinetic and static friction that resist the relative motion between two surfaces in contact we can derive the coefficient of friction µ. The coefficient of friction is the ratio between the force required to overcome friction and the normal force. The normal force represents the reaction force to the object’s weight, and acts perpendicular to the contact force between the two objects. The kinetic friction fk in Newtons is computed as: Fk = µ kN (equation 1)
The static friction in Newtons is given by:
Fs = µ sN (equation 2)
where µ k (no units) is the coefficient of kinetic friction and µs (no units) is the coefficient of static friction and N is the Normal Force in Newtons. This derives the equation coefficient of kinetic friction from equation 1 as µ k = fk / N (no units) (equation 3) and the coefficient of static friction from equation 2 as µ s = fs / N (no units) (equation 4) Considering the free-body diagram Figure.1, of a block falling down an incline, the sum of the force acting in the y-direction is zero as the object is at rest. The Normal Force N in Newtons can be calculated as N=mg (cos q) (equation 5) Where m is the mass in kg and g is the acceleration 9.8m/s2 due to gravity. In the x-direction there is a component force acting mg sin (q), when the object is at rest and at motion the Ffriction force in Newtons acts on it in the x-direction up the incline. Which derives the equation Ffriction= mg sin (q) (equation 6) Where m is the mass in kg and g is the acceleration due to gravity.
. Figure 1:Diagrams illustrating a free body diagram of an object sliding down an inclined plane.
Equation 1, 5 ,6 can be used to derive the equation µ k =Fk/N= mg sin qk/ mg cos qk which is equivalent to µ k =Fk/N= W sin qk/ W cos qk where W represents the weight, in a triangle that has two sides perpendicular to each other, the having equal angles through trigonometry µ k =h/b=tan qc (equation 7) where h and b represents the height and base respectively in centimeters. This is the ratio of the Sin angle q and Cos q to give the tangent of angle q. In this experiment, the frictional forces/ pulling force between the plane and the block will be derived and measured to determine whether the coefficient of friction is affected by the mass, angle of inline and surface area. In our experiment, we made the use of dynes instead of Newtons where 1N=105 dyne
Apparatus The apparatus used were as follows: An adjustable inclined plane was used in all the experiments. A wooden block with precut holes for holding weights. The block would be used to slide on the adjustable inclined plane. A 500 g and 1000g mass to be used to adjust the weight of the wooden block. 1 meter long string used to attach the wooden block on one end and a weight hanger on the other end. A set of gram weights to measure the pulling force and two meter sticks to be used to measure the height and base of the incline. Experimental Procedure
In the first part of the experiment, a wooden block was measured and placed on a parallel horizontal plane using two different methods. In the first method, the wide side was facing down and in the second the narrow side was facing down. The height of the pulley at the end of the plane was placed such that the top of the pulley was the same height as the eyelet on the wooden block. Weights were then added to the weight hanger on the string until the block moved at a constant velocity. The mass of the weights required to cause a motion on the block was recorded as the pulling force F in dynes in Table 1. The procedure was repeated with an increment of 500g and 1000g respectively. Unfortunately, the string kept touching the end of the incline plane and had to constantly be adjusted. The coefficient of friction was calculated using equation 3. The experiment was repeated using a block only without adding any masses on the block at angles of 15° , 30° , and 45° and the pulling force was tabulated in Table 2. In Table 2, the Normal Force in dynes was calculated using equation 5 and the parallel force in Newtons was calculated using equation 6. In the second part of the experiment, a wooden block with the wide side facing down was placed on an inclined plane. The inclined plane was raised to an angle until the block moved at a constant velocity as it slid down the incline. The resulting angle was recorded in Table 3. This was repeated for two trials with an addition of 500 grams and 1000g masses added to the block. The height and the base length of the incline was recorded in cm. The coefficient of friction was calculated using equations 7. Our meter sticks had a lot of wear at the ends, we decided to rectify this problem by beginning our measurement from 1cm instead of 0cm. Data We measured the Pulling Force(dynes) of the Wide and Narrow block with varying masses and calculated the coefficient of friction µ k (no unit). Table 1: The effect of a wide and narrow block on the coefficient of friction with increasing mass
Object moved Weight W(dynes)
Side Used Angle (q)
Pulling Force F(dynes)
Coefficient of Friction µk
=fk/N Block Only .91 1 Wide 0 9.80E+04 0.51
Block + 500g 6.81E+05 Wide 0 1.96E+05 0.29 Block + 1kg 1.17E+06 Wide 0 2.94E+05 0.25
Block Only 1.91E+05 Narrow 0 6.86E+04 0.36 Block + 500g 6.81E+06 Narrow 0 2.16E+05 0.32 Block + 1kg 1.17E+06 Narrow 0 3.53E+05 0.30
We measured the Pulling Force(dynes) of the [Block Only] at angles of 15°, 30°, 45° and
calculated the Normal Force(dynes), Parallel Force(dynes), Frictional Force(dynes) and coefficient of friction µ k (no unit). Table 2:Data table showing the effect of an increase in the angle of incline on the coefficient of friction
on a block of equal mass.
Object Moved
Pulling Force F (dynes)
Angle Normal
Force N=W cos
Parallel Force
W sin (q)
Frictional Force
f=F-W sin (q )
Coefficient of Friction µk =fk/N
Block Only 8.82E+04 15 1.84E+05 4.94E+04 3.88E+04 0.21
Block Only 1.47E+05 30 1.65E+05 9.54E+04 5.16E+04 0.31
Block Only 1.86E+05 45 1.35E+05 1.35E+05 5.12E+04 0.38 We measured the Angle at which a block slid down the incline and obtained the h and b in cm. The coefficient of friction µ k was calculated with no units. Table 3:Data table showing the effect of an angle on the incline and mass on the coefficient of friction
using blocks of varying mass. Object Moved
Weight W (dynes)
Angle q(q )
h (cm)
b (cm)
µk µk =h/b
µk µk = tan q
Block Only 1.91E+05 13 18 60 0.31 0.23
Block + 500g 6.81E+05 14 19 60 0.32 0.25 Block + 1kg 1.17E+06 12 18 60 0.30 0.21
Calculations and Graphs
The coefficient of kinetic friction µk from Table 1. was computed by dividing the Pulling Force F in dynes by the Weight in dynes for all the blocks. The relationship between the Weight and Pulling Force is represented in Figure 1. Coefficient of Friction for [Block Only] =µk =fk/N =9.80E+04/1.91E+05 =0.51(no units)
The resulting coefficients for all the blocks plot a linear graph of Weight vs Pulling Force for the wide and narrow block.
Graph 1: The weight(dynes) and the Pulling Force(dynes) for a narrow and wide block. The slope of
the best fit line gives the coefficient of friction for each block
The best-line of fit from Graph 1 between the Narrow Block and the Wide block are close to one another and using a linear relationship y=mx +b. The slope of the graph m in the equation y=0.29x+ 14485 for the narrow block and y=0.2x+59829 for the wide block provides us with a coefficient of friction of 0.29 and 0.2 respectively. This goes to show that the surface area does not have an impact on the coefficient of friction of the block on the wood.
The percentage difference in the coefficient of friction in Table 4, of the blocks is used to show if there is a big difference between the surface area and the coefficient of friction. The average coefficient of friction is obtained by finding the average coefficient between all trials from Table 1. The resulting average can then be used to calculate the percentage difference between the two coefficients i.e. coefficient of friction of the wide block and the coefficient of friction of the narrow block to get 5.9%. The small percentage difference goes to show that the surface area does not affect the coefficient of friction. Percent difference = (|value1-value2|/average value) *100 (equation 8)
=|Average Coefficient of Wide Block-Average Coefficient of Narrow Block|/Average*100
Percent difference = (|0.35-0.33|/ (1/2*(0.35+0.33))) *100
=5.9% Table 4:The percentage difference in the coefficient of friction between the narrow and wide block at specific masses and the percentage difference of the average coefficient of friction
Object moved Weight
W(dynes)
Coefficient of friction
Wide Block
Coefficient of friction
Narrow Block
Percent difference %
Block Only 1.91E+05 0.51 0.36 34
Block + 500g 6.81E+05 0.29 0.32 9.8
Block + 1kg 1.17E+06 0.25 0.30 18
Average 0.35 0.33 5.9 Similarly, the percentage difference in coefficient of friction between the three trials for the Wide-Side of the block was calculated using equation 8 and recorded in table 5 while that of the Narrow-side was recorded in table 6. The results show that the coefficient of frictions was close to each other between the three results in each method. Therefore, the small percentage in coefficient of friction from Figure 1, Table 4, 5 and 6 support our claim that surface area does not affect the coefficient of friction. The analysis of Table 5 and 6 shows that mass has no impact on the coefficient of friction because varying masses were used in each set of experiments. In the analysis of Table 6, there are lower values in percentage difference of 8.8%, 3.1% and 9.5. the results of Table 5 are offset by the high coefficient of friction of 0.51 recorded for Block Only. This deviation can be attributed to frictional force on the string. Table 5:The percentage difference of Table 6:The percentage difference of coefficient of friction from their average value coefficient of friction on the narrow block from of 0.35. Notice the high coefficient of the block the average value of 0.33. only.
Wide Side Coefficient of Friction µk =fk/N
Percentage difference
Narrow Side
Coefficient of Friction µk =fk/N
Percentage difference
Block Only 0.51 37 Block Only 0.36 8.8 Block +500g 0.29 19 Block + 500g 0.32 3.1 Block + 1kg 0.25 33 Block + 1kg 0.30 9.5 Average 0.35 Average 0.33
The results from the second part of experiment 1 derives a graph from Table 2 that shows the relationship between the angle of incline at 15° , 30° , 45° and the coefficient of friction. By applying
the linear relationship of the best line of fit from the equation of y=mx+ b where m is the slope we find that y=0.0056x+0.1315. This gives us a slope of 0.0056 which is considered small and therefore shows that the angle of an inclined plane does not have an effect on the coefficient of friction. In the experiment, we used a block of equal mass to make sure that only the angle of incline was being tested for.
Graph 2: Graph showing the relationship between the coefficient of friction and the angle of
incline from Table 2. From the resulting Pulling Force obtained in Table 2, we calculated the Normal Force in dynes for all values using N=W cos (q). N=190855 cos (15° ) where W is the previously recorded Weight from Table 1. =1.84E+05 dynes. The Parallel Force in dynes was calculated using Parallel Force =W Sin (q) =190855 Sin (15° ) = 4.94E+04 dynes. The Friction Force was calculated as
f=F-W sin (q ) =(8.82E+04)- 190855 Sin (15° ) =3.88E+04 dynes The coefficient of friction was then computed as
µk =f/N = (3.88E+04 dynes)/ (1.84E+05 dynes.)
=0.21 From comparing the coefficient of friction from Table
3, we can deduce that the mass and the angle of incline do not affect the coefficient of incline. We can use the µ k obtained by measuring the height and base of the meter stick to obtain a standard deviation of 0.01 which is low thus showing that the mass and angle of incline do not have an effect on the coefficient of friction. The standard deviation can be calculated using the formula.
The standard deviation is computed using the three coefficients from the four-different set of experiments. The standard deviation of all the coefficient of friction recorded from the experiment is 0.05 which is low enough to show that the coefficients are close together thus showing that neither mass, angle of incline or surface area affect the coefficient of friction. Considering the fact that our meter stick had wear at the beginning of the marking, we might have been able to obtain more precise data for the height and base in cm which would have given us a coefficient of friction that was more precise. Table 7:The standard deviation of all coefficient of friction obtained from all the experiments. The
standard deviation for all values is 0.05
Object Moved
Coefficient of Friction
Wide Block
Coefficient of Friction Narrow Block
µk µk =fk/N µk =tanq Average
Standard Deviation
Block Only 0.51 0.36 0.31 0.23 0.35 0.12 Block + 500g 0.29 0.32 0.32 0.25 0.29 0.03 Block + 1kg 0.25 0.30 0.30 0.21 0.27 0.04 Average 0.35 0.33 0.31 0.23 0.30 0.05 standard deviation 0.14 0.03 0.01 0.02 0.04 0.05
The resulting standard deviation from Table 7 of the specific mass is minimal from all sets of
masses apart from the [Block Only] which is 0.12. Despite the fact that its standard deviation is higher than those from the other set of data [Block + 500g] and [Block +1 kg] it is safe to assume that the mass has no effect on coefficient of friction.
Discussion of Results and Error Analysis The best line of fit from graph 1. showed a coefficient of friction with a 5.9 % percent difference.
In an ideal situation, we would expect to get similar coefficient of friction from all data sets with no percent difference. This is because the material in contact would still be the same. The amount of contact surface would not be a factor because the static friction would not change with increase in surface area because friction is only proportional to the total normal force. “The magnitude of kinetic friction force usually increases when the normal force increases” (Young et al). However, the Pulling Force obtained varied from each other which can be attributed to experimental error. In our setup, the string that was attached to the block was in direct contact with the surface at the end of the plane. This would have increased the amount of friction because the block was not the only object in contact with the string. If we compare the percentage difference of the Narrow Facing block in Table 6 there is a decreased variation compared to those of the Wide faced block in Table 5. The higher average percentage of 34% recorded in Table 4 can be attributed to our failure to minimize the amount of contact of the string with the plane. There is a decrease in the percentage difference among the other values which assisted us in comparing the other values. The high percentage difference of all the values in Table 4 can be attributed to the results of the Wide-facing block [Block Only] of 0.51 which increases the average value significantly. If we used the results of the [Block+ 500g] and [Block + 1kg] that are closer to each other we are able to conclude that the surface area does not have an effect on the coefficient of friction.
From our experiment, it is noted that the angle of the incline does not affect the coefficient of friction based on Graph 2. The given slope between the three inclination angles 15°, 30°, 45° y=0.0056x +0.1315 is minimal which supported our claim. According to our claim that the angle of incline does not affect the coefficient of friction, we cannot validly claim that we had placed our incline at the wrong angle. However, our source of error can be blamed on friction on the pulley system. This would result in an increase in the coefficient of friction across the table. While performing the experiment we made the mistake of tapping on the block to start any motion. The action of tapping on the block might have resulted in recording the wrong value for when we observed that the object was in motion. In a different set-up, it would be advisable to avoid tapping on the blocks and allow for the Pulling Force created by the blocks to cause motion. As mentioned, our meter stick had worn out ends which impacted the measurements obtained for the height and base.
It would be impossible to calculate the percentage error because we were not given the accepted values for the block or the coefficient of friction. However, if we were to assume that the coefficient of friction of wood on wood was .25 (Segway) we can calculate the percentage error for all average coefficients from all the data sets using the formula % error=((|exp value- accepted value|)/accepted value) * 100%
= (|µ k tan (q ) -accepted value|/accepted value|) * 100% = (|0.23-0.25|)/ 0.25) *100 =8%
Table 8:The percentage error using a coefficient of friction of wood on wood of 0.25
Object Moved
Coefficient of Friction µk =fk/N
Coefficient of Friction µk =fk/N
µk µk =h/b µk =tan
Average Coefficient of Friction 0.35 0.33 0.31 0.23
Percentage error 40% 32% 24% 8%
From Table 8. We can conclude that using the angle alone and the equation 7 had less
experimental error. In the future, it might be important to provide values for an ideal system to compute experimental error for all data sets. Despite the high experimental error using a proposed coefficient of friction we can support our claim that mass, angle of incline and surface area do not affect the coefficient of friction because as seen in Table 7 our standard deviation of 0.05 is a low number. Conclusion In conclusion, mass, the angle of an incline and surface area of the moving object do not affect the coefficient of friction. We were able to examine that the coefficient of kinetic friction was similar between the Wide and Narrow blocks achieving a percentage difference of 5.9%. Secondly, we performed the experiment using different angles of inclines on similar masses and had a slope of 0.0056 which represents the coefficient of friction. Thus, we can support our claim that the angle of incline does not affect the coefficient of friction. Lastly, we can conclude that the mass has no effect on the coefficient of friction. After comparing the coefficient of friction between all the masses from procedure 1 and 2 in all possible inclines, we recorded an average standard deviation of 0.04 which further supports our claim. While this system was efficient, it is not the most ideal method to use. The experiment can be improved by applying lubricant on the pulley system to reduce the amount of friction on the pulley itself because we only want to derive the coefficient of friction between the block and the inclined plane. Second, we would have to avoid tapping the block to facilitate motion, this would prevent improperly recording a value as if it was having moving at constant velocity. The block needed to move in constant velocity because any acceleration would result in addition of a force. Third, we should invest in using better calibrated meter sticks to make sure that we record proper values. Lastly, instead of performing a single trial for each block, we should perform three trials for each block and work with the average values. We noticed that our method was not precise and accurate which can be seen in our varying percentage differences. References. Serway, Raymond A. Physics for scientists and engineers. Brooks Cole, 2014, pp 132. Young, Hugh D., et al. Sears and Zemanskys university physics: with modern physics. Pearson, 2016, pp 147. University of North Texas Physics 1730 Lab Manual, 2017, pp 69-78.
