Friction 58294i

Static and Kinetic Friction

Abstract The coefficients of static and kinetic friction for wood-on-wood was measured experimentally by observing the friction force and normal force due to gravity. The relationship between the angle of inclination of a plane and the coefficient of static and kinetic friction was also explored. Additionally explored was the relationship between surface area of and the coefficient of static and kinetic friction. It was determined experimentally that the coefficient of static friction for wood was .4321 ± 2.6% and the coefficient of kinetic friction for wood was .2982 ± 3.8%. STUDENT:

Huard, Andrew

PARTNER:

Cosco, Andrea

INSTRUCTOR:

Mr. Becht

DATE:

October 9, 2005

MOORPARK COLLEGE

THEORY The force of friction varies directly with the normal force due to gravity. Static friction is defined as: f s = µ s Fn

(1)

where µ s is the coefficient of static friction and Fn is the normal force. Kinetic friction is defined as: f k = µ k Fn

(2)

where µ k is the coefficient of kinetic friction. Alternatively, the coefficient of static friction is defined as: tan θ = µ

(3)

s

where theta (θ ) is the angle between an incline and a horizontal surface at which an object will just start to slide along the incline from rest. Additionally, the coefficient of kinetic friction is defined as: tan θ = µ

(4)

k

where theta is the angle between an incline and a horizontal surface at which an object will slide along the incline at constant velocity. THE EXPERIMENT Part 1:

Static VS Kinetic Friction

A wooden block weighing (524.7 ± .1) grams is placed on a smooth wooden surface of the same type. A 400-gram weight is placed on top of the block to make measurements easier. This total weight equals (924.7 ± .1) grams. The grain pattern of the wood in the block is made parallel with the grain pattern of the wooden surface and a small piece of tape is used to mark the starting position of the block. The block and wooden surface are sanded using steel wood and sandpaper to remove any foreign inconsistencies as well as to insure the smoothest conditions possible. A string is then attached to an eye-bolt at one end of the block and the other end is fastened to a spring force scale. An Ohaus Pull-Type Spring Scale (Model 8003-MN) was used in this

Source: Ohaus Corp. www.ohaus.com

The intent of this part of the experiment is to calculate the coefficients of static and kinetic friction, µ s and µ k, between a wooden block and wooden surface.

Figure 1.1

experiment, which has a least count of 25 grams (see Figure 1.1.) Figure 1.2

Once the apparatus is set up, the minimum force required to overcome static friction is measured by pulling on the spring scale as shown in figure 1.2. Five data points are collected and then averaged to find the average static friction force, fs . The data is shown in table 1. This experiment measures the friction force in grams for simplified calculations. Both Newtons and grams are graduated on the Ohaus (Model 8003-MN) spring scale. fs

(± 25) grams 400 400 400 400 380

Figure 1.3

Force vector diagram on the wooden block system

Table 1

Averaging the values for fs in table 1,

fs

is evaluated to (396 ± 25) grams.

To find the average kinetic friction force, fk , read the spring scale as the block is moving across the wooden surface with zero acceleration (constant velocity). The data is shown in table 2. fk

(± 25) grams 200 180 200 200 190 Table 2

Averaging the values for fk in table 2,

fk

is evaluated to (194 ± 25) grams.

ANALYSIS Equation (1) may be written in the form,

µs =

fs Fn

(5)

which shows µ s is directly proportional to fs and Fn. Furthermore, since fs and Fn are both known experimentally, µ s can be evaluated. 396 .0 g 924 .7 g µs =.4282 ± 6.3%

µs =

Equation (2) may also be written in of µ k,

µk =

fk Fn

(6)

and evaluated, µk =

194 .0 g 924 .7 g

µk =.2098 ± 13%

ERRORS / CONCLUSION The main source of error in this experiment is the innate non-precision of the spring scale for measuring force. Because the scale “bounces” as the block comes out of static friction into kinetic friction, it is impossible to read accurately the magnitude of friction force, even to the least count, and an estimation must be performed. A more appealing alternative may be to use a computer-recorded sensor that may provide a much smaller least count. Other sources of error include the inherently rough, inconsistent, and uneven properties of wood as wells as other foreign impurities such as dust and oil from human skin deposited from handling the wood. Another potential error may be from holding the string/scale slightly above or below the horizontal, which creates an angled vector force not aligned with the x-axis. Uncertainty in µ s and µ k was evaluated using the propagation of errors method. The following formula was used to evaluate the uncertainty in µ s: ∂µs

µs (7)

Since,

(100 ) = (100 ) (

∂µs 1 = ∂fs Fn

and

∂µs df s ∂fs

µs

∂µs dF n ) 2 + ( ∂Fn )2

∂µs − fs = ∂Fn Fn 2

µs

1 − fs df s dF n 2 ∂µs (100 ) = (100 ) ( Fn ) 2 + ( Fn )2 f s f s µs Fn Fn ∂µs

µs

df s 2 −dF n 2 ) +( ) fs Fn

(100 ) = (100 ) (

(8) ∂µs

(100 ) = (100 ) (

25 g 2 − 0.1g 2 ) +( ) 924 .75 g

µs 396 g ∂µs (100 ) = (100 ) 0.003986 µs ∂µs (100 ) = (100 )( 0.631 ) µs ∂µs (100 ) = 6.3% µs

Similarly, µ k is also evaluated by: ∂µk ∂µk df k dF n ∂µk ∂fk (100 ) = (100 ) ( ) 2 + ( ∂Fn )2 µk µk µk (9) Since,

∂µk 1 = ∂fk Fn

∂µk − fk = ∂Fn Fn 2

and

1 − fk df k dF n 2 2 F n F n (100 ) = (100 ) ( ) +( )2 fk fk µk Fn Fn

∂µk

∂µk

µk

(100 ) = (100 ) (

df k 2 −dF n 2 ) +( ) fk Fn

(10) ∂µk

(100 ) = (100 ) (

25 g 2 − 0.1g 2 ) +( )

µk 194 g 924 .75 g ∂µk (100 ) = (100 ) 0.01666 µk ∂µk (100 ) = (100 )( 0.128866 ) µk ∂µk (100 ) =13 % µk

Part 2:

The Affect of Area on the Force of Friction

This part of the experiment is intended to determine the properties of friction in relation to surface area. The procedure and apparatus is the same as in part 1, except the surface area of between the wooden block and surface has been reduced to half of its original area (see figure 2.1).

Sample block with ½ surface area on one side

Table 3 shows the data collected: Iteration 1 2 3 4 5

Figure 2.1

fs

± 25 (grams) 360 350 360 360 360

fk

± 25 (grams) 180 200 200 200 200

Table 3

ANALYSIS Evaluating the averages from table 3, f s =(358 ±25 ) g

and

f k =(196 ±25 ) g

Using equations (5) and (6), µ s and µ k can be evaluated respectively. µs =

358 .0 g 924 .7 g

(using equation 5)

µs = .3872 ± 7.0%

µk =

196 .0 g 924 .7 g

µk = .2120 ±13 %

ERRORS / CONCLUSION The coefficient of static friction in part 1 is just outside the uncertainty of static friction of part 2. The additional error is most likely due to other factors of uncertainty not calculated in the experiment. The percent difference of µs is,

E1 − E 2

[ ](100 ) % Difference in µs = E1 + E 2

(11)

2 .4282 −.3872

[ ](100 ) % Difference in µs = .4282 +.3872 2

% Difference in µs = 10% The percent difference in µs makes this test for a relationship between surface area and static frictional force inconclusive. The percent difference in µk is, .2098 −.2120

[ ](100 ) % Difference in µk = .2098 +.2120

(from equation

2

11) % Difference in µk = 0.10% Seeing that there is a very low percent difference in µk , it is conclusive that there is no relationship between kinetic friction and surface area of . However, since static friction and kinetic friction are closely related by similar equations, it is also conclusive that the same must be true for static friction. Major sources of error were the same as part 1. However, there was a greater possibility of depositing human skin oils on the ing surface of the block due to handling the top of the block during part 1 of the lab. Also, since the surface of the block has changed, there is also a possibility that the “thinner” side of the block might be more rough and inconsistent than its “fatter” side, which was tested in part 1. This might be prevented in the future by more rigorously sanding the “thin” side to make sure it is of equal experimental quality to the “fat” side. Washing hands before the lab might also help eliminate the possibility of depositing oils.

Part 3:

The Affect of Normal Force on the Force of Kinetic Friction

Figure 3.1

This part of the experiment will calculate the coefficients of static and kinetic friction, µ and µ k, between a wooden block and a wooden surface using variable weights for Fn and tension in the string. The tension force in the string where the block just begins to slide will be the force of maximum static friction (fs), and the tension force in the string where the block slides at constant velocity and zero acceleration will be the force of kinetic friction (fk). Both weights will be measured in grams for simplicity.

s

The apparatus involves a wooden block, wooden surface, and known weights just as in part 1 and 2, except now the string has one end fastened to an eye screw with the other end attached to a variable weight. A pulley is used to change the direction of the string as shown in figure 3.1. Data is taken by changing the weight of the wooden block and then adding weight to the string until tension equals the frictional force and either the block begins to slide (for fs) or the block slides with zero acceleration (for fk). Before testing static friction, the block is moved around to first break any additional static friction from the block remaining on the table too long. The block is then placed back in its original position marked by a piece of tape and 5 seconds is counted before testing the friction of the block. Table 4 shows the data collected:

Static Fn (g) 524 624 724 824 924 1024 1124

Kinetic fs (g) 245 275 330 380 420 445 505

Fn (g) 524 624 724 824 924 1024 1124

fk (g) 170 195 225 257 300 320 340

Table 4

Graphs of the data in table 4 are located in the appendix under “Normal Force VS Static Friction” and “Normal Force VS Kinetic Friction.” ANALYSIS Since the data in table 4 is linear, the least squares fit method may be used to find the slope of the line. Because equation (1) and (2) are linear, x and y may be substituted for the normal force and frictional force respectively. The slope of the line is either µs or µk depending on whether static or kinetic friction is being tested. Using equation one, fs = µsFn ⇒y = µsx

where µs is the slope of the line. The least squares fit method may be used to evaluate slope m ( µs ), m=

m=

N ∑xiyi − ∑xi ∑yi N ∑xi 2 − (∑xi ) 2

(12)

7( 2263400 ) − (5768 )( 2600 ) 7(5032832 ) − 33269824

m = .432143

The standard deviation in the slope (Sm) may be calculated using the equation, Sm = Sy

Sm = Sy

N N ∑xi 2 −(∑xi ) 2

7(5032832

7 ) −33269824

(13)

Sm = (0.00189 ) Sy

where,

∑δy

i

Sy =

2

N −2

(14) 7(52 ) 5

Sy =

Sy = 5.92

therefore, Sm = .01119 Sm

% error = µs (100 ) % error = 2.6% and, µs =.4321 ± 2.6%

Using equation (12) to evaluate µk , m=

7(1572468 ) − (5768 )(1807 ) 7(5032832 ) − 33269824

(using equation 12)

m = .2982

Using equations (13) and (14) to evaluate the standard deviation of the slope, Sy =

7(52 ) 5

(using equation 14)

Sy = 5.92 Sm = Sy

7(5032832

7 ) −33269824

Sm = (. 00189 ) Sy

(using equation 13)

Sm = .01119 Sm

% error = µk (100 ) % error = 3.75% therefore, µk = .2982 ± 3.8%

ERRORS / CONCLUSION The percent difference in µs and µk from part 1 and part 3 can be evaluated using equation (11), % difference in

µs = [

.4321 −.4282 ](100 ) .4321 + .4282 2

% difference in µs = 0.92 % % difference in

µk = [

.2982 −.2098 ](100 ) .2982 + .2098 2

% difference in µk = 35 % While the difference in µs is acceptable, the difference in µk makes this value inconclusive. Since part 1 of the experiment had a much higher percentage of error, the value from part 3 is more certain than part 1. Major sources of error were also reduced in part 3, which gives its value for µk more validity. Sources of error include inconsistencies within the block and wooden surface, dust, and foreign human skin oils coming into with the wood prior to the experiment. Although much better than the spring scale, the weight and pulley system still has error; the pulley has some friction within its bearings, which although negligible in magnitude, may have an infinitesimal effect on µk and µs . Also, when testing µs in part 3, the experiment requires that the block be moved from its original position and replaced in order to break any additional forces of static friction built up over time. This procedure is not done in part 1 when testing µs and may for some of the percent difference in the two values. Part 4:

Finding Coefficients of Friction by Inclination Angle

Figure 4.1

The final part of the experiment will explore the relationship between the angle of inclination of a plane and the coefficients of friction. Start by placing a wood block on an inclined wooden surface. Weights may be added to the wood block as shown in figure 4.1. Data points are taken for static friction by observing at what angle the block just begins to slide and for kinetic friction by observing at what angle the block will slide at constant velocity. Three different weights for the wooden block are used to collect data. Table 5 shows the data collected: Static Weight ± .1 (grams) 524 624 724

Kinetic Weight ± .1 (grams) 524 624 724

θ ± 2° 22 23 23 Table 5

ANALYSIS Taking the average angles, θs =22 .67 ° and

θk =18 °

Using equation (3) and (4), µs =.4177

and

µk =.3249

Evaluating the absolute error in the static coefficient of friction, dµs =

∂µs dθ ∂θ

θ ± 2° 18 18 18

dµs =

π csc 2 (22 .67 ) 180

dµs =.02498

Evaluating for the percent uncertainty of µs , dµs

µs

dµs

µs

(100 ) =

.02498 (100 ) .4177

(100 ) = 6.0%

Calculating the percent uncertainty in the kinetic coefficient of friction, dµk =

dµk =

∂µk dθ ∂θ

π csc 2 (18 ) 180

dµk =.0401

dµk

µk

dµk

µk

(100 ) =

.0401 (100 ) .3249

(100 ) = 12 %

Therefore, µs = .4177 ± 6.0%

and, µk = .3249 ±12 %

ERRORS / CONCLUSION The experimental resultant values could have been affected by some of the same sources of error as in previous parts; the wood could be inconsistent, dust might accumulate on the surface, and foreign skin oils may make with the surface. Unique to part 4 however, the inclined plane itself must be moved. Careful attention must be paid to the current angle of the plane to facilitate adjustments. Errors due to buildup of static friction were avoided by moving the block before testing µs . A

gentle nudge was given for µk , and a guess was made as to when it was sliding at constant velocity down the plane. Because of the difficulty in adjusting the plane with the equipment used, and guesses made for when the block was traveling with zero acceleration, a relatively high 2 degree uncertainty was recorded.