35
COEFFICIENT OF STATIC FRICTION:
A SIMPLE EXPERIMENT FOR APPLIED
MECHANICS STUDENTS
Sarah E. Leach
Mechanical Engineering Technology
Purdue University
1733 Northside Boulevard
South Bend, Indiana 46634-7111
574-237-4142
[email protected]
COEFFICIENT OF STATIC FRICTION: SIMPLE EXPERIMENT
FOR APPLIED MECHANICS STUDENTS
Sarah E. Leach
Purdue University School of Technology
South Bend, IN 46634
Key Words: static friction, equilibrium, friction coefficient, sliding
Prerequisite Knowledge: Students should have an understanding of basic mechanics including;
the concept of static equilibrium, free-body-diagrams, the location(s) of frictional forces, the
equation for calculating a force resulting from friction, and basic algebra and trigonometry.
Objective: Provide students with an opportunity to demonstrate the relationship between the
“angle of repose” and coefficient of static friction for sliding surfaces. This experiment also
provides an opportunity to experimentally determine the coefficient of static friction for several
pairs of material surfaces, and to evaluate the relationship between the area of surface in contact
and the coefficient of friction.
Equipment and Materials:
Short lengths and blocks of 2”x 4” dimensional lumber
rigid aluminum (or other metal) slab approximately 8” in length
abrasive paper (400 grit carbide paper or other fine grit)
bond paper
optional, other papers or sheets of material
yard or meter stick
Introduction:
Static friction creates a force between two surfaces which acts to impede motion. The magnitude
of the friction force depends on the normal force and on the nature of the surfaces in contact. The
N
maximum friction force that can be developed is given by the equation: F max =
µ
where N = normal force and µ = coefficient of static friction.
In a simple system of two unlubricated surfaces, between a block and a horizontal plane, the
maximum static friction force will be present when an external force, P, is equal in magnitude to
the friction force. This condition of static equilibrium is one of impending motion, since the
application of additional external force cannot be balanced by additional friction, motion
(sliding) of the block will occur. Note that there is no variable in this equation for the area of the
surfaces in contact between the block and the plane. As first proposed by Amontons in 1699, the
friction force is directly proportional to the normal force and is independent of the area in
contact1. The normal force, N, has a magnitude equal to the weight of the block, W, for a
horizontal plane.
Figure 1
Forces for horizontal sliding.
If the external force P is removed and the plane is tilted to some degree Θ , the conditions for
static equilibrium are altered to include both the normal component of the weight of the block
(W sin Θ ), and the component of the weight acting parallel to the plane (W cos Θ ).
Figure 2
Forces for sliding on an inclined plane.
In this arrangement, motion will be impending when the maximum static friction force is equal
in magnitude to the component of the weight of the block acting parallel to the tilted plane. An
equilibrium equation can be written which relates the coefficient of static friction to the angle of
the tilted plane.
F max = W cos Θ = µ (W sin Θ)
Rearranging gives:
µ=
W cos Θ
W sin Θ
then:
µ=
cos Θ
sin Θ
and finally:
µ = tan Θ
Using this final equation, students can use the angle of tilt for impending sliding to calculate the
coefficient of static friction between the surfaces in contact. The angle of tilt is found by
measuring the length of the sliding plane, L, and the height of the free end of the plane, h, when
motion is impending.
⎛ h⎞
Θ = sin−1 ⎜ ⎟
⎝ L⎠
Figure 3
Experimental setup for wooden block on inclined plane, showing a large area of contact.
Figure 4
Experimental setup for wooden block on inclined plane, showing a small area of contact.
Simple equipment and materials are sufficient for measurement of the coefficient of static
friction, as determined by the “angle of repose” method. Standard pine “2 x 4s” can be cut into
4” and 12” pieces. The longer pieces can be used as inclined planes, and the shorter pieces can be
used as blocks. A block can be placed on its broad surface, as shown in Figure 3, or on its narrow
surface, as shown in Figure 4, allowing comparison of the measured coefficient with two
different areas of contact. Sandpaper, bond paper, or other sheets of material can be used to
supply different experimental conditions. A thick aluminum sheet or slab can provide another
inclined plane surface for evaluation.
Procedure:
It is valuable for this experiment to perform a preliminary exercise to review the development of
the “angle of repose” method and give the students time to prepare a spreadsheet for data
collection and analysis3. Preparing a spreadsheet in advance will help them organize their
laboratory activities, clarify the requirements for experimental data, and allow rapid calculations
for comparison of experimental data and reference materials.
Students should be organized into teams of two to four members each. Each team should be
supplied with materials for evaluation. Students should work together to measure the length of
the inclined plane, and the height of the end of the plane when the block has impending sliding
motion. Teams should be required to collect at least three “trials” for each material combination,
and to use an average height for calculating each coefficient of friction. Note: when testing
wood-on-wood, the grain of the block should be parallel to the grain of the plane. The block
should not be placed on a cut surface or “across” the grain.
Photo 1
Experimental setup showing inclined plane, block, and measuring device.
Coefficient of State Friction from Angle of Repose
Materials
L, inches
Wood-on-wood, lg. surface area
12
Wood-on-wood, lg. surface area
12
Wood-on-wood, lg. surface area
12
h, inches
6.75
6.5
6
Average
Angle, theta,
degrees
(sin-1 (h/L))
34.23
32.80
30.00
Coefficient of Static
Friction
(tan theta)
0.68
0.64
0.58
0.63
Table 1
Sample spreadsheet for data collection and calculations.
After collecting data and calculating coefficients of static friction, students should be required to
compare their experimental data to published results, and to discuss their experimental results.
Students should determine whether or not their data support the theory that the coefficient of
friction is independent of the size of the area in contact. They should consider possible sources of
error; measurement, surface condition, surface defects, humidity, etc.
Comments:
This experiment was performed by five student teams in Spring 2004. The table below lists
materials combinations and the coefficients of static friction measured experimentally by the
students.
Materials
Wood on Wood,
large surface area
Wood on Wood,
small surface area
Wood on 400 grit
paper
Wood on
aluminum
Wood on bond
paper
Reference
Coefficient of
Static Friction4
0.25 - 0.5
Experimental Coefficient of Static Friction
Team 1
Team 2
Team 3
Team 4
Team 5
.57
.58
.63
.69
.52
0.25 – 0.5
.52
.42
.55
.57
.40
-
1.1
.98
.80
1.0
.89
0.6
.52
.35
.56
.48
.35
-
.52
.54
.59
.54
.44
Table 2
Summary of student experimental results.
Students were required to submit lab reports, in which they reported that the measured
coefficients were close to published data. Despite the differences shown in the table above, three
of the teams stated that the data confirmed that the coefficient was independent of the area of
contact. Team 2 reported that their data did not support the concept that the coefficient was
independent of area of contact, but offered no explanation. Team 4 attributed the apparent
discrepancy between the large and small areas of wood-on-wood contact to a surface defect in
their sample. Two teams reported coefficient values equal to or greater than 1 for the 400 grit
paper. Since a coefficient larger than 1.0 implies that additional interactions are occurring at the
interface, abrasion in this case, this result is higher than anticipated. No wood particles were
observed on the abrasive paper. The lower than expected results for wood-on-aluminum are
probably attributable to surface contamination of the aluminum. No solvents or other cleaners
were used to clean the surface, either before or during the testing. Even though the measured data
showed differences from the reference information, students seemed to gain an appreciation for
the obvious differences between the pairs of materials.
Analysis of data can be simple as in the results reported above, or can include a statistical
approach requiring students to determine variance and the significance of measured differences.
Each experimental “trial” can be completed in a few minutes, so many repetitions are possible,
and would probably improve the quality of the overall results.
In order to demonstrate the independence of friction on area of contact, it is helpful to provide
examples of data for other materials the students are familiar with, like automobile tires. Snow
tires have increased friction because of the tread design, not because they have a greater load per
unit area. The coefficient of static friction (no slipping) for a smooth automobile tire4 on wet
brick pavement at 5 miles/hour is 0.49, for a tire with circumferential grooves it is 0.58, and for
angular grooves at 45 degrees it is 0.77 .
References:
1. as cited in Bowden, F.P. and Tabor, D., The Friction and Lubrication of Solids,
Clarendon Press, Oxford, 1950, pg. 87
2. Walker, Keith M., Applied Mechanics for Engineering Technology, 7th Edition, Pearson
Prentice Hall, Upper Saddle River, New Jersey, 2004.
3. Meier, Mike, Integrating Spreadsheets into an Indroductory Materials Course,
Proceedings of the 2004 American Society for Engineering Education Annual
Conference, Salt Lake City, UT, 2004.
4. From Bowden and Tabor, as referenced in Machinery’s Handbook, Industrial Press, New
York.
5. Baumeister, T., Avallone, E.A., and Baumeister, T. III, Editors, Marks’ Standard
Handbook for Mechanical Engineers, Eighth Edition, McGraw-Hill, New York, 1978.
Biography:
Sarah Leach, P.E., is an assistant professor of Mechanical Engineering Technology at Purdue
University Elkhart/South Bend. Her primary teaching responsibilities are in materials and
applied mechanics.