Friction
Introduction
Frictional forces always act parallel to the contact surface and perpendicular to the normal force. The static frictional force will take the magnitude (up to a maximum of μsN) and direction necessary to hold the object in place relative to the surface. The kinetic frictional force will always have a magnitude of μkN and a direction opposite the relative motion.
Note that the coefficients of friction are empirical. They cannot be derived from first principles of physics. Don’t bet your life on them and don’t expect experiment to always match theory. In other words, don’t get mad if this experiment doesn’t work out as you expect.
There are four different tests (plus variations thereof) you can use in this lab to investigate friction: inclined plane test for static friction, inclined plane test for kinetic friction, hanging mass test for static friction, and hanging mass test for kinetic friction.
Inclined plane test for static friction
If you place a block on an inclined plane, then slowly and steadily raise the ramp until the block breaks static friction, then the angle of the incline at which this event occurs may be used to calculate the coefficient of static friction. Use the typical rotated reference frame in this type of problem (put the normal force on the y-axis and the friction force on the x-axis) to obtain the following equations based on Newton’s first law:
ΣFx = 0
W*sinθ − fs = 0 (based on free-body diagram)
mg*sinθ − μsN = 0 (insert formulas for weight and maximum static friction)
ΣFy = 0
N − W*cosθ = 0 (based on free-body diagram)
N − mg*cosθ = 0 (insert formula for weight)
Solve these simultaneous equations for μs
N = mg*cosθ (the y equation solved for N)
mg*sinθ − μsmg*cosθ = 0 (the x equation with the substitution for N)
μsmg*cosθ = mg*sinθ
μs = sinθ/cosθ
μs = tanθ [1]
Inclined plane test for kinetic friction
If you place a block on an inclined plane, give the block a nudge down the ramp, and observe that the block moves with constant speed, then the angle of the incline at which this event occurs may be used to calculate the coefficient of kinetic friction. Use the typical rotated reference frame in this type of problem to obtain the following equations based on Newton’s first law:
ΣFx = 0
W*sinθ – fk = 0 (based on free-body diagram)
mg*sinθ – μkN = 0 (insert formulas for weight and kinetic friction)
ΣFy = 0
N − W*cosθ = 0 (based on free-body diagram)
N − mg*cosθ = 0 (insert formula for weight)
Solve these simultaneous equations for μk
N = mg*cosθ (the y equation solved for N)
mg*sinθ – μkmg*cosθ = 0 (the x equation with the substitution for N)
μkmg*cosθ = mg*sinθ
μk = sinθ/cosθ
μk = tanθ [2]
You can vary the mass, initial velocity, and contact area of the block in this particular test.
Note that while equations [1] and [2] are deceptively similar, the physical conditions under which you are allowed to use them are different. With equation [1], you use the angle at which the block begins to move on its own. With equation [2], you use the angle at which the block will move with constant velocity when given a nudge.
Hanging mass test for static friction
Another test for static friction is to put a block on a horizontal surface, then to connect it by a horizontal string over a pulley to a hanging mass. If you very slowly add mass to the hanger until the block breaks static friction, the masses of the block and hanger can be used to calculate the coefficient of static friction. A free body diagram of the block and Newton’s first law yield the following equations:
ΣFx = 0
T − fs = 0
T − μsN = 0 (formula for maximum static friction)
T = μsN
ΣFy = 0
N – W = 0
N − mblockg = 0 (formula for weight)
N = mblockg
Newton’s first law in the y-direction for the hanging mass yields the following:
ΣFy = 0
T – W = 0
T – mhangg = 0 (formula for weight)
T = mhangg
The three equations can be combined to solve for the coefficient of friction as follows:
T = μsmblockg (combining y and x equations for box)
mhangg = μsmblockg (substitute for T from hanging mass)
μs = mhang/mblock [3]
Hanging mass test for kinetic friction
Another test for kinetic friction is to put a block on a horizontal surface, then to connect it by a horizontal string over a pulley to a hanging mass. If you find the hanging mass that will maintain the velocity of the block after you give it a nudge, the masses of the block and hanger can be used to calculate the coefficient of kinetic friction. A free body diagram of the block and Newton’s first law yield the following equations:
ΣFx = 0
T – fk = 0
T – μkN = 0 (formula for kinetic friction)
T = μkN
ΣFy = 0
N – W = 0
N − mblockg = 0 (formula for weight)
N = mblockg
Newton’s first law in the y-direction for the hanging mass yields the following:
ΣFy = 0
T – W = 0
T – mhangg = 0 (formula for weight)
T = mhangg
The three equations can be combined to solve for the coefficient of friction as follows:
T = μkmblockg (combining y and x equations for box)
mhangg = μkmblockg (substitute for T from hanging mass)
μk = mhang/mblock [4]
You can vary the mass, initial velocity, and contact area of the block in this particular test.
Note that while equations [3] and [4] are deceptively similar, the physical conditions under which you are allowed to use them are different. With equation [3], you use the hanging mass for which the block begins to move on its own. With equation [4], you use the hanging mass for which the block will move with constant velocity when given a nudge.
Experimental Procedures
You will invent your own procedure to answer the following questions based on data, calculations of µ, error propagation for µ, and specific comparisons of µ:
1) Is μs > μk as predicted by theory? Provide evidence for your answer in the form of comparisons of µ with their errors.
2) Is μk constant for different tests (inclined plane versus hanging mass) as predicted by theory? Be sure to keep the string horizontal in all hanging mass experiments. Provide evidence for your answer in the form of comparisons of µ with their errors.
3) Is μk constant for different masses (not different substances!) as predicted by theory? Provide evidence for your answer in the form of comparisons of µ with their errors.
4) Is μk constant for different speeds as predicted by theory? The speeds will depend on how hard you nudge the block. The speeds need not be quantified, but they should be observed as different. Provide evidence for your answer in the form of calculations and comparisons of µ with their errors.
5) Is μk constant for different amounts of contact area (not different substances!) as predicted by theory? Provide evidence for your answer in the form of comparisons of µ with their errors.
All 5 questions must be answered based on data, calculations of µ, error propagation for µ, and specific comparisons of µ. If any of the 5 questions are not answered based on data, calculations, error propagation, and specific comparisons of µ you may have your report returned for resubmission. Hint: you will need to perform at least 6 tests to make 5 comparisons of µ. Did I mention the importance of comparisons of µ?
Please include a brief description of your procedure in the raw data section. As usual, you must estimate and justify error in your raw data, propagate that error in your calculations, and make scientifically defensible comparisons based on the results of your error propagation. Think very carefully about the estimated error for the angles and hanging masses. Multiple trials will not necessarily yield the same results, so your estimated error should attempt to encompass this uncertainty. This is certainly a case where the error in the raw data is not limited to the error in the measuring device. Please see the general lab instructions if you are unclear on this important point.
Comparisons of µ!