experiment 06: COEFFICIENTS OF STATIC AND DYNAMICS FRICTION

Purpose

 

In this experiment we will determine the coefficients of kinetic and static friction for a block of wood on a wood plank, and the static friction coefficients for glass and other materials.  We will also test the hypothesis that the force of friction does not depend on the amount of surface area in contact. 

 

Principles

 

Whenever two surfaces touch, they exert forces on each other.  The ultimate source of these surface or “contact” forces is the electrical attraction or repulsion between the charged particles – electrons and protons – of which all matter is made. The vector sum of all the sub-microscopic forces between the particles in the surfaces is a macroscopic force that we can measure in the laboratory. 

 

Diagram 1 illustrates two surfaces in contact.  Each surface exerts an equal but oppositely directed force on the other.  The total force, F₁₂, that surface 2 exerts on surface 1 is in some arbitrary direction of space, as illustrated.  For convenience, we break F₁₂ up into components parallel and perpendicular to the surface.  We call the perpendicular component the normal force (F_N in the diagram) and the parallel component the frictional force (f).

 

 

 

The normal force tends to keep the two surfaces apart: it resists surface 1 pressing into surface 2.  The frictional force resists any relative motion between the surfaces and is always directed opposite to that motion, if the surfaces are moving, or opposite to any potential motion, if the surfaces are at rest.

 

Static and Kinetic Friction

 

Frictional forces are of two types, depending on whether or not the surfaces are moving relative to each other.

 

Static friction acts when the two surfaces are at rest relative to each other and resists any sliding.  Static friction (f_s) is a bonding force that tries to keep the surfaces together.  The direction of f_s is opposite to that of any external force that tries to make the surfaces move. Up to a certain limit, the magnitude of f_s is equal to the external force, so that the net force on the surface remains at zero and the surface does not move.  Static friction is thus a variable force, taking on any value necessary to keep the surface in equilibrium.

 

However, f_s cannot exceed a certain maximum value.  When the external force exceeds this maximum value, the surface bonding between the materials breaks and sliding begins.  The maximum value of f_s depends on the chemical makeup of the surfaces and must be determined by experiment.

 

Once sliding begins, kinetic friction (f_k) takes over.  Kinetic friction is directed opposite to the relative velocity between the two surfaces, resisting the motion and slowing its speed.  Kinetic friction is approximately constant under normal conditions.  Like static friction, its magnitude depends on the surfaces involved and must be determined by experiment.

 

The Coefficient of Friction

 

Experimentally, we find that both static and kinetic frictional forces are, to a good approximation, proportional to the normal force between the surfaces.  The ratio of the frictional force to the normal force is called the coefficient of friction, (m):

 

(1)                                                       

 

Mu is a unit-less number and is usually less than one in value.

 

For any given surfaces the coefficients of static and kinetic friction are different, so we must distinguish between m_s (for static friction) and m_k (for kinetic friction).  If we know the coefficients and the normal force, then we can calculate the frictional forces:

 

(2)                                                       

 

(where the inequality reflects the fact that f_s is variable; the maximum value of f_s is given by the equal sign;)

 

(3)                                                       

 

Usually, m_s is greater than m_k, so the maximum strength of static friction is greater than the strength of kinetic friction.

 

Determining the Frictional Force

 

In our experiment we will first determine the coefficients of static and kinetic friction between two wood surfaces (a wood block and a wood plank).  Diagram 2 illustrates the set-up we will use.  The block (M₁) is placed on a horizontal plank and is connected by a string to a weight (M₂), which is hung vertically by means of a pulley.  To find the magnitude of kinetic friction, we hang just enough weight on the string so that if the block is set in motion, it slides at constant speed.  (We must first set it in motion by giving it a little shove, since static friction will otherwise hold it in place.)

 

Since the block’s acceleration is zero in both the horizontal and vertical directions, Newton’s second law of motion tells us that the sum of the forces in each of these directions must be zero.  In the horizontal direction, the tension in the string (which equals the weight of the hanging mass) pulls the block and kinetic friction resists the pull.  In the vertical direction, the block’s own weight pulls down and the normal force pushes up.  Each of these pairs of forces must be equal and opposite, so we must have:

 

(4)                                          and                   

 

where W₂ is the weight of the hanging mass and W₁ is the weight of the block (and any mass on top of it).  Since we know the masses, we can calculate the frictional and normal forces.

 

We can determine the coefficient of kinetic friction by running several trials and calculating the ratio of the masses for each trial:

 

(5)                               

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Diagram 2

 

 

The coefficient of static friction can be determined in much the same way.  In this case we leave the block at rest and look for that hanging mass which just starts the block sliding.  When the block just begins to slide we have

 

(6)                                          and     

 

just as in the kinetic case.  We will determine m_s by graphing f_s as a function of F_N.   The slope of the graph is m_s.

 

Friction and Surface Area

 

Since frictional forces arise when two surfaces are in contact with each other, it might seem reasonable to expect that the magnitude of the frictional force would decrease as the amount of surface area in contact decreases.  However, we find that frictional forces are roughly independent of the size of the area in contact.

 

As an explanation for this, consider that the weight of the block is the same, regardless of whether the wide or the narrow face rests on the plank.  For the narrow face, we have the same force (the weight of the block) pressing into a smaller area of the plank.  This means the pressure is greater (pressure is force divided by area) and the surfaces are squeezed together more tightly.  This increases the frictional force in the same proportion as the surface area has decreased.  The result is that the magnitude of the frictional force is the same as it was with the wide face down.  We will test this idea by re-performing our experiment for kinetic friction using the narrow face of the block and comparing results in the two cases.

 

Limiting angle of repose

 

There is a second way of determining the coefficient of static friction.  Refer to diagram 3.  We put the block on the plank and raise one end of the plank so that it makes an angle q with the horizontal.  When the angle is large enough, the block will slide down the incline.

 

 

 

 

 

 

 

 

 

 

 

           

 

 

 

Diagram 3

 

 

At an angle just before the block begins to slide (called the limiting angle of repose), the forces are still balanced and we have:

 

                                       and                 

 

Taking the ratio of these two equations gives us m_s:

 

(7)                                           

 

 

.

The procedures below consist of four related experiments:

 

1.  Determine the coefficient of kinetic friction between a wooden block and a wooden plank by finding the force (weight of hanging mass) necessary to keep the block moving at constant speed.  We will simply “eyeball” the motion of the block to estimate when it is moving at constant speed.

 

2.  Perform the same experiment with a smaller area of contact between the block and plank to determine what effect surface area has on the force of kinetic friction.

 

3.  Determine the coefficient of static friction between the block and the plank.  This uses the same setup as in part 1; here one simply determines what weight of hanging mass is necessary to start the block moving from rest.

 

4.  Determine the coefficient of static friction from the limiting angle of repose.  Here, we use an inclined plank and find the maximum angle to which the plank can be raised before the block begins sliding.

 

 

Equipment

 

 

·         Wood block

·         Glass block

·         Block with other material (optional)

·         Wood plank

·         Pulley

·         Mass hanger

·         Mass set, including 5 100-gram masses as weights

·         String

·         Table stand

·         Right angle clamp

·         Metal rod

·         Inclinometer

 

Kinetic Friction – Wide Face of Block

 

1.      Weigh the block and record its mass.

2.      Wipe the surfaces of the plank and block with a moist paper towel.  Make sure both are free of dirt and grit.

3.      Clamp the pulley at the end of the plank and place the plank at the edge of a lab table.  Place the block on the far end of the plank and attach a length of string to it.  Drape the string over the pulley and hang the mass hanger from its end.  The string should be short enough so that the block can slide the length of the plank before the mass hanger hits the floor.

4.      Determine what weight must be added to the hanger so that the system (block, string and hanger with mass) moves at constant speed:

 

·         Add a little mass to the hanger.  Give the block a slight push to start it moving.  If the block accelerates (speeds up), take a little mass off and try again.  If the block decelerates (slows down), add a little more mass and try again.  (Note:  If the block accelerates from the weight of the mass hanger alone, put 20 or so grams on top of the block and treat that mass as part of the block.)

·         If the block moves the length of the plank at roughly the same speed, you have found the necessary mass.

·         Record the total hanging mass and its weight.  The weight of the hanging mass is equivalent to the force of friction on the block (see equation 4 above).

·         Record the total mass of and on the block and its weight.  The weight on the block is equivalent to the normal force, F_N.

 

6.      Repeat the above process, adding 100 grams on top of the block for each new trial, for a total of six trials.

 

Kinetic Friction - Narrow Face of Block

 

7.      To test the hypothesis that the force of friction is independent of the surface area in contact, repeat the above experiment using the narrow face of the block:

 

• Set the block at the end of the plank on its narrow face and reconnect the string to the lower hook so that the string is horizontal.
• Determine the amount of hanging mass necessary to keep the block moving at constant speed after an initial push.  Do this for 0 – 500 grams placed on the block, in 100-gram increments, for a total of six trials.

 

 

Analysis

 

1.  Calculate the normal force, F_n, and the frictional force, f_k, for each trial.  Note that the normal forces are the same for the wide & narrow faces.

2.  Calculate the coefficient of friction μ_k for each trial.

3.  Find the average values, the deviations, and the standard deviation for both data sets.

4.  Calculate the percent difference between the average values for the wide & narrow faces.

5.  Questions:

a)  In determining the kinetic friction force, f, why was it necessary that the block move at constant speed?

b)  Using Newton’s Laws, show how you would measure μ_k for an accelerating block.

 

Static Friction – Flat Plank

 

1.      Use the same set-up as in the kinetic friction experiments, with the wide face down.  This time, however, determine what hanging mass is necessary to just start the block moving without a push.

2.      Test five different masses on the block, starting with no mass on the block and adding 100 grams for each new trial. Determine the average hanging mass necessary to start the system moving from rest for each mass set.  Make at least 3 trials for each mass set, by the following procedure:

 

·         First, wipe the plank again with a dry cloth or paper towel.

·         For each new trial, take all mass off the mass hanger.  Lift the block off the plank and replace it firmly on the board at the same starting position.

·         Build up the mass on the hanger using mass increments so that the block slides after you have put at most a 5-gram piece on the hanger.  This means you will need to first find the general mass range in which the block will slide.  Then work up to the sliding mass using small mass increments.  If the block slides after you have placed, for instance, 10 grams on the hanger, do not use that trial: you have overshot the mark.  Ideally, you want the last mass placed on the hanger to be a 1- or 2-gram piece.

 

This procedure is necessary because the block tends to become “cold-welded” to the plank while you are slowly adding mass to the hanger.  You will no doubt notice this effect as you perform the experiment.  Doing it this way keeps each trial consistent with the others.

 

1.      Record the total mass on the block (m₁) and the hanging mass, including the hanger (m₂), for each trial.

 

 

Analysis

 

1.  Calculate the average hanging mass for each normal force and from this the average maximum static friction force for each mass set.

2.  Graph the frictional force as a function of the normal force.  Assume that if the normal force is zero (a massless block), the frictional force will also be zero, so use the origin (0,0) as the first point of the graph.  Using a straight edge, draw the best line you can determine from the origin and through the data points.

3.  Take the slope of the graph and write down the equation of the graph.  The slope is your experimental value for the coefficient of static friction.

4.  Question:  In determining the static friction force for the block on the plank, if too much mass were placed on the hanger, the block would jerk off suddenly.  Use your results for μ_k and μ_s to explain why.

 

 

Static Friction:  Limiting Angle of Repose

 

1.      Clamp the right-angle clamp to the table stand.  Insert the metal rod into the hole in the side of the plank and clamp the end of the rod with the right angle clamp.  By raising and lowering the clamp, you can increase or decrease the angle the plank makes with the horizontal.  Start with the plank all the way down (nearly horizontal.)

2.      Remove the string from the block and place the block on the plank near the clamp.

3.      Slowly raise the clamp (and the plank) until the block begins to slide.  Tighten the clamp as soon as sliding begins.

4.      With the inclinometer, measure the angle the plank makes with the horizontal.  This is the limiting angle of repose for the block.  Record the angle in Table 4.

5.      Repeat the above two more times so that you have three separate measurements of the limiting angle.  Start with the plank horizontal each time.

6.      Perform the above procedures three times using the slab of glass or other materials provided.  Record the limiting angles in Table 4.

 

 

Analysis

 

1.  Find the average of the three recorded angles. This is q_L, the limiting angle of repose.  2.  Find the deviations from the average for each trial.

3.  Calculate coefficient of static friction for each material using q_L equation (7).

4.  Find the uncertainty in the coefficient, Δμ_s, using the average deviation in  the limiting angle as Δq_L.  (Refer to the section in the introduction, Calculating with Errors, if necessary.)

5.  Take the percent difference between μ_s for the block found here and that found by the flat-plank method.

6.  Question:  Calculate the acceleration of the block on the incline once it started moving.  Use your value of μ_k from Part 1 and the average angle of repose.

 

 

 

Note:  Your data should be recorded in your lab notebook.  The following is a guide only

 

 

Mass of block (g):   ___________________

 

 

Kinetic Friction – Wide Face
Trial	Mass on block	Total Mass (m1)	Normal Force (FN)	Hanging mass (m2)	Friction Force (fk)	μk	Dev.
1
2
3
4
5
6

 

                                                                                                                Average:  _______  _______

 

Standard Deviation:           _______

 

 

Kinetic Friction – Narrow Face
Trial	Mass on block	Total Mass (m1)	Normal Force (FN)	Hanging mass (m2)	Friction Force (fk)	μk	Dev.
1
2
3
4
5
6

 

 

                                                                                                                Average:  _______  _______

 

Standard Deviation:           _______

 

Flat Plank Method

 

 

Mass of block (g):   ___________________

 

 

 

Trial	Mass on block	Total Mass (m1)	Normal Force (FN)	Hanging mass (m2)	Friction Force (fs)
1				1.
				2.
				3.
				Av:
2				1.
				2.
				3.
				Av:
3				1.
				2.
				3.
				Av:
4				1.
				2.
				3.
				Av:
5				1.
				2.
				3.
				Av:

 

 

Equation of Graph:                                                              μ_s :  __________

 

Limiting Angle of Repose

 

 

Trial	Limiting angle (q)
	Wood	Glass	Other
1
2
3
Average
ms

 

 

 

 

                                                            Percent Difference:               ___________

                                                            (m_s wide & narrow faces)