# Coefficient of Friction

Experiment Type: Cookbook

Overview

In this experiment, students will determine the coefficient of static friction and the coefficient of kinetic friction for a weighted cart bottomed with felt sliding on aluminum.  Students will determine what dependence the coefficients have upon the contact area of the cart and the force pressing the two surfaces together.

Key Concepts

Newton’s laws, coefficient of static friction, coefficient of kinetic friction

Objectives

On completion of this experiment, students should be able to:

1)      explain what dependence the coefficient of friction has upon the contact area of the cart

2)      explain what dependence the coefficient of friction has upon the forces between the surfaces

3)      explain the concept of friction

4)      explain how we can use Newton’s laws and kinematics to find the frictional force

Review of concepts:

Friction is the culprit

In the lecture part of the course, you have learned Newton’s laws.  Newton’s second law can be written (in component form):

(5-1)

What these equations say is that if you apply a net force to an object, that object will accelerate.  They also say that if the force is zero, the object will have zero acceleration—its velocity will not change!  That seems a bit counter-intuitive to say the least!  If you push a block on the floor, it will come to a stop almost immediately after you stop pushing—it won’t keep moving across the floor forever.  Are Newton’s laws wrong?

Well, there’s another force present: friction!  Friction causes the block to slow down and eventually stop.  Friction is present in almost every real experiment.  Sometimes friction is a good thing (brakes on your car) and other times we would like to minimize it (grease motor parts).  In order to design any real machinery, we need to understand the frictional force better.  What variables is it dependent upon?  How does it work?

In this experiment, we will use Newton’s laws to see what factors affect the frictional force.  We will look at a 1D motion of a block sliding on a plane.  Consider Fig. 5-1.

Figure 51 A cart with mass mc sliding on a table and attached to a freely hanging mass mh

What forces are on each block? What is the acceleration of each block?

 (a) Fhc:    the force of the hanging mass on the cart Fg:     the force of gravity on the cart FN:    the normal force from the table on the cart fk:      the frictional force from the table on the cart (b) Fch:    the force of the cart on the hanging mass Fg:     the force of gravity on the hanging mass

Figure 52 Free body diagrams of (a) the cart and (b) the hanging mass

The sum of the forces on the cart is:

(5-2)

Since there is no acceleration in the y direction, FN = Fg = mcg.

The sum of the forces on the hanging mass is:

(5-3)

Since the two objects are tied together, they must move together.  Thus, they must have the same acceleration, ax.  Also, Fhc must be equal in magnitude to Fch if the pulley between them is assumed to be massless.  Solving for the frictional force in equation (5-1) yields:

(5-4)

Substituting (5-3) into (5-4) yields:

(5-5)

Thus, we can find the force of friction if we can measure both masses and the acceleration of the cart (or equivalently the acceleration of the hanging mass).  We can try varying the contact area of the cart to see if this has any effect on this frictional force.  We can try varying the amount of mass the cart has to see if increasing the normal force has any effect on this frictional force.  We could also see what effect changing the angle of the plane has on the frictional force.  To do this last part, however, we need to rework the equations in the same manner as we did for a flat plane.  The free body diagram for the cart on a plane which is inclined by an angle θ is shown below in Fig. 5-3.

Figure 53 The free body diagram for the cart on an inclined plane

The Coefficient of Friction

How do we normally model friction?  In your textbook, you undoubtedly have the following equation:

(5-6)

The symbol μk stands for the coefficient of kinetic friction.  This coefficient is a constant of proportionality between the frictional force and the normal force.

If we plot fk vs. FN, the slope should give us the coefficient of kinetic friction.

Figure 54 A graph of frictional force versus normal force

The coefficient of kinetic friction in the graph shown in Fig. 5-4 is μk = 0.2065 ± 0.0025 (no units).

Is fk = μkFN “the answer”?

Are we doing this experiment for nothing? Do we really need to measure anything at all?  After all, equation (5-6) seems to tell us everything that we want to know.  It says that the frictional force is only dependent on the normal force (not angle or area or anything else).  It also says that the frictional force is constant for a given normal force. It implies that for a given set of surfaces there is a constant μk which describes the “roughness” of the two surfaces.  Can’t we just look this up in a table?

The operative word in this equation is “model”.  Equation (5-6) is a model of how friction works—sometimes.  Let me see if I can poke some holes in this model to show you that things are not as easy as it seems.  Are wide tires better than narrow tires?  Equation (5-6) says that they should give the same frictional force (at least when they are sliding).  Why don’t we use bicycle tires for our cars?  What if we put sticky tape on the bottom of our cart?  Have you ever seen something with sticky tape accelerate (or decelerate) smoothly (with a constant acceleration)?  What exactly does lubricant do?  I know that sometimes we model friction as a function of velocity.  (In fact, we will do so in the damped oscillations lab!)  However, this equation says nothing of velocity!  Probably the biggest question we should ask is: why?  Why is friction proportional to the normal force?  Why is this a valid model—even sometimes?  Where does friction come from in the first place?

Part of your assignment in this lab is to see how well this model describes this particular situation.  However, you should be aware that even if it does, that does not prove that the model will be valid in other situations.

Procedure

As always, the procedure listed in Part I – V is not set in stone.  If you can think of a way to improve the procedures while still meeting the objectives, you can feel free to modify them; let your TA know if you have decided to deviate from the procedure.

The apparatus in detail

We will perform today’s experiment with the following equipment:

1. Two plastic carts, one red and one blue
2. One aluminum track with inlaid ruler and stationary backstop at end of track
3. Movable stop block (attaches to track)
4. Fishing wire
5. One plastic pulley wheel
6. Hanging weight set (mass hanger and weights)
7. Rectangular masses for carts
8. A stopwatch
9. Video camera (will be handled by TA)

Place the aluminum track so that it lies a few inches in from the long edge of your table. The end of the track should overhang the table slightly so that the table does not interfere with the pulley and hanging mass. Use a bubble lever to determine if the track is level; if it isn’t, use the screw at the end of the track under the backstop to adjust the title of the track. Attach the pulley in the center of the track with the clamp screw downward; you may wish to use a ruler to verify your placement.

Attach the hanging masses to the cart in such a way that a) the force of the masses is directed only along the horizontal and b) the hanging mass doesn’t impact with the floor. We suggest that you cut the string so that the when the cart is as far back along the track as it can go the mass hanger is just shy of the pulley. This will give you the longest distance for measurements (from highest starting position of the masses to the ground) and an easy reference for where to start the multiple trials for this experiment (against the backstop).

Now adjust the height and/or angle of the pulley so that the fishing wire is parallel to the ground. Non-horizontal fishing wire will add a vertical component to the force that will increase as the cart moves down the track. You can check your wire by using a ruler to see how far above the track the wire is at the cart and at the pulley. If you are not satisfied that the heights are equal, you’ll have to readjust the pulley.

Slide the cart down the track until the masses almost hit the floor. Place the movable stop here and record the distance from this point to the front of the cart when is abuts the backstop. This is the distance that the cart travels.

Make a final inspection of your setup to be sure that nothing except friction impedes the cart’s progress along the track and that the system’s behavior is not obviously inconsistent with what one would expect. Some slight transverse wobble of the cart is acceptable, and there may be some jerk in the motion of the cart. This will be explored later; just make sure it is minor and isn’t due to obvious sources.

Part 0: The Estimate Of Static Friction

Each time the cart slides down the track starting from rest it has to first overcome the force of static friction. We can estimate this force, and thus estimate how much weight will be needed to get the cart moving, by doing the following:

1.      Remove the mass hanger from the wire

2.      SLOWLY tilt the track until the cart begins to move.

3.      Record this angle and use the fact that

and the fact that at this angle force of gravity along the track is approximately equal to the static friction force.

Part I: The coefficient of kinetic friction (on a level surface)

1.      Measure the mass of the cart alone.

2.      Make a few trial runs with 250gm in the cart if you start with only one cart mass, 750gm if you start with three cart masses.  Find out how much weight you have to add to the hanging mass to get the cart to slide.

3.      Measure the length of motion.

4.      Using the masses you have measured in step 2, allow the hanging mass to fall and measure the time of fall (i.e. measure the amount of time it takes from the moment you release the cart to the instant it hits the stop block).

5.      Repeat step 4 at least five (5) times. Periodically check that the length of motion has not changed, as the stop block can move slightly as a result of repeated collisions.

6.      Steps 1 – 5 describe the procedure you should follow to find the frictional force for a single, set normal force.  Next, you want to change the normal force.  Do this by adding weight to the cart.  Get at least three more sets of data (three different normal forces) by repeating steps 1 – 5 with different amounts of mass in the cart. You will have to give/take a mass from another group that has less/more.

7.      To analyze your data, see the Analysis section of this chapter.

Part II: The coefficient of kinetic friction (changing the area of contact)

There are two carts: one cart has a single wide swath of material underneath, the other has two smaller patches. Switch the cart you have for a cart with a different area of contact. Use the procedure described in Part II to determine the coefficient of friction for this other cart.  Is it the same as before?

Part III: The coefficient of static friction

1.      Restore the apparatus to its original configuration (Fig. 5-5).

2.      Determine the coefficient of static friction by finding the amount of weight you need to add to the hanging mass such that the cart just starts moving.  Record this weight.  Also record the amount of weight in the cart.  Make sure that when you are doing this you do not jostle the hardwood board.

3.      Repeat step 2 for at least three other normal forces (add different weights to the cart).

Part IV: Is the acceleration constant?

You should have enough experience with determining kinematical variables from Chapter 3 and 4 such that you can come up with a procedure to determine whether or not the acceleration of the cart is constant. The equation fk = μkFN implies that the force of friction is constant if the normal force is held constant.  How true is this? Please record your procedures and techniques.

Part V: (Optional) The coefficient of kinetic friction (using an inclined surface)

1.      Attached to the 2x4 is a small metal bracket.  This bracket holds a stand.  Attach the stand to this metal bracket.  Also, you will notice that the hardwood board has a hole drilled through its width.  There is a metal bar which fits into this hole.  A clamp at the end of the bar attaches to the stand.  With these two additions, you can elevate one end of the hardwood board.

2.      Repeat the procedure in Part II, but with one difference.  Instead of adding weight to the cart, increase the angle of incline (by raising the end of the board).

3.      Find the frictional force for at least 5 angles. (Note: the apparatus can only be raised to about 40° at maximum.)  Be sure to measure the angle of incline for every set of data.

Analysis

Parts II-III

For parts II-III, we will assume that the acceleration is nearly constant.  You can tell me how good—or how bad—this assumption is using your data from Part I!

If the acceleration is constant and the initial velocity is zero, then the length of motion, L, as it is defined in Fig. 5.5 is related to the time-of-fall, t, by the equation:

(5-7)

You can rearrange this equation to solve for the acceleration:

(5-8)

Which t should you use?  You measured the time-of-fall at least five times.  You have two options here.  You can either:

1.      Solve for the acceleration in each trial, then average the acceleration and find the appropriate statistical error in this acceleration

2.      Average the time-of-fall and propagate the error from the statistical error in the average t and the estimated error in L.

In either case, you can simplify your life by reading Appendix A: Using Spreadsheets in Experiments. You can now plug this acceleration (and associated error) into equation (5-5).  The normal force can be found using Eq. (5-9) for Part II and Eq. (5-10) for Part III.

(5-9)

(5-10)

(θ has been previously defined in Fig. 5-3)

Plotting fk vs. FN (as in Fig. 5-4) will yield a graph which tells you the coefficient of kinetic friction and it will also tell you how well a linear function fits the data.  You must perform a least squares fit (linear fit) to this data.  Do not take the difference between points and get the “average” slope! We want to know how well a line fits the data.  If you average, you are already explicitly assuming that the friction is proportional to the normal force—but that’s exactly what we are trying to verify!

Part IV

In part IV, we assume that right before the cart starts moving the weight you added to the hanging mass was just enough to overcome static friction.  In other words, right before kinetic friction kicked in, the static frictional force had reached its maximum.  In the static case, the acceleration is zero, so that the force from the hanging mass equals the force of static friction.

(5-11)

If you find the normal force for each trial, you should be able to plot fs,max vs. FN in the same manner you did for Parts I – III.  Is the coefficient of static friction the same as the coefficient of kinetic friction?

Assignment

Make sure you follow the instructions for cookbook style reports.  I have some additional hints for you below:

In your data and analysis sections, please show one example of the equations you needed to perform in order to determine the quantities which needed to be calculated, even if you used a spreadsheet to determine them.

In your analysis section, the aim is to show how well the model of friction fits our observations.  These questions may help you understand what is needed.

1.      How well does the equation fk = μkFN fit the data in part II and III?  Does the function fit the data within error bars (i.e. within 1σ)?

2.      How well does the coefficient that is determined in part II match the coefficient determined in part III?  Do they match within 1σ?

3.      (Optional) How well does the equation fs,max = μs,maxFN fit the data in part V?  Does the function fit the data within error bars?

Answer any questions assigned by the TA.

Questions

1. Based upon your observations, discuss the model of friction from your textbook (fk = μkFN).  How well did this model hold up in our experiment?  Can you think of some other functions which might also fit the data?

1. Suppose that the frictional force had a quadratic relationship with the normal force (fk =  μkFN + C(FN)2, where C is some constant of the appropriate unit).  Fit this function to your part II data.  Do all of the data points fit within error bars to this fit line?  What is the value of C?  Under what conditions would you be able to distinguish the two fit lines (significantly)? (i.e. Could you perform an experiment which would give data showing that the linear fit was incompatible?)

1. Even though we give names to the forces in this lab (e.g. “friction”, “normal force”), these forces are really just manifestations of the fundamental forces.  If you consider that there are only three forces in the universe (or less if you ascribe to any of the Grand Unified Theories), the electromagnetic force, the weak nuclear force, and the strong nuclear force, which force gives rise to friction?  Which gives rise to the normal force?

(Note: Electromagnetic force = force between charges, i.e. like charges repel. Strong nuclear force = force that binds the nucleus together.  Weak force = force that produces instability in nuclei, e.g. in radioactive substances.)

1. Kinetic or static: What kind of friction is acting on your tires when you are skidding on an icy road?  What kind of friction is acting when you have good traction on a road?  Which one would you want to use when you are braking to avoid collision?  Why?