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
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
Review of concepts:
Friction is the culprit
In the lecture
part of the course, you have learned
_{} (51)
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
counterintuitive 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
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
Figure 5‑1 A cart with mass m_{c} sliding on a table and attached to a freely hanging mass m_{h}
What forces are on each block? What is the acceleration of each block?
(a) 
F_{hc}: the force of the hanging mass on the cart F_{g}: the force of gravity on the cart F_{N}: the normal force from the table on the cart f_{k}: the frictional force from the table on the cart 

(b) 
F_{ch}: the force of the cart on the hanging mass F_{g}: the force of gravity on the hanging mass 
Figure 5‑2 Free body diagrams of (a) the cart and (b) the hanging mass
The sum of the forces on the cart is:
_{} (52)
Since there is no acceleration in the y direction, F_{N} = F_{g} = m_{c}g.
The sum of the forces on the hanging mass is:
_{} (53)
Since the two objects are tied together, they must move together. Thus, they must have the same acceleration, a_{x}. Also, F_{hc} must be equal in magnitude to F_{ch} if the pulley between them is assumed to be massless. Solving for the frictional force in equation (51) yields:
_{} (54)
Substituting (53) into (54) yields:
_{} (55)
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. 53.
Figure 5‑3 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:
_{} (56)
The symbol μ_{k} 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 f_{k} vs. F_{N}, the slope should give us the coefficient of kinetic friction.
Figure 5‑4 A graph of frictional force versus normal force
The coefficient of kinetic friction in the graph shown in Fig. 54 is μ_{k} = 0.2065 ± 0.0025 (no units).
Is f_{k} = μ_{k}F_{N} “the answer”?
Are we doing this experiment for nothing? Do we really need to measure anything at all? After all, equation (56) 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 μ_{k} 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 (56) 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 (56) 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:
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. Nonhorizontal 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. 55).
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 f_{k} = μ_{k}F_{N} 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 IIIII
For parts IIIII, 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 timeoffall, t, by the equation:
_{} (57)
You can rearrange this equation to solve for the acceleration:
_{} (58)
Which t should you use? You measured the timeoffall 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 timeoffall 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 (55). The normal force can be found using Eq. (59) for Part II and Eq. (510) for Part III.
_{} (59)
_{} (510)
(θ has been previously defined in Fig. 53)
Plotting f_{k} vs. F_{N} (as in Fig. 54) 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.
_{} (511)
If you find the normal force for each trial, you should be able to plot f_{s,max} vs. F_{N} 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 f_{k} = μ_{k}F_{N} 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 f_{s,max} = μ_{s,max}F_{N}_{ }fit the data in part V? Does the function fit the data within error bars?
Answer any questions assigned by the TA.
Questions
(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.)