L06.  The Nature of Static Friction

About your lab report:  This could be called an "answer as you go" lab.  Simply answer the questions or do the problems as they're presented to you, giving each the same number as in the instructions.  Here are some reminders about frequently-overlooked rules for recording data. These aren't just rules for this class but are accepted (and expected) ways of recording data in the scientific community.

1. Don't erase, blot out, or scratch out data even you think it's wrong. Cross out the data with a single line so that the original data can be seen. This is done so that no data is lost. It's not unusual for an experiment to decide that the original data was correct.
2. Record all data that would be required for someone to duplicate your experiment.
3. Whenever you record data, identify what you're recording. For example, when you record the weight of the box in this experiment, write "Weight of box = _______."
4. When you record data in a table, provide descriptive column headings. Include the units in the headings but not beside each number.

Remember that as your lab reports are returned to you, you should be keeping them in a notebook. You should print out copies of electronic files. This collection will be your lab portfolio. You may find that some college physics departments may require that you provide your portfolio for review in order to place out of or exempt their introductory physics course. They may require this in addition to a passing AP exam score.

Goals

1. to understand the nature of the static friction inequality, f_s ≤ µ_sN
2. to measure the coefficient of static friction of an object on a surface

Preparation

You should have read section 6.1 and completed E.6.1abc.  You'll need the following items of equipment. See this page for more information about equipment.

1. Inclined plane (If you don't have this item, you can improvise by using a board and propping up one end.)
2. Friction box (If you don't have this item, you may to improvise by using a box (cardboard, for example) to put weights into.)
3. Laboratory weights
4. Textbook
5. Spring scale that measures up to 20 N.
6. Protractor (You won't need this if your inclined plane has a protractor mounted on it.)
7. Supplies:  string, tape

Part A.  Static friction on a horizontal surface

1. Begin by writing the usual things on your paper:  Name, date, title, goal. 

2. Place your textbook on your desk.  Push gently in a horizontal direction on one side.  Increase the force of your push more and more until the book begins to move.  Before the book moved, it was in equilibrium.  This means the horizontal forces added to 0 as did the vertical forces.  Since you were increasing the magnitude of your push, that meant the resisting static friction force was increasing in magnitude to counter your push.  When the book started to move, though, the static friction could increase no more.  This implies that there is a maximum value for static friction force.  This value is given by f_s,max = µ_sN, where µ_s is the coefficient of static friction between the book and the desk, and N is the normal force exerted on the book by the table.  It's important to realize that this equality only applies to the maximum value of f_s, which is µ_sN.  Until the maximum is reached, f_s is less than µ_sN  .  A more general relationship is

f_s ≤ µ_sN.

Never write the relationship between f_s and µ_sN as an equality.

Now explain why force diagram A below could never describe the physical situation discussed above.

Force Diagram A

3. Force diagrams  B and C show the cases where i) the book hasn't begun to slide and ii) the book is accelerating in the direction of the push force.  Explain why the static friction force is labeled f_s in one diagram and f_s,max in the other.  Note that the lengths of the arrows representing the vectors are drawn to scale.

Force Diagram B	Force Diagram C

4. Here's how one way to measure the coefficient of static friction.  Use the box and place about a kilogram of mass in it.  You could use a textbook, brick, or laboratory mass.  Place the box with its contents on the surface that you'll be using for your inclined plane. In this case, though, make sure the plane is horizontal. You'll need a way to pull the box across the surface with the spring scale.  If you're using a standard friction box designed for such experiments, it should already have a hook on it.  Otherwise, attach a string to the front of the box.  You could just wrap the string around the box in a large loop.  Use your ingenuity.  When you're ready, pull the box horizontally but not with enough force to move it.  Now increase the force gradually while watching the scale reading.  Read the scale just before the box slips.  Repeat the measurement four times.  Record your readings in a table to the nearest 0.1 N. Include a column for deviations. Find the mean, mean deviation, and percentage mean deviation as you did in L05. Consider this to be standard procedure whenever you take repeated trials of a measurement. (For future reference, a standard procedure is one that you're expected to follow without being told to do so. That's part of learning to be a good experimenter.)

5. Before going on to measure the weight, describe how the scale reading changed as you approached the point where the box slipped and then went past it.  Explain why this occurred.  In your answer, we expect to see application of something you learned in the section 6.1 reading.

6. Now measure the weight of the box with the book in it.  Simply hang the combination vertically from the scale.  Record the reading on the spring scale to the nearest 0.1 N.  Now it's time for some theory.

7. Which force diagram above describes most nearly the situation of the box just before it started moving?  Explain.

8. We used the phrase most nearly above, because there's a way to improve the force diagram so that it describes the situation more precisely.  What is that improvement and why?

9. Now it's time to do the net force problem that will allow you to express the coefficient of static friction in terms of what you measured.  The things you measured were the pull force, P, and the weight, W.  Theoretically, you know that f_s,max = µ_sN or µ_s= f_s,max/N.  You didn't measure either N or f_s,max directly.  However, you can set up and solve the net force equations that will allow you to relate N and f_s,max to P and W. 

1. Select the +x axis to the right and the +y axis up.  Write the horizontal and vertical net force equations.  Use f_s,max rather than f_s for the friction force.

2. The fact that the block moves at constant velocity tells you what the accelerations are.  Substitute those.  You should now see how the forces relate.

10. Solve the two net force equations together with f_s,max = µ_sN in order to obtain µ_s in terms of P and W.

11. Substitute the mean pull force from your table to obtain a value for µ_s.  Comment on whether your value makes sense and why. Again, we expect to see application of the section 6.1 material.

12. Suppose you had placed twice as much mass in the box.  Make an hypothesis about how that would affect the value of µ_s.  Explain your hypothesis.

13. Test your hypothesis by adding mass to the box and repeating step 4. Do your results support your prediction?  A complete answer to the last question requires quantitative support. That means that you need to look at the mean deviations of your measurements to determine whether the difference in your two values of µ_s can be accounted for by the mean deviations. Consider this to be standard procedure.

Part B.  Static friction on an incline

There's a way to measure the coefficient of static friction without using a spring scale.  All you need is your inclined plane (You'll tilt it this time.), the box with weights, and a protractor.  You'll use that method in this section, but first there's theory to do.  We'll start by drawing a picture of the box on the inclined plane and then drawing a force diagram beside it.

Picture	Force Diagram

Here are some things to note about the force diagram.

1. The normal force is perpendicular to the plane and is therefore not directly opposite the weight force.  As a result, the magnitudes of those forces aren't equal.  (This should make it clear that the assumption N = W should never be made.  The relationship between the two forces must always be determined from a net force equation.)
2. There's no need for an extra push force in this situation.  The component of the weight parallel to the plane supplies the push (or pull).
3. We've set up axes parallel and perpendicular to the plane.  That makes sense, because the acceleration, if there is one, will be along the plane.

Now here are things for you to do.

1. The angle of inclination, q, of the plane is shown in the picture and the force diagram.  Two other angles, a and b, are shown.  One of them equals q.  Which one?  Explain using your knowledge of geometry and a diagram.  (If you're not sure about this, you should know where to go in your textbook to find the answer.)

2. The net force equations are the following:

F_net,x = Wsinq - f_s

F_net,y = N - Wcosq

We're assuming the block is at rest.  That means the accelerations and net forces are zero.  With that substitution, we simplify the equations to the following.

f_s = Wsinq

N = Wcosq

Here's a useful technique when we have equations in the form above.  Simply divide them.  That eliminates one of the forces.  Mathematically speaking, we need to realize that this technique yields an undefined result when the denominator is zero.  Keeping in mind that limitation, we go ahead and do it anyway.

f_s/N = sinq/cosq = tanq,

where we've used a well-known trig identity.  It would be tempting at this point to say f_s/N = tanq = µ_s . Tell why that would be an incorrect statement. What could you change in the statement to make it correct?

3. Now take some time to examine how the forces on the box change as the angle of the plane increases from 0 to 90°. For each of the following forces or force components, tell specifically how big the force is at 0° and 90° and how it changes from 0 to 90°.  a. normal    b. component of weight parallel to the plane      c. static friction

4. Describe what experimental procedure you could use in order to be able to say with certainty (as certain as a scientist can be, anyway) that µ_s has the value of tanq. 

5. Carry out your procedure and record your measurements.  Take several trials, as you did in Part A. Determine your best value for the coefficient of static friction of the plane.

6. Compare the value of the coefficient of static friction that you just determined to the value that you determined in Part A. Can you say they're the same to within experimental error? Explain.

Follow up

We hope that you understand the importance of clearly specifying whether the static friction force is at its maximum value.  When it is, you can use the equality, f_s,max = µ_sN.  When it isn't, the inequality applies:  f_s ≤ µ_sN.  If you're having trouble with the concept, though, the following graph may help.  The two forces that act parallel to the plane are plotted vs. the angle of inclination of the plane.  The equations plotted are these:

W-parallel = Wsinq  (red line)

f_s,max = µ_sN = µ_sWcosq   (green line)

By the way, we're assuming W = 10 N and µ_s = 0.5 for the graph.  In order to hold the block motionless, the amount of force required is just the amount necessary to resist W-parallel.  The vertical blue line is drawn through the intersection of the two curves.  The intersection occurs at the angle q_c = tan^-1(µ_s) = 26.6°.  This angle--called the critical angle--is the angle at which the object just begins to slide and for which the static friction force being supplied is the most it can be at that angle.  To the left of the blue line, the amount of available friction force, so to speak, is greater than the force needed to hold the block motionless.  To the right of the blue line, the situation is reversed.