| Week of | Aug 22 | Sep 5 | Oct 1 | Nov 5
| Phy 205 page |
| Aug 27 | Sep 10 | Oct 8 | Nov 12 | ||
| Sep 17 | Oct 15 | Nov 19 | |||
| Sep 24 | Oct 22 | Nov 26 | |||
| Oct 29 | Notes for Fall, 2000 |
1 Oct
Work out three examples of the application of F = m a and
to the extent possible, compare with experiment. All examples were done
the same way:
1. Sketch
2. List things applying forces to each mass involved
3. Choose a basis and write the total force in this basis
4. Determine what I can about the acceleration and write it in the same basis
5. Apply Ftotal = m a
6. Break into components
7. See if any more equations are needed and if so, find them
8. Solve simultaneous equations
9. Check results
10. Compare with experiment in one case.
The three examples were
(1) Swing a bucket with water in it over my head. How fast must the
bucket move so I don't get wet?
(2) If I take a scale to weight myself onto an elevator, what will the
scale read as the elevator accelerates?
(3) The conical pendulum. This is one for which I can do a quantitative
comparison with experiment.
3 Oct One of the homework problems raised some conceptual questions, not just technical ones of how to do it. This provided an entree to a discussion of Newton's 1st law.
A common way that I've seen Newtons's laws written is something like
1. If no force acts on a body then its velocity stays constant.
2. Ftotal = m a
3. Fon 1 by 2 =
- Fon 2 by 1
If the total force on an object is zero, then the 2nd law says that its acceleration is zero. That means that it moves with constant velocity and that's the 1st law. This seems to say that I can throw away one of these and now I have Newton's two laws of motion!
Newton was no fool. It's unlikely that this analysis is correct, and it isn't. The problem is that these aren't Newton's laws of motion. A more correct statement is
1. Definition: An "inertial system" ("inertial coordinate system," "inertial frame," "inertial reference system") is one in which IF there is no total force on a mass THEN its acceleration equals zero.
2. IF you are in an inertial system, THEN F = ma (or more generally = dp/ dt).
Now it's apparent that you can't derive the 1st law from the second because the 1st law is a definition needed to state the 2nd.
To understand the significance of all this you need an example of a non-inertial system. Standing on the Earth you can see the stars rotate around us daily. Every 24 hours or so a distant star will circle the Earth. Its acceleration is the usual v2/r toward us, but there's no force pushing it this way. The Earth is not an inertial system; it's rotating.
If I'm at a carnival on a merry-go-round, I'm sitting on a wooden horse noticing all the people around me are going in circles and oscillating up and down. I of course take the ego-centric point of view that I'm standing still and the world rotates around me. There's nothing wrong with this, it's just not inertial.
If you want to understand the Earth's weather or the oceans, you'd better not ignore the Earth's rotation. What is done is to patch up F=ma by adding a couple of terms to account for the transformation to the rotating coordinate system. These added terms are called "centrifugal force" and "coriolis force" because they get added to the "F" side of F=ma and look just like real forces. We won't do this.
Friction: Kinetic (or sliding) friction can be described to a decent approximation by a single equation,
When two objects are sliding past each other this relates the two components of the contact force -- the component normal (perpendicular) to the surface and the component tangent to the surface (the part from friction). The numerical value of μk is found experimentally, and its value can range from 0.01 or so up to the neighborhood of 3 or so. For common automobile tires on a dry road a value of 0.7 is about what you'd expect. You can get more, but only at a cost in money and in durability.
Static friction is handled differently. Instead of an equality it uses an inequality:
I started to show how to apply this and even how to measure it, but I didn't finish.
5 Oct Set up a measurement of static friction and carry out all the F=ma calculations. A mass is on an incline and is not moving. Its acceleration is zero, so the total force on it is zero too. I write down these equations and add the equation (inequality) for static friction to derive a relation between the coefficient of static friction and the maximum angle at which the mass won't slip on the incline.
For a mass sliding downhill about 3/4 of the preceeding calculation is repeated unchanged. The two differences are that the acceleration isn't zero and the relation for the frictional force is an equality and not an inequality.
Take the basis vectors i down the slope and j perpendicular to the slope, which is at an angle θ to the horizontal. The total force on m is
This much is the same for the static and the sliding case. When it is sliding the acceleration is downhill, = +ax i. Write the vector equation, F = m a and break the result into components for ease of manipulation:
The auxiliary equation involving the coefficient of kinetic friction is
These three equations involve the three unknowns Ffr, FN, and ax. Solve for the latter to obtain
Check the dimensions and both sides are length over time-squared.
Check a couple of special cases for the angle at 90 degrees and at zero
degrees. It takes a little thought to understand what's happening in
each case and to see for example that the signs are right, but both
cases make sense.
