HW: New and Old

New homework is posted, as you may have noticed.
Here are the answers to the textbook version of the even-numbered "random-#" problems.
7.26: -2.0,2.0 m
7.28: (a) -6.7 N, (b) 0 N, (c) 4.5 N
10.22: 1.2 m

Here are some hints.

7.41: Simple energy conservation problem.
7.56: How much mechanical energy does the bug lose, each time it passes the sticky region? Think the work-energy theorem.
7.59: This is a GREAT problem. Energy conservation + Example 5.7. I.e., is the speed (and thus K) 0 or not, at top of the loop?
7.64: Simple energy conservation problem.
7.66: The power, the unit, and the work-energy theorem.
7.26: Energy conservation. What is the total mechanical energy, given v_max? What is the value of K at turning points?
Potential Energy Graphs and Motion: Even if you do not do this problem, please do note the following point. As we learned in class, an equilibrium point for a stable object in nature is described as the minimum point of a potential energy curve. However, this is not the only equilibrium point that can happen in nature. There can be a maximum point of a potential energy curve. This is called an unstable equilibrium point, as opposed to the former kind, which is a stable equilibrium point. Why are both of these called "equilibrium" points? It is because F(x)=-dU/dx=0, whether it is a minimum point, or a maximum point (or else). So, an object at that value of x, for which F(x)=0, with zero initial velocity will stay there forever. Thus, an equilibrium. Now, ask the question, what happens if the initial velocity is very very small, but still finite. [Strictly a side remark (outside the scope of this course): Indeed, you might ask "how can any object in nature maintain a truly zero velocity?" This is a deep question. Answering that question from any point of view you might have, the answer comes down to "not possible at all."] For a stable equilibrium point, the object oscillates back and forth around the initial point (e.g. a mass on a string, a child on a swing, etc.). For an unstable equilibrium point, the object never comes back to the equilibrium point again (e.g. that bug who starts from the top of that bald man's head; the top is an equilibrium point). I will say a few words about this in Wednesday's lecture.
7.28: How do we get F(x) from U(x)? (See the previous hint, if you don't remember.)
7.51: What is the equilibrium point? (Read that hint again, if you don't remember, interpreting r as x.)
Graphs of Linear and Rotational Quantities Conceptual Question: Remember that homework problem where you derived the centripetal acceleration (again) from scratch using your differential calculus technique alone?
10.19,46: 1D kinematics in the "angle language." (like Example 10.2)
10.22: Definition of torque. (Assume angle = 90 degrees, i.e. the most efficient angle.)
10.24: Definition of torque.
10.26: Definition of I.
The Parallel Axis Theorem: Another simple example for this cool theorem.
10.65: Identical to what I did in class, except the hole punched is a circle this time. This is one of those "must-do" problems in the first course in physics.

I have also updated files in the "sols+" folder of the course website. Solutions to HW #4 are uploaded. Also, please note that there are files, Sols-HW01-MP.pdf etc. (They were there before, too, up to HW03. Now they are up to HW04, and some of the printing problems of all previous files are corrected.) These "Sols...MP" files are the print-outs of the MP version of the solutions. Why do I have them there? It is because you can review those files to review past homework in preparation for exams. Let me know if these "Sols...MP" files serve this purpose for you.