Here are several challenging problems that you can attempt for extra credit. (How much extra credit? That's up to me!) If you are at all interested in the fundamentals of how nature works (in particular, physics) I suggest that you work a few of them.
I've labeled the problems according to their difficulty - as I see it. You may have a different idea about which ones are more difficult than others. The three levels of difficulty are
[hard]
[harder]
[hardest]
Credit
where credit is due: I've obtained these from many sources (and modified
them to fit my own rules for problem statements), but the main sources are David
Morin's textbook 'There once was a classical theory ...' and Marion's textbook
'Classical Dynamics.'
[harder] A block of mass m rests on a horizontal surface and you pull on the block with a force F at an angle q above the horizontal. The coefficient of static friction between the block and the surface is m. The magnitude of the force F needed to make the block slip depends on the angle q. Sketch F versus q, and find the angle q for which F is a minimum.
[hardest] Consider the infinite Atwood machine shown in the figure. A string passes over each pulley, with one end attached to a mass and the other end attached to another pulley. All the masses are equal to m and all the pulleys and strings are massless. What is the acceleration of the top mass?
HINT: Consider the system to be made of N pulleys, with a non-zero mass replacing what would have been the (N + 1)st pulley. Then take the limit as N goes to infinity.
[harder] A ball is dropped from rest at height h, and it bounces off a surface at height y (with no loss in speed). The surface is inclined so that the ball bounces off at an angle of q with respect to the horizontal. What should y and q be so that the ball travels the maximum horizontal distance?
[hardest] A ball is thrown at speed v0 at an angle q above horizontal from zero height on level ground. We already know that in order for the range (horizontal distance traveled) to be a maximum, the angle must be q = p/4. In this problem, however, I want you to answer the following question: What must q be so that the distance traveled through the air is a maximum?
Show that q satisfies the equation
sinq ln [(1 + sinq)/cosq] = 1,
and you can show numerically that q = 56.5o.
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ABCDEFGHIJKLMNOPQRSTUVWXYZ