In this article, I show how it is easy to solve the problems of rotational motion of fundamentals. It is a continuation of the last two articles on rolling motion. The notation used is summarized in section "teaching dynamic rotation. As usual, I describe the method in terms of an example.
Problem. A solid ball of mass m and radius r is rolling through a horizontal speed v surface when it encounters an inclined at an angle th. Distance d along the inclined ball moves to stop and go back down? Assume that the ball moves without dragging?
Analysis. Since the ball moves without dragging, mechanical energy is conserved. We will use a frame of reference which the origin is a distance r above at the bottom of the slope. This is to centre the ball as it starts up the ramp, SW = 0 Yi. When assimilate us the mechanical energy of the ball at the bottom of the slope (where Yi = 0 and Vi = V) and to the point where it stops (Yu = h and Vu = 0), we
Mechanical energy conservation
Mechanical energy = mechanical energy final initials
M(VI**2)/2 Icm(Wi**2)/2 + MGYi = M(Vu**2)/2 Icm(Wu**2)/2 + MGYu
M(V**2)/2 + + MG (0) Icm(W**2)/2 = M(0**2)/2 Icm(0**2)/2 + MGH;
where h is the vertical displacement of the ball stops on the slope at the moment. If d is the distance the ball moves along the slope h = d sin (th). This insertion with W = V/R and Icm = 2 M(R**2)/5 in energy equation, we find, after some simplification, the ball moves the slope distance
d = 7 (V * 2) / (10Gsin (th))
to rotate and position downwards.
This solution of the problem is exceptionally easy. Once again the same message: start all solutions of problem with a fundamental principle. When you do, your ability to solve problems is greatly improved.
Dr. William Moebs is physical retired professor who has taught at both universities: Indiana - Purdue Fort Wayne and Loyola Marymount University. You can see hundreds of examples illustrating how stressed the fundamental principles by visiting physical support.
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