Now, I go to the conservation of mechanical energy. Again, I'll show you how I explain the solutions of the physical problem in terms of fundamental principles. As always, the statement of principles will be the first line of the solution to the problem. The problem that I have chosen to illustrate the method requires the application of the second law of Newton and mechanical energy conservation.
Notation I used of this series described in the preceding articles, specifically "kinematics of teaching", "Teaching the second law of Newton" and "the problems of labour-power."
Problem. A small box of mass m starts rest and sliding down the surface friction of a cylinder of RADIUS r-free show box leaves the surface when the angle between the line of radiial in the box and the vertical axis is th = arccos(2/3).
Analysis. The box is just affected the frictionless cylindrical surface, exercising a normal force outwards n on it. The only other force on the box is its weight MG. Box moves on a circular path, so we apply Newton's second law in radial direction (positive outward). With the help of a free body diagram, we have
... Newton's second law
... Sum (fr) = MAr
...-MGcos (th) + N = M(-V**2), r.
Since the cylindrical surface can only push (he can not draw), box cannot remain on the surface, unless force normal n is greater than zero. As a result, box leaves the surface at the point where N = 0. Since the last equation corresponds to
... cos (th) = (V * 2) / RG.
But this tells us much if we do not know the speed v box, we are going to see what we can learn from the application of the conservation of mechanical energy. We use a framework co-ordinated by inertia with axis y vertical and the origin at the center of the circular cylinder. Assimilate us mechanical energy box at the top of the bottle and the time it leaves the cylinder. The initial position of the box is Yi = R, its muzzle velocity is Vi = 0, its final position is Yu = Rcos (th) and its final speed is given = V, speed when he leaves the surface. Now with the conservation of mechanical energy.
... Mechanical energy conservation
... (IVM * 2) / 2 + MGYi = (MVu * 2) / 2 + MGYu
... (M0 * 2) / 2 + MGR = (MV * 2) / 2 + MGRcos (th);
therefore... 2 * V = 2 GR (1 - cos (th)).
Finally, connect this result in cos equation (th) that we found earlier, we
... cos (th) = 2 GR (1 - cos (th)) /RG = 2 - 2cos (th);
and... Th = arccos(2/3).
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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