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Unformatted text preview: Lecture notes for Lecture #5, October21th, 2010 Definitions Force (gravitational): The magnitude of the force due to gravity is given by mass times the gravitational acceleration, g. F = mg Force (spring): The magnitude of a force resulting from compressing or stretching a spring is given by k, the spring constant times its displacement from the equilibrium position. F = kx Work: Work is defined as the total force applied through a distance. Most of us have seen Work given as, W = Force X Distance. This is only true in the case when the force is constant and the distance traveled is parallel to the direction of the applied force. A more general expression for work, as long as the force is applied parallel to the motion, is given by: Work = Units Work, like energy, is measured in joules. [force * distance] = [Newtons * meters] = [(kg-meters)/sec 2 * meters] = [Joules] Force, work, and potential energy. Work as the Area under the curve defined by the graph of Force versus displacement In class we did the following: 1) plotted force versus displacement, 2) looked at the area under the curve defined by plotting the force versus displacement, 3) verified that the area represented the work done, 4) and showed that this area also represented the potential energy associated with that force. We will do the same again here: We take the expression for F due to gravity, that is, F = mg, and graph this as a function of displacement or distance from some reference point. If the reference point is the ground, we can safely graph F as a function of height with no confusion. Since the force due to gravity, F=mg, as you can clearly see, has no dependence at all on height, the graph of F=mg as a function of height is just a horizontal straight line as seen below: mg Height (distance from the ground as a reference point) Force Now let us pick two points on the line F = mg. We will chose the first point, A (x 1 ,y 1 ) = (0, mg), and we will chose as our second point, B (x 2 ,y 2 ) = (h, mg). If we try to find an expression that gives us the area under the line that connect these two points, A and B, it is just the the distance between A and B, times the distance between the X-axis and the line mg. So the area of the rectangle above is simply Area = mg* h. But we should immediately recognize the product mgh as a very familiar quantity! This is just the expression for the potential energy in a gravitational field. This is not a coincidence. Let us see if we can figure out why. Any integral, we can just call it I for simplicity, is really just another way of summing a series of rectangles that are defined by the height of any arbitrary function, f, times the small differential thickness, as shown by the illustration below: I = . You will probably get a much more rigorous explanation in your calculus class, but the description below is really all you need to know for this class....
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