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Ch0127 - Chapter 27 Oscillations Introduction Mass on a...

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Chapter 27 Oscillations: Introduction, Mass on a Spring 185 27 Oscillations: Introduction, Mass on a Spring If a simple harmonic oscillation problem does not involve the time, you should probably be using conservation of energy to solve it. A common “tactical error” in problems involving oscillations is to manipulate the equations giving the position and velocity as a function of time, ) 2 cos( max t x x f π = and ) 2 sin( max t f v v π = rather than applying the principle of conservation of energy. This turns an easy five-minute problem into a difficult fifteen-minute problem. When something goes back and forth we say it vibrates or oscillates. In many cases oscillations involve an object whose position as a function of time is well characterized by the sine or cosine function of the product of a constant and elapsed time. Such motion is referred to as sinusoidal oscillation. It is also referred to as simple harmonic motion. Math Aside: The Cosine Function By now, you have had a great deal of experience with the cosine function of an angle as the ratio of the adjacent to the hypotenuse of a right triangle. This definition covers angles from 0 radians to 2 π radians (0 ° to 90 ° ). In applying the cosine function to simple harmonic motion, we use the extended definition which covers all angles. The extended definition of the cosine of the angle θ is that the cosine of an angle is the x component of a unit vector, the tail of which is on the origin of an x -y coordinate system; a unit vector that originally pointed in the + x direction but has since been rotated counterclockwise from our viewpoint, through the angle θ , about the origin. Here we show that the extended definition is consistent with the “adjacent over hypotenuse” definition, for angles between 0 radians and 2 π radians. For such angles, we have: in which, u , being the magnitude of a unit vector, is of course equal to 1, the pure number 1 with no units. Now, according to the ordinary definition of the cosine of θ as the adjacent over the hypotenuse: θ u y u x u y x
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Chapter 27 Oscillations: Introduction, Mass on a Spring 186 u u x = θ cos Solving this for u x we see that θ cos u u x = Recalling that u = 1, this means that θ cos = x u Recalling that our extended definition of θ cos is, that it is the x component of the unit vector u ˆ when u ˆ makes an angle θ with the x -axis, this last equation is just saying that, for the case at hand ( θ between 0 and 2 π radians) our extended definition of θ cos is equivalent to our ordinary definition. At angles between 2 π and 2 3 π radians (90 ° and 270 ° ) we see that u x takes on negative values (when the x component vector is pointing in the negative x direction, the x component value is, by definition, negative). According to our extended definition, cos θ takes on negative values at such angles as well.
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