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Unformatted text preview: 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 = and ) 2 sin( max t f v v = 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 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 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 , 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 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 as the adjacent over the hypotenuse: u y u x u y x Chapter 27 Oscillations: Introduction, Mass on a Spring 186 u u x = cos Solving this for u x we see that cos u u x = Recalling that u = 1, this means that cos = x u Recalling that our extended definition of cos is, that it is the x component of the unit vector u when u makes an angle with the x-axis, this last equation is just saying that, for the case at hand ( between 0 and 2 radians) our extended definition of cos is equivalent to our ordinary definition. At angles between 2 and 2 3 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 takes on negative values at such angles as well....
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