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North Seattle College PHYSICS 223: Task 1: Simple Harmonic Motion

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Physics 223 Name: Task 1: Simple Harmonic Motion Simple harmonic motion (SHM) is the description of rhythmic cyclical motion, such as exhibited by a pendulum or a weight on a stretched spring. For such motion, the amount of distance away from the "middle point" of the motion (for instance, the weight hanging straight down from a spring that will be stretched) is called the displacement, and it is given by the scalar equation: where y is the displacement, A is the amplitude (a constant), ù is the angular frequency (another constant), t is the time after motion begins (assume motion begins at t = 0) and ö is the phase angle (still another constant, measured in radians). Note that this equation can be considered a vector equation, if the sign of y is used. a. Use a piece of engineering graph paper to plot the time versus the displacement for one complete oscillation, which means from the time when the motion begins to the time when the motion brings the object back to the same displacement it had when it started. A, ù and ö are arbitrary constants (in other words, no actual numbers). Be careful in putting the independent variable along the x-axis, and how you label the axes of your graph. b. Find and express in terms of y. Recall that means !!!!!!. Don't forget about the chain rule. c. The period T is the time required to complete one oscillation. Find T of an object which obeys the following equation of motion: y = 0.60 cos (15t – ð/2) where y is measured in centimeters and t is measured in seconds. d. The phase angle is often not included in the displacement equation because it can be “folded” into a sine or cosine term. Using trigonometric identities, recast the equation for y in part c such that the right side of the equation does not contain the ð/2, or any radian-based constants at all. Note that the graph of the function should remain the same! e. Show that ! = !!!!(!"!!) + !!!!!(!"!!) (or, if you prefer, y = C1 exp[i(ùt + ö)] + C2 exp[–i(ùt + ö)]) is a solution for the differential equation in part b. Again, be careful about the chain rule. Note: the “i” refers to – 1 ; that is, !! = -1. f. Using the complex plane construction, show that ! = !!!!(!"!!) + !!!!!(!"!!)is equivalent to y = A cos (ùt + ö) for a judicious choice of C1, C2 and A.

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