p>There are two common forms of the equations used to model simple harmonic motion (SHM), which is the motion of springs, swings, tides,
and many other periodic phenomena. These equations are and , where:
y(t) = distance of weight from equilibrium position
=angular frequency (measured in radians per second)
A= amplitude
=phase (depends on initial conditions)
c1=A?sin
c2=A?cos
Suppose you are an engineer trying to recreate an experiment involving a weight on the end of a spring. This simulation will give you
an idea of what the experiment will look like. For more information, you can visit this simple harmonic motion website.
You are given the equation , which models the position of the weight with respect to time. You need to find the amplitude of the
oscillation, the angular frequency, and theinitial conditions of the motion. You will also be required to find the time(s) at which the
weight is at a particular position. To find this information, you need to convert the equation to the first form, .
a. Use the information above and the trigonometric identities to prove that .
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b. To rewrite in the form , you must first find the amplitude, A. Use the given values and , along with the Pythagorean
identity, to solve for A.
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c. To rewrite in the form , solve for .
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d. Write in the form and identify the amplitude, angular frequency, and the phase shift of the spring motion.
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e. The angle represents the phase shift, determined by the initial conditions of the experiment or the position of the weight at
t = 0. If the weight is at its maximum positive position (weight is above equilibrium) at t = 0, then = 0. If the weight is at its
maximum negative position (spring is stretched and weight is below equilibrium) at t = 0, then . If the weight is traveling in the
negative direction and passing through equilibrium at t = 0, then . Describe the initial condition of our experiment; specifically,
describe the position of the weight and the direction in which it was traveling.
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f. Find the times (to the nearest hundredth of a second) that the weight is halfway to its maximum negative position over the
interval . Solve algebraically. Hint: Use the amplitude to determine what y(t) must be when the weight is halfway to its maximum
negative position. Graph the equation and explain how itconfirms your solution(s).
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