Question 1

Which equation best models the position of an object attached to a spring, pulled down 5 units from equilibrium and released, if the period of oscillation is 4 seconds?

Accepted Answer

$$s(t) = -5 \cos \left(\frac{\pi}{2} t \right)$$

Question 2

Given the equation $$s(t) = 8 \sin(3t)$$ for simple harmonic motion, what is the amplitude of the oscillation?

Accepted Answer

$$8$$

Question 3

For the function $$y = e$$^{-x}$$ \sin x$$, what happens to the amplitude of the oscillation as $$x$$ increases?

Accepted Answer

The amplitude decreases exponentially.

Question 4

If the period of a spring's oscillation is 4 seconds, what is the frequency?

Accepted Answer

$$\frac{1}{4}$$ cycles per second

Question 5

Which of the following best describes the envelopes for the graph of $$y = e$$^{-x}$$ \cos(2\pi x)$$?

Accepted Answer

The graphs of $$y = e$$^{-x}$$ and y$$ = $$-e^{-x}$$

Question 6

At what values of $$x$$ does the graph of $$y = e$$^{-x}$$ \cos(2\pi x)$$ touch the graph of $$y = e$$^{-x}$$?

Accepted Answer

$$x = 0, 1, 2, 3$$

Question 7

Which of the following is the correct formula for the period of simple harmonic motion modeled by $$s(t) = a \cos(\omega t)$$?

Accepted Answer

$$T = \frac{2\pi}{\omega}$$

Question 8

A weight oscillates according to $$s(t) = 6 \cos(2t). $$What is the frequency of the oscillation?

Accepted Answer

$$\frac{2}{2\pi}$$ cycles per second

Question 9

If an object oscillates according to $$s(t) = 10 \sin(\omega t)$$ and completes 5 cycles in 10 seconds, what is the value of $$\omega$$?

Accepted Answer

$$\omega = \pi$$

Question 10

Which of the following best describes damped oscillatory motion?

Accepted Answer

Oscillations where amplitude decreases over time due to friction

Question 11

For the function $$y = e$$^{-x}$$ \cos(2\pi x)$$, at which values of $$x$$ does the graph intersect the $$x-$$axis?

Accepted Answer

$$x = \frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}$$

Question 12

A sound wave is modeled by $$y = 3 \sin(5t). $$What is the period of the wave?

Accepted Answer

$$\frac{2\pi}{5}$$ seconds

Question 13

If a spring's motion is modeled by $$s(t) = -7 \cos(\omega t)$$ and at $$t = 0$$ the displacement is $$-7$$, which function type is most appropriate for modeling the motion?

Accepted Answer

Cosine function

Question 14

Which of the following statements about frequency is correct?

Accepted Answer

Frequency is the number of cycles per unit of time and equals the reciprocal of the period.

Question 15

If the amplitude of a spring's motion is $$|a|$$, what does this represent physically?

Accepted Answer

The maximum displacement from equilibrium

Question 16

Which equation models damped oscillatory motion?

Accepted Answer

$$y = e$$^{-x}$$ \sin x$$

Question 17

If a spring is stretched and released, and friction is present, what happens to the motion over time?

Accepted Answer

The amplitude decreases until the object comes to rest.

Question 18

For the function $$y = e$$^{-x}$$ \cos(2\pi x)$$, what is the effect of the $$e^{-x}$$ term?

Accepted Answer

It decreases the amplitude as $$x$$ increases.

Question 19

If the period of a harmonic motion is $$T$$, what is the formula for frequency $$f$$?

Accepted Answer

$$f = \frac{1}{T}$$

9.8: The Unit Circle and the Functions of Trigonometry – Harmonic and Damped Oscillatory Motion

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