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Chapter Review

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In this chapter we study what happens when objects in equilibrium (chapter 11) are disturbed slightly away from this equilibrium. The most common result of such disturbances is that objects oscillate about (around) the equilibrium position (or configuration) from which it was disturbed. Here we study this oscillatory motion.

13-1 - 13-2 Periodic and Simple Harmonic Motion

An object's motion is referred to as periodic motion when it repeats itself over and over again. When the motion comes exactly back to its original state (of position and velocity) we say that the object has gone through a complete cycle or complete oscillation. The amount of time it takes for one complete cycle to occure is called the period of the motion T.

While the period is one way to characterize the speed fo the periodic motion, another way is through the frequency f. Frequency is a measure of how frequently the motion repeats; it is most commonly quoted as the number of oscillations (or cycles for short) per second. The frequency relates directly to the period by

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Since period measures an amount of time its SI unit is seconds; therefore, the SI unit of frequency must be the inverse second s^-1. When dealing with frequencies only, the special name of hertz (Hz) is given to s^-1. One hertz refers to one oscillation cycle per second.

The above discussion applies to any type of periodic motion. One particular type of motion that is very important in physics is called simple harmonic motion (SHM). This latter type of periodic motion occurs as a result of Hookes's law force. Hence SHM is the motion experienced by an object on the end of a spring and, to a close approximation, by media that are disturbed slightly away from equilibrium.

The position of an object undergoing SHM changes, with time, sinsuoidally. This fact means that it can be described using a sine or cosine function (we'll use cosine)

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In the above equation, T is the period and A is called the amplitude of the motion. The amplitude represents the maximum distance the object gets from equilibrium as it moves back-n-forth. The above expression assumes the object starts its motion at x = A. It is to be understood that the argument of the cosine function is an angle in radians; therefore, your calculator should be in radian mode when using the above expression.

Physlet Illustration: Periodic Motion of a Mass/Spring System
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A block is attached to a wall by a horizontal spring. The block rests on a frictionless floor. The distance grid is in meters and the times are shown in seconds. Pull or push the block away from its equilibrium location, and then push "Play" to watch it move back and forth. Study the graph of the block's displacement from equilibrium as a function of time. Determine the period and frequency of the block's motion. Start
Hints Using either the clock or the graph, measure the time that it takes the block to undergo one complete oscillation. For example, how long does it take the block to move from its point of maximum displacement on the right, back to the left and then back to its maximum displacement on the right again? The frequency (in Hz) is the inverse of the period (in seconds).

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Exercise 13.1 Simple Harmonic Motion: An object undergoes simple harmonic motion with a frequency of 3.50 Hz and an amplitude of 0.850 m. If its equilibrium position is x = 0, wehre is it (a) when the motion starts, and (b) 1.055 seconds later? (c) Write the complete equation of the motion.

Solution: We are given the following information.

Given: f = 3.50 Hz, A = 0.850 m

Find: (a) xi (b) xf (c) write the equation

(a) At the start of the motion is when we begin timing, so right at the beginning t = 0. Since the problem doesn't state the initial conditions of the motion we use the above expression for position vs. time in SHM

x = Acos(0) = A = 0.850 m.

(b) To get the position at t = 1.055 s, again we use the general expression for SHM together with the relationship between frequency and period

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(c) Here, we only need to insert all of the numerical values into the expression for x. First, let's find the period

Now that we have the period, we can write out the full formula for the motion

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In part (c) we could have used the given value for the frequency instead of first finding the period.

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Physlet Illustration: Velocity and Acceleration in Harmonic Motion
A mass/spring system oscillates back and forth. Study the graphs of the block's displacement from equilibrium,  velocity and acceleration as functions of time. How are they related? Start
Hints How do the periods (or frequencies) of oscillation of these three quantities compare? At what point in the motion is each of these quantities a maximum? A minimum? Zero? Do these results agree with the mathematical expressions for the displacement, velocity, and acceleration as given in your text?

Practice Quiz

Most generally, simple harmonic motion occurs... when a mass is attached to the end of a spring. when an object move back-n-forth when the motion of an object repeats itself. when an object is acted upon by a Hooke's law force. [None of the above.]

If the period of an oscillating body is 2.3 seconds, what is its frequency? 2.3 Hz 0.43 Hz 2.3 s 0.43 s 2.7 Hz

An object oscillates with SHM according to the equation x = (0.15 m)cos[(3.7 rad/s)t]. What is the amplitude of this motion? 0.15 m 3.7 m 1.7 m 0.94 m 0.59 s

An object oscillates with SHM according to the equation x = (0.15 m)cos[(3.7 rad/s)t]. What is the period of this motion? 0.15 s 3.7 m 1.7 m 0.94 s 0.59 s

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