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In fact, the mass $m$ and the force constant $k$ are the only factors that affect the period and frequency of simple harmonic motion.
The period of a simple harmonic oscillator is given by
and, because $f=1/T$ , the frequency of a simple harmonic oscillator is
Note that neither $T$ nor $f$ has any dependence on amplitude.
Find two identical wooden or plastic rulers. Tape one end of each ruler firmly to the edge of a table so that the length of each ruler that protrudes from the table is the same. On the free end of one ruler tape a heavy object such as a few large coins. Pluck the ends of the rulers at the same time and observe which one undergoes more cycles in a time period, and measure the period of oscillation of each of the rulers.
If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping (See [link] ). Calculate the frequency and period of these oscillations for such a car if the car’s mass (including its load) is 900 kg and the force constant ( $k$ ) of the suspension system is $6\text{.}\text{53}\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ .
Strategy
The frequency of the car’s oscillations will be that of a simple harmonic oscillator as given in the equation $f=\frac{1}{\mathrm{2\pi}}\sqrt{\frac{k}{m}}$ . The mass and the force constant are both given.
Solution
Discussion
The values of $T$ and $f$ both seem about right for a bouncing car. You can observe these oscillations if you push down hard on the end of a car and let go.
If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in [link] . Similarly, [link] shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.
The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by
where $X$ is amplitude. At $t=0$ , the initial position is ${x}_{0}=X$ , and the displacement oscillates back and forth with a period $T$ . (When $t=T$ , we get $x=X$ again because $\text{cos}\phantom{\rule{0.25em}{0ex}}\mathrm{2\pi}=1$ .). Furthermore, from this expression for $x$ , the velocity $v$ as a function of time is given by:
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