Chapter 7: Problem 3

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance $s$ from the origin is the given function. $$ s=\frac{1}{2} \cos (\pi t-8) $$

Short Answer

Expert verified

Amplitude: $\frac{1}{2}$, Period: 2, Frequency: $\frac{1}{2}$ Hz, Velocity Amplitude: $\frac{\pi}{2}$.

Step by step solution

Identifying the Amplitude

The amplitude of a trigonometric function in the form of \[ s = A \, \cos(\omega t + \phi) \]is the coefficient in front of the cosine function. Here, it is \[ \frac{1}{2} \]. Thus, the amplitude is \frac{1}{2}.

Determining the Angular Frequency

In the function \[ s = \frac{1}{2} \, \cos(\pi t - 8) \] the angular frequency $ \omega $ is the coefficient of $ t $ inside the cosine function. Here, it is $ \pi $.

Calculating the Period

The period $ T $ of the function is given by the formula \[ T = \frac{2\pi}{\omega} \]. Since $ \omega = \pi $, we get \[ T = \frac{2\pi}{\pi} = 2 \].

Finding the Frequency

The frequency $ f $ is the reciprocal of the period: \[ f = \frac{1}{T} \]. Since the period $ T = 2 $, we have \[ f = \frac{1}{2} \, \text{Hz} \].

Calculating the Velocity Amplitude

The velocity amplitude $ v_{\text{max}} $ is given by the product of the amplitude and the angular frequency: \[ v_{\text{max}} = A \omega \]. Using $ A = \frac{1}{2} $ and $ \omega = \pi $, we find \[ v_{\text{max}} = \frac{1}{2} \cdot \pi = \frac{\pi}{2} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude

In harmonic motion, the amplitude represents the maximum distance a particle moves from its equilibrium (rest) position.
For the function given,

Period

The period (T) of a harmonic motion is the time it takes for the motion to complete one full cycle. This means it's the time taken for the particle to return to its original position and velocity.

To determine the period, use the formula:

Frequency

Frequency (f) describes how often the cyclic motion occurs per unit of time. It is directly related to the period and is calculated as the reciprocal of the period:
When the period is calculated as discussed previously, you can find the frequency:
When T is 2 seconds, the frequency is:

Velocity Amplitude

The velocity amplitude is the maximum speed attained by the particle in motion. It's directly linked to both the amplitude of motion and the angular frequency.
To calculate the velocity amplitude:

So, for the given function:

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Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$ s=\frac{1}{2} \cos (\pi t-8) $$