Problem 27 - College Physics

Question

\(\bullet\) The displacement of an object is given by \(y=(5.0 \mathrm{~cm}) \cos [(20 \pi \mathrm{rad} / \mathrm{s}) t] .\) What are the object's (a) amplitude, (b) frequency, and (c) period of oscillation?

Step-by-Step Solution

Answer

(a) Amplitude: 5.0 cm, (b) Frequency: 10 Hz, (c) Period: 0.1 s.

Step 1: Determine the Amplitude

The amplitude of a cosinusoidal function in the form \(y = A \cos(Bt)\) is simply the coefficient of the cosine function, which is \(A\). In the given equation \(y=(5.0 \mathrm{~cm}) \cos[(20 \pi \mathrm{rad} / \mathrm{s}) t]\), the amplitude \(A\) is \(5.0 \mathrm{~cm}\).

Step 2: Find the Angular Frequency

The coefficient of \(t\) inside the cosine function, which is \(20 \pi \mathrm{rad/s}\) in this case, represents the angular frequency \(\omega\).

Step 3: Calculate the Frequency

To find the frequency \(f\), use the relationship between angular frequency and frequency: \[ f = \frac{\omega}{2\pi} \]Substitute \(\omega = 20 \pi \mathrm{rad/s}\):\[ f = \frac{20 \pi}{2\pi} = 10 \mathrm{~Hz} \]

Step 4: Calculate the Period

The period \(T\) of an oscillation is the inverse of the frequency:\[ T = \frac{1}{f} \]With \(f = 10 \mathrm{~Hz}\), the period is:\[ T = \frac{1}{10} = 0.1 \mathrm{~s} \]

Key Concepts

Oscillation Frequency

In Simple Harmonic Motion, the oscillation frequency refers to how often the object completes one full cycle of movement in a given period of time, typically one second.
The frequency is measured in Hertz (Hz), which is equivalent to cycles per second.
To find the frequency from a trigonometric equation like a cosine function, you need to understand the concept of angular frequency.
**Understanding Angular Frequency**
The angular frequency is the expression that describes how quickly the cosine function progresses through its cycle in terms of radians per second.
In the given trigonometric function, the angular frequency is represented by the coefficient of time (\( t \)) within the cosine function.
In this exercise, the coefficient is \( 20 \pi \) rad/s.
**Calculating the Frequency**
The direct relationship between angular frequency (\( \omega \)) and frequency (\( f \)) is given by the formula:
\[ f = \frac{\omega}{2\pi} \]
By substituting \( \omega = 20 \pi \) rad/s, we find:

• \( f = \frac{20 \pi}{2\pi} = 10 \text{ Hz} \)

This tells us that the object completes 10 cycles every second, which is the oscillation frequency.

Amplitude

Amplitude is a fundamental aspect of Simple Harmonic Motion because it describes the maximum extent of oscillation or displacement from the equilibrium point.
Think of it as how far the object moves away from the center point, either in the positive or negative direction.
In trigonometric terms, it is the coefficient in front of the cosine (or sine) function.
**Identifying Amplitude in an Equation**
In a trigonometric equation like \( y = A \cos(Bt) \), \( A \) is the amplitude.
It is a positive value that represents the maximum distance from the horizontal axis (equilibrium position) to the peak or trough of a wave.
For the given equation \( y = (5.0 \text{ cm}) \cos[(20 \pi \text{ rad/s}) t] \), the amplitude \( A \) is very straightforward to identify. It is \( 5.0 \text{ cm} \). Thus, the object reaches a maximum displacement of 5.0 cm from its equilibrium point in one direction, either up or down.

Oscillation Period

The oscillation period is another core element of Simple Harmonic Motion. It tells you the time it takes for the object to complete one full cycle of oscillation.
This means that from a starting point, moving to a peak, returning through equilibrium, hitting the trough, and coming back to equilibrium, takes one period.
The unit of period is seconds (s).
**Calculating the Period**
The period (\( T \)) is inversely related to the frequency (\( f \)), given by the formula:

• \[ T = \frac{1}{f} \]

By knowing the frequency, you can easily determine the period.
Using the previously calculated frequency of the object, \( f = 10 \text{ Hz} \), we can find the period:

• \[ T = \frac{1}{10} = 0.1 \text{ s} \]

This value means that the object completes one full cycle of oscillation every 0.1 seconds, making it a relatively fast oscillation.
Understanding how frequency and period relate helps you to predict and measure the behavior of oscillating systems in real-world scenarios.