Oscillator Problem: Period And Frequency Calculation

Hey guys! Today, we're diving into an interesting physics problem involving an oscillator. This isn't just any oscillator; it's a mass-spring system, a classic example in mechanics. We'll break down the problem step-by-step, making sure everyone understands the concepts involved. So, let's jump right in!

Understanding the Problem

We have a point mass, specifically m = 1.6 * 10^2 kg, which is attached to a spring. This mass oscillates due to the elastic force exerted by the spring. The motion of this oscillator is described by the equation y = 10^-1 sin(π/8 t + π/8) (m). Our main task is to determine two key characteristics of this oscillation: the period and the frequency. These two parameters give us a complete picture of how the oscillator moves over time.

Breaking Down the Equation

The equation y = 10^-1 sin(π/8 t + π/8) is a sinusoidal function, which is typical for simple harmonic motion. Let's dissect it:

• y: This represents the displacement of the mass from its equilibrium position at any given time t. The unit here is meters (m).
• 10^-1: This is the amplitude of the oscillation. Amplitude tells us the maximum displacement of the mass from its equilibrium position. In this case, the mass moves a maximum of 0.1 meters away from its resting point.
• sin(π/8 t + π/8): This is the sinusoidal part of the equation. The sine function describes the oscillatory nature of the motion. The argument inside the sine function, (π/8 t + π/8), contains information about the angular frequency and the phase constant.

Why This Matters

Understanding oscillators is fundamental in physics and engineering. They appear in various systems, from pendulum clocks to musical instruments to the vibrations of molecules. By solving this problem, we're not just doing a textbook exercise; we're gaining insights into the behavior of systems that exhibit periodic motion. This knowledge is crucial for designing and analyzing systems that rely on oscillatory behavior, such as shock absorbers in cars or resonant circuits in electronics.

Calculating the Period

The period (T) is the time it takes for one complete oscillation cycle. In other words, it's the time it takes for the mass to return to its starting position and direction. To find the period, we need to look at the angular frequency within our equation.

Angular Frequency

The angular frequency (ω) is the rate of change of the phase of the sinusoidal function. It's the term that multiplies t inside the sine function. From our equation, y = 10^-1 sin(π/8 t + π/8), we can identify the angular frequency as π/8 radians per second (rad/s).

The relationship between angular frequency (ω) and period (T) is given by the formula:

ω = 2π / T

This formula is a cornerstone in understanding oscillatory motion. It tells us that the angular frequency is inversely proportional to the period. A higher angular frequency means a shorter period (faster oscillations), and vice versa.

Finding the Period

Now that we know the angular frequency and the formula relating it to the period, we can solve for T:

T = 2π / ω

Substituting ω = π/8 rad/s into the formula, we get:

T = 2π / (π/8) = 2π * (8/π) = 16 seconds

So, the period of the oscillation is 16 seconds. This means it takes 16 seconds for the mass to complete one full cycle of its motion. This is a relatively slow oscillation, which makes sense given the small angular frequency.

Determining the Frequency

The frequency (f) is the number of complete oscillation cycles per unit time. It's essentially the inverse of the period. If the period tells us how long one cycle takes, the frequency tells us how many cycles occur in a second.

Relationship Between Frequency and Period

The relationship between frequency (f) and period (T) is beautifully simple:

f = 1 / T

This formula is a fundamental concept in wave mechanics and oscillatory motion. It highlights the inverse relationship between these two quantities. If the period is long, the frequency is low, and vice versa.

Calculating the Frequency

Now that we know the period T = 16 seconds, we can easily calculate the frequency:

f = 1 / T = 1 / 16 Hz

So, the frequency of the oscillation is 1/16 Hertz (Hz). Hertz is the unit of frequency, representing cycles per second. A frequency of 1/16 Hz means that the mass completes one-sixteenth of an oscillation cycle every second. This is a very low frequency, indicating a slow oscillation.

Why Frequency Matters

The frequency of an oscillation is crucial in many applications. In music, frequency corresponds to pitch – higher frequencies mean higher notes. In electronics, the frequency of an alternating current determines how many times the current changes direction per second. In mechanical systems, the frequency of vibration can influence stability and resonance. Understanding and calculating frequency is therefore essential in various fields of science and engineering.

Connecting the Dots: Period and Frequency

Let's recap what we've found: the period (T) is 16 seconds, and the frequency (f) is 1/16 Hz. These two values provide a comprehensive description of the oscillatory motion of our mass-spring system. The period tells us the duration of one complete cycle, while the frequency tells us how many cycles occur per second. They are two sides of the same coin, linked by the simple inverse relationship f = 1 / T.

The Importance of Units

It's crucial to pay attention to units when dealing with physical quantities. The period is measured in seconds (s), which is a unit of time. The frequency is measured in Hertz (Hz), which is equivalent to inverse seconds (s^-1). Using the correct units ensures that our calculations are consistent and our results are meaningful.

Visualizing the Oscillation

Imagine the mass attached to the spring oscillating back and forth. The amplitude of 0.1 meters tells us how far the mass travels in each direction. The period of 16 seconds tells us that each complete back-and-forth motion takes a relatively long time. The low frequency of 1/16 Hz reinforces the idea that this is a slow oscillation.

Practical Implications

Understanding the period and frequency of an oscillator has numerous practical applications. Consider a pendulum clock: the period of the pendulum's swing determines the clock's accuracy. If the period changes (due to changes in pendulum length or gravitational acceleration), the clock will gain or lose time. In mechanical engineering, understanding the natural frequencies of structures is crucial to prevent resonance, which can lead to catastrophic failures.

Examples in Engineering

• Suspension Systems in Vehicles: The suspension system in a car is designed to oscillate at a certain frequency to absorb shocks and provide a smooth ride. The period and frequency of these oscillations are carefully tuned to match typical road conditions.
• Electronic Oscillators: Oscillators are fundamental components in electronic circuits, generating periodic signals used in radio communication, signal processing, and digital clocks. The frequency of these oscillators is precisely controlled to ensure proper operation.
• Vibration Dampers in Buildings: Tall buildings are susceptible to vibrations caused by wind or earthquakes. Vibration dampers, often tuned mass dampers, are used to counteract these vibrations by oscillating at a specific frequency, reducing the building's sway.

Conclusion

So, there you have it! We've successfully calculated the period and frequency of our oscillator using the given equation. Remember, guys, the key takeaway here is understanding how the equation of motion relates to these fundamental properties. By identifying the angular frequency, we were able to determine the period and subsequently the frequency. These concepts are not only essential for solving physics problems but also for understanding and designing real-world systems that involve oscillatory motion. Keep practicing, and you'll become oscillation masters in no time!