Oscillator Period & Frequency: Calculation Explained

Hey guys! Let's dive into the fascinating world of oscillators and tackle a problem that involves calculating the period and frequency of a specific oscillator. We're given a scenario where a point mass is attached to a spring, and it's vibrating under the spring's elastic force. The motion is described by a mathematical equation, and our mission is to extract the period and frequency from it. Sounds exciting, right? Let’s break it down step by step so it’s super clear.

To really understand what's going on, let's first talk about the basic concepts. An oscillator is essentially anything that moves back and forth in a rhythmic way. Think of a pendulum swinging, a guitar string vibrating, or even your heart beating! These systems have a natural tendency to repeat their motion over a certain time interval. This brings us to the concept of the period, which is the time it takes for one complete cycle of the oscillation. Imagine watching our mass-spring system go from its starting point, moving to its maximum displacement, then back through the starting point, and finally to the opposite maximum displacement before returning to the start. That whole trip is one cycle, and the time it takes is the period. We usually measure period in seconds. Now, the frequency is like the period's energetic cousin. It tells us how many cycles happen in one second. So, if our oscillator completes 2 cycles every second, its frequency is 2 Hertz (Hz). Frequency and period are actually closely related – they're inverses of each other. If you know one, you can easily find the other using a simple formula: frequency = 1 / period. In our specific case, we have a mass attached to a spring, which is a classic example of a simple harmonic oscillator. The spring exerts a force that tries to pull the mass back to its equilibrium position, and this restoring force is what causes the oscillation. The equation that describes the motion of this system is a sinusoidal function, which is a fancy way of saying it looks like a wave. Understanding these fundamentals will make deciphering the given equation and extracting the period and frequency much easier. So, with these concepts in mind, let's get into the nitty-gritty of the problem!

Decoding the Oscillation Equation

Okay, now let's get into the heart of the problem! We're given the equation describing the oscillator's motion: y = 10^-1 sin(π/8 t + π/8) (m). This equation might look a bit intimidating at first, but don't worry, we'll break it down piece by piece. Think of it as a secret code we're about to crack. The equation represents the displacement (y) of the mass from its equilibrium position at any given time (t). The equation is in the form of a sine function, which, as we discussed, is characteristic of simple harmonic motion. The general form of a sinusoidal equation for simple harmonic motion is y = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. Let's see how this general form helps us understand our specific equation.

First, let's identify the different parts of our equation and relate them to the general form. The term 10^-1 in front of the sine function represents the amplitude (A). The amplitude is the maximum displacement of the mass from its equilibrium position. In simpler terms, it's how far the mass swings back and forth. So, in our case, the amplitude is 0.1 meters. Next, we have the sine function itself, sin(π/8 t + π/8). Inside the sine function, we have π/8 t, which is the term that involves time (t). The coefficient of t, which is π/8, represents the angular frequency (ω). Angular frequency tells us how fast the oscillation is happening in terms of radians per second. Radians are just another way to measure angles, and they're particularly useful in dealing with circular and oscillatory motion. Finally, we have the term π/8 that's added inside the sine function. This is called the phase constant (φ). The phase constant tells us about the initial position of the oscillator at time t = 0. It essentially shifts the sine wave horizontally. However, for our purpose of finding the period and frequency, the phase constant doesn't directly affect our calculations, so we can set it aside for now. The key takeaway here is that we've successfully identified the amplitude and the angular frequency from the given equation. The angular frequency, ω = π/8, is crucial because it's directly related to the period and frequency, which are what we're trying to find. So, with this information in hand, let's move on to the next step: calculating the period and frequency.

Calculating Period and Frequency

Alright, we've successfully decoded the oscillation equation and pinpointed the angular frequency (ω), which is π/8 radians per second. Now comes the fun part: using this information to calculate the period and frequency of the oscillator. Remember, the period (T) is the time it takes for one complete oscillation, and the frequency (f) is the number of oscillations per second. These two quantities are intimately related to the angular frequency (ω). The formulas that connect them are quite elegant and straightforward. The relationship between angular frequency and period is given by: ω = 2π / T. This formula tells us that the angular frequency is equal to 2π divided by the period. It makes sense if you think about it: 2π radians represents one full circle, and dividing that by the period gives us the rate at which the oscillator is going through its cycle. To find the period (T), we can rearrange this formula to get: T = 2π / ω. Now, we can plug in the value of ω that we found earlier, which is π/8 radians per second. So, T = 2π / (π/8). When we divide by a fraction, it's the same as multiplying by its reciprocal. So, T = 2π * (8/π). The π terms cancel out, leaving us with T = 2 * 8 = 16 seconds. This means that it takes 16 seconds for the mass-spring system to complete one full oscillation. That's a pretty slow oscillation! Now that we've found the period, calculating the frequency is a breeze. Remember that the frequency (f) is the inverse of the period (T): f = 1 / T. So, f = 1 / 16. This gives us a frequency of 0.0625 Hertz (Hz). Hertz, as we mentioned earlier, is the unit of frequency and represents cycles per second. So, our oscillator is completing 0.0625 cycles every second. To recap, we used the angular frequency that we extracted from the equation of motion, along with the formulas relating angular frequency, period, and frequency, to calculate the period and frequency of the oscillator. We found that the period is 16 seconds and the frequency is 0.0625 Hz. Awesome! We've successfully solved the problem. But let's take a step back and think about what these values actually mean in the context of the physical system.

Interpreting the Results

So, we've crunched the numbers and found that the period of our oscillator is 16 seconds and the frequency is 0.0625 Hz. But what do these numbers really tell us about the motion of the mass-spring system? Understanding the physical meaning of these values is crucial for truly grasping the behavior of the oscillator. A period of 16 seconds means that it takes a full 16 seconds for the mass to complete one back-and-forth oscillation. Imagine watching the mass swing – it moves slowly, taking a considerable amount of time to go from one extreme position to the other and back again. This relatively long period suggests that the oscillations are quite slow. On the other hand, a frequency of 0.0625 Hz tells us that only a small fraction of an oscillation is completed each second. In other words, the mass barely moves through a complete cycle within one second. This low frequency reinforces the idea that the oscillations are slow and leisurely. Now, let's think about what factors might be contributing to this slow oscillation. We were given that the mass attached to the spring is m = 1.6 * 10^2 kg, which is 160 kg. That's a pretty hefty mass! A large mass will naturally resist changes in its motion, so it will take more force and more time to get it moving and to bring it to a stop. This inertia contributes to the slow oscillation. Another factor that influences the period and frequency of a mass-spring system is the stiffness of the spring. A stiffer spring will exert a stronger restoring force, causing the mass to oscillate faster, resulting in a shorter period and higher frequency. Conversely, a weaker spring will lead to slower oscillations, a longer period, and a lower frequency. However, we weren't given any information about the spring constant in this problem, so we can't directly assess its impact. But it's important to remember that the spring's stiffness plays a crucial role in determining the oscillatory behavior. In summary, the long period and low frequency we calculated tell us that the oscillator is moving slowly. This is likely due, at least in part, to the large mass attached to the spring. The interplay between the mass and the spring's stiffness determines the overall characteristics of the oscillation. By interpreting the results in this way, we gain a deeper understanding of the physical system and its behavior. It's not just about getting the right numbers; it's about making sense of those numbers in the real world.

Key Takeaways and Real-World Connections

Alright guys, we've journeyed through the problem of calculating the period and frequency of an oscillator, and we've learned a ton along the way! Let's recap the key takeaways and then explore some real-world connections to solidify our understanding. First and foremost, we learned how to decode the equation of motion for a simple harmonic oscillator. We identified the amplitude, angular frequency, and phase constant, and we understood how each of these parameters contributes to the overall motion. We then focused on the angular frequency, recognizing its crucial role in determining the period and frequency. We used the formulas T = 2π / ω and f = 1 / T to calculate the period and frequency from the angular frequency. Remember, the period is the time for one complete oscillation, and the frequency is the number of oscillations per second. These two quantities are inversely related, meaning that a longer period corresponds to a lower frequency, and vice versa. Finally, and perhaps most importantly, we emphasized the importance of interpreting the results in the context of the physical system. We discussed how the mass of the object and the stiffness of the spring influence the period and frequency of the oscillations. A large mass tends to slow down the oscillations, while a stiffer spring tends to speed them up. Now, let's bring these concepts to life with some real-world examples. Oscillations are everywhere around us! Think about a swing set in a park. When you push someone on a swing, you're essentially creating an oscillating system. The swing moves back and forth with a certain period and frequency, determined by the length of the swing's chains and the force of gravity. Another example is a grandfather clock. The pendulum in a grandfather clock swings back and forth with a very precise period, and this regular oscillation is used to keep time. The length of the pendulum is carefully chosen to achieve the desired period. Musical instruments also rely heavily on oscillations. When you pluck a guitar string, it vibrates at a specific frequency, which determines the pitch of the note you hear. Different strings have different masses and tensions, which result in different frequencies and therefore different notes. Even at the atomic level, oscillations play a crucial role. Atoms in a solid material vibrate around their equilibrium positions, and these vibrations contribute to the material's thermal energy. Understanding these atomic vibrations is essential in fields like materials science and nanotechnology. So, as you can see, the concepts of period, frequency, and oscillation are not just abstract mathematical ideas; they are fundamental to understanding the world around us. By mastering these concepts, you'll gain a deeper appreciation for the rhythmic and vibratory nature of the universe. Keep exploring, keep questioning, and keep oscillating! You've got this!