Simple Harmonic Motion: Displacement, Amplitude & Phase Angle

by Andrew McMorgan 62 views

Alright, guys! Let's dive into a classic physics problem involving simple harmonic motion. We've got an object bouncing back and forth, and we need to figure out its exact position at any given time, its maximum displacement, and how it all starts. Buckle up, it's gonna be a fun ride!

Understanding Simple Harmonic Motion

So, what exactly is simple harmonic motion? Well, imagine a spring with a mass attached to it. When you pull the mass and release it, it oscillates back and forth. That, in a nutshell, is simple harmonic motion. The key characteristic is that the restoring force (the force pulling the mass back to its equilibrium position) is proportional to the displacement from equilibrium. This leads to sinusoidal motion, meaning the position of the object can be described by sine or cosine functions.

Frequency and Period: The frequency (often denoted by 'f') tells us how many complete oscillations occur per unit of time (usually seconds). The period (often denoted by 'T') is the time it takes for one complete oscillation. They are inversely related: T = 1/f. In our case, the frequency is given as 8, which means the object completes 8 oscillations every second.

Initial Conditions: To fully describe the motion, we need to know where the object starts and how fast it's moving at the beginning. These are called initial conditions. We're given that y(0) = -4, which means at time t=0, the object's displacement is -4 units. We're also given that y'(0) = 5, which means at time t=0, the object's velocity is 5 units per second. These initial conditions are crucial for determining the specific equation that describes the object's motion.

General Equation: The general equation for simple harmonic motion can be written in a couple of ways. One common form uses cosine and sine functions:

y(t) = A * cos(ωt + φ)

where:

  • y(t) is the displacement at time t
  • A is the amplitude (the maximum displacement from equilibrium)
  • ω (omega) is the angular frequency (related to the frequency by ω = 2πf)
  • φ (phi) is the initial phase angle (which tells us the initial position of the object within its oscillation cycle)

Our goal is to find the values of A and φ that fit our specific problem. We can also express the solution as a sum of sine and cosine terms with different coefficients. This form is particularly useful when we have initial conditions for both position and velocity:

y(t) = C1 * cos(ωt) + C2 * sin(ωt)

where C1 and C2 are constants that we need to determine based on the initial conditions. Both forms of the general equation are equivalent, and we'll use the second form to solve this problem.

Finding the Displacement, y(t)

Okay, let's get down to business! We know the frequency (f = 8), so we can calculate the angular frequency:

ω = 2πf = 2π * 8 = 16π

Now, we can write the general solution as:

y(t) = C1 * cos(16πt) + C2 * sin(16πt)

We need to find C1 and C2 using the initial conditions. First, let's use y(0) = -4:

-4 = C1 * cos(16π * 0) + C2 * sin(16π * 0) -4 = C1 * cos(0) + C2 * sin(0) -4 = C1 * 1 + C2 * 0 -4 = C1

So, we've found that C1 = -4. Now, let's use the second initial condition, y'(0) = 5. We need to find the derivative of y(t) first:

y'(t) = -16π * C1 * sin(16πt) + 16π * C2 * cos(16πt)

Now, plug in t = 0 and y'(0) = 5:

5 = -16π * (-4) * sin(16π * 0) + 16π * C2 * cos(16π * 0) 5 = -16π * (-4) * 0 + 16π * C2 * 1 5 = 16π * C2 C2 = 5 / (16π)

Great! We've found C1 and C2. Now we can write the complete equation for the displacement:

y(t) = -4 * cos(16πt) + (5 / (16π)) * sin(16πt)

That's it! This equation tells us the position of the object at any time t.

Determining the Amplitude (A)

The amplitude (A) represents the maximum displacement of the object from its equilibrium position. We can find it using the values of C1 and C2 that we just calculated.

The relationship between A, C1, and C2 is:

A = √(C1² + C2²)

Plugging in our values:

A = √((-4)² + (5 / (16π))²) A = √(16 + (25 / (256π²))) A ≈ √(16 + 0.00988) A ≈ √16.00988 A ≈ 4.0012

So, the amplitude of the oscillation is approximately 4.0012 units. Notice that it's very close to the absolute value of C1. This makes sense because the contribution of C2 is relatively small in this case.

Calculating the Initial Phase Angle (φ)

The initial phase angle (φ) tells us the initial position of the object within its oscillation cycle. We can find it using the following relationships:

C1 = A * cos(φ) C2 = -A * sin(φ)

From these, we can derive:

tan(φ) = -C2 / C1

Plugging in our values:

tan(φ) = -(5 / (16π)) / (-4) tan(φ) = 5 / (64π)

Now, we need to find the angle φ whose tangent is 5 / (64π). We can use the arctangent function (atan or tan⁻¹) for this:

φ = atan(5 / (64π)) φ ≈ 0.0248 radians

However, we need to be careful about the quadrant of φ. Since C1 is negative and C2 is positive, φ must be in the second or third quadrant. The arctangent function only returns values in the first and fourth quadrants. To get the correct angle, we need to add π to our result:

φ ≈ 0.0248 + π φ ≈ 3.1664 radians

Now, let's find the sine and cosine of the initial phase angle:

sin(φ) = sin(3.1664) ≈ -0.0247 cos(φ) = cos(3.1664) ≈ -0.9997

These values are consistent with our earlier calculations: C1 = A * cos(φ) ≈ 4.0012 * (-0.9997) ≈ -4 and C2 = -A * sin(φ) ≈ -4.0012 * (-0.0247) ≈ 0.0988 ≈ 5/(16π).

Summary

So, to recap, we've found the following:

  • Displacement: y(t) = -4 * cos(16πt) + (5 / (16π)) * sin(16πt)
  • Amplitude: A ≈ 4.0012
  • Initial Phase Angle: φ ≈ 3.1664 radians
  • Sine of Initial Phase Angle: sin(φ) ≈ -0.0247
  • Cosine of Initial Phase Angle: cos(φ) ≈ -0.9997

We successfully determined the displacement equation, amplitude, and initial phase angle for the object undergoing simple harmonic motion! Remember, understanding the concepts of frequency, initial conditions, and the general equation for SHM is key to solving these types of problems. Keep practicing, and you'll become a simple harmonic motion master in no time!