To analyze the damped harmonic oscillation described by the equation of motion, we need to break down the components and apply the relevant formulas. The equation you provided is a second-order linear differential equation that describes the motion of a damped oscillator. Let's dive into the calculations step by step.
Calculating the Period of Motion
The period of a damped harmonic oscillator can be determined using the formula:
T = 2π / ω_d
where ω_d is the damped angular frequency, given by:
ω_d = √(k/m - (γ/(2m))²)
Step 1: Calculate ω_d
Given:
- m = 0.025 kg
- γ = 0.07 kg/s
- k = 85 N/m
First, we calculate the term inside the square root:
k/m = 85 N/m / 0.025 kg = 3400
Next, we calculate the damping term:
γ/(2m) = 0.07 kg/s / (2 * 0.025 kg) = 1.4
Now, we can find ω_d:
ω_d = √(3400 - 1.4²) = √(3400 - 1.96) = √(3398.04) ≈ 58.2 rad/s
Step 2: Calculate the Period T
Now we can substitute ω_d into the period formula:
T = 2π / 58.2 ≈ 0.108 s
Finding the Number of Oscillations for Amplitude Reduction
To determine how many oscillations it takes for the amplitude to reduce to half its initial value, we can use the formula for the amplitude of a damped oscillator:
A(t) = A_0 * e^(-γt/2m)
We want to find the time t when A(t) = A_0 / 2. Setting up the equation:
A_0 / 2 = A_0 * e^(-γt/2m)
Dividing both sides by A_0 gives:
1/2 = e^(-γt/2m)
Taking the natural logarithm of both sides:
ln(1/2) = -γt/2m
Solving for t:
t = -2m * ln(1/2) / γ
Substituting the values:
t = -2 * 0.025 kg * ln(1/2) / 0.07 kg/s
t ≈ 0.0714 s
Step 3: Calculate the Number of Oscillations
Now, to find the number of oscillations N in this time period:
N = t / T
Substituting the values:
N = 0.0714 s / 0.108 s ≈ 0.66
This means it takes approximately 0.66 oscillations for the amplitude to reduce to half its initial value.
Summary of Results
- Period of Motion (T): Approximately 0.108 seconds
- Number of Oscillations for Amplitude to Halve: Approximately 0.66 oscillations
These calculations illustrate the behavior of a damped harmonic oscillator, showing how the damping factor influences the period and amplitude over time. If you have any further questions or need clarification on any part of this process, feel free to ask!
