How do I show mathematically that Q is equal to the following in a damped oscillator?
The quality factor Q of a damped oscillator is defined as Q=2pi τ(tau) /T. Show mathematically that Q is equal to 2pi*)energy of oscillator/energy lost by oscillator over one period)
To prove that Q is equal to the expression you provided for a damped oscillator, we first need to understand the concepts and equations involved.
1. Energy of the Oscillator (E):
The energy of an oscillator can be expressed as E = 1/2 kA², where k is the spring constant and A is the maximum amplitude of oscillation.
2. Energy Loss per Period (∆E):
The energy lost by the oscillator over one complete period can be expressed as ∆E = E - E', where E' is the energy of the oscillator after one period.
3. Time Period (T):
The time period T of an oscillator is the time taken for one complete cycle of oscillation.
4. Time Constant (τ):
In a damped oscillator, the time constant τ represents the time it takes for the amplitude of the oscillation to decrease by a factor of 1/e (where e is Euler's number, approximately 2.71828).
Now, let's understand how to mathematically prove the expression for Q.
Step 1: Express the time constant (τ) in terms of the damping factor (b) and the mass (m) of the oscillator.
- In a damped harmonic oscillator, d²x/dt² + 2b(dx/dt) + (k/m)x = 0, where x is the displacement of the oscillator.
- The damping factor b is related to the time constant τ by the equation b = 1/τ.
Step 2: Find the value of the damping factor (b) in terms of ω₀, τ, and Q.
- In a damped oscillator, the angular frequency ω of the oscillator is related to the spring constant k and the mass m by the equation ω₀ = sqrt(k/m).
- The damping factor can be expressed as b = ω₀ / Q.
Step 3: Calculate the total energy (E) of the oscillator.
- Using the equation E = 1/2 kA², where k is the spring constant and A is the maximum amplitude of oscillation, we can find the energy of the oscillator.
Step 4: Calculate the energy loss (∆E) over one period (T).
- The energy lost (∆E) can be expressed as ∆E = E - E', where E' is the energy of the oscillator after one period (T).
Step 5: Show that Q is equal to the expression (2π) * (Energy of Oscillator / Energy Lost per Period).
- Substitute the equations for b and ∆E into Q = 2πτ / T.
- Simplify the expression, making use of the relationship between b and Q obtained in Step 2, and the calculated energy loss ∆E.
By following these steps and substituting the appropriate equations into each step, you can mathematically show that Q is equal to the given expression for a damped harmonic oscillator.