A sinusoidally varying driving force is applied to a damped harmonic oscillator of force constant k and mass m. If the damping constant has a value b1, the amplitude is A1, when the driving angular frequency equals √(k/m). In terms of A1, what is the amplitude for the same driving frequency and the same driving force amplitude Fmax, if the damping constant is (a) 3b1, and (b) b1/2.
Well, sinusoidally varying driving forces and damped harmonic oscillators sure know how to party! Now, let's tackle the question at hand.
(a) If the damping constant is 3b1, we're talking about some serious damping going on. It's like having three clowns trying to exit a tiny car - it's gonna take some effort! In this case, the amplitude A1 will be reduced. How reduced, you ask? Well, since we're talking about amplitude, it will be reduced to 1/3 of the original value. So, the amplitude for this case will be A1/3. It's like having a laughter explosion that's a little less intense.
(b) Now, if the damping constant is only b1/2, we have a less intense party going on. It's like having half a bottle of squirting flower water instead of the full thing. In this case, the amplitude A1 will be increased. How increased, you ask? Well, since we're talking about amplitude again, it will be increased to 2 times the original value. So, the amplitude for this case will be 2A1. It's like having twice the laughter, twice the fun!
So, to sum it up: for a damping constant of 3b1, the amplitude will be A1/3, and for a damping constant of b1/2, the amplitude will be 2A1. Keep on oscillating and clowning around!
To determine the amplitude for the same driving frequency and driving force amplitude Fmax, we can use the concept of resonance in a damped harmonic oscillator.
Resonance occurs when the driving angular frequency matches the natural frequency of the system (√(k/m)) and the amplitude is maximized.
Given:
Damping constant b1
Amplitude A1 at driving angular frequency √(k/m)
Amplitude Fmax for the same driving frequency
We need to find the amplitudes for damping constants (a) 3b1 and (b) b1/2.
(a) Damping constant = 3b1
The amplitude for the same driving frequency and driving force amplitude can be found using the following formula:
A2 = (A1 * Fmax) / (k / ((√(k/m))^2 + (3b1 / (√(k/m)))^2))
Simplifying the expression:
A2 = (A1 * Fmax) / (1 + 9 * (b1^2 / k))
(b) Damping constant = b1/2
The amplitude for the same driving frequency and driving force amplitude can be found using the following formula:
A3 = (A1 * Fmax) / (k / ((√(k/m))^2 + ((b1/2) / (√(k/m)))^2))
Simplifying the expression:
A3 = (A1 * Fmax) / (1 + (b1^2 / 4k))
So, the amplitude for the same driving frequency and driving force amplitude Fmax, with damping constants (a) 3b1 and (b) b1/2, are A2 and A3 respectively, as given by the above formulas.
To find the amplitude for the same driving frequency and driving force amplitude Fmax, we can use the equation for the amplitude of a driven damped harmonic oscillator. The equation is given by:
A = Fmax / (m * sqrt((ω0^2 - ω^2)^2 + (2βω)^2))
where A is the amplitude, Fmax is the maximum driving force amplitude, m is the mass of the oscillator, ω0 is the natural angular frequency of the oscillator (ω0 = sqrt(k/m)), ω is the driving angular frequency, and β is the damping constant.
Let's proceed to find the amplitudes for the given cases:
(a) If the damping constant is 3b1:
We know that ω0 = sqrt(k/m) and ω = sqrt(k/m). Substituting these values in the equation, we get:
A = Fmax / (m * sqrt((ω0^2 - ω^2)^2 + (2βω)^2))
= Fmax / (m * sqrt((k/m - k/m)^2 + (2*3b1*sqrt(k/m))^2))
= Fmax / (m * sqrt((0^2 + (6b1*sqrt(k/m))^2))
= Fmax / (m * 6b1 * sqrt(k/m))
= Fmax / (6 * b1 * sqrt(k * m^3 / m^3))
= Fmax / (6 * b1 * sqrt(k * m^3) / m)
= Fmax * (m / (6 * b1 * sqrt(k * m^3)))
Therefore, the amplitude when the damping constant is 3b1 is A = Fmax * (m / (6 * b1 * sqrt(k * m^3))).
(b) If the damping constant is b1/2:
Using the same approach as above, we can substitute the given values in the equation:
A = Fmax / (m * sqrt((ω0^2 - ω^2)^2 + (2βω)^2))
= Fmax / (m * sqrt((k/m - k/m)^2 + (2*(b1/2)*sqrt(k/m))^2))
= Fmax / (m * sqrt((0^2 + ((b1/2)*sqrt(k/m))^2))
= Fmax / (m * sqrt((b1^2 * k/m) / 4))
= Fmax * (4 * sqrt(k * m) / (m * b1))
Therefore, the amplitude when the damping constant is b1/2 is A = Fmax * (4 * sqrt(k * m) / (m * b1)).
These are the expressions for the amplitudes in terms of A1, Fmax, k, m, and the damping constant values b1, 3b1, and b1/2, respectively.