Correct Answer - Option 1 : Driving frequency is equal to the natural frequency
CONCEPT:
Forced oscillations and resonance:
- When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency ω, and the oscillations are called free oscillations.
- All free oscillations eventually die out because of the ever-present damping forces. However, an external agency can maintain these oscillations. These are called forced or driven oscillations.
- The most familiar example of forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations.
- Suppose an external force F(t) of amplitude Fo that varies periodically with time is applied to a damped oscillator. Such a force can be represented as,
⇒ F(t) = Fo.cos(ωdt)
Where ωd = driving angular frequency
- When we apply the external periodic force to the oscillation, the oscillations with the natural frequency die out, and then the body oscillates with the angular frequency of the external periodic force. Its displacement at any time t, after the natural oscillations die out, is given as,
⇒ x(t) = A.cos(ωdt + ϕ)
Where A = amplitude and t is the time measured from the moment when we apply the periodic force
- The amplitude of the oscillations is given as,
\(⇒ A=\frac{F_o}{[m^2(ω^2-ω_d^2)^2+ω_d^2b^2]^{1/2}}\)
a) Small damping (Driving frequency far from natural frequency):
- In this case, ωdb will be much smaller than \(m(ω^2-ω_d^2)\).
- We can neglect the term ωdb from the amplitude.
- The amplitude of the oscillations is given as,
\(⇒ A=\frac{F_o}{m(ω^2-ω_d^2)}\)
- The amplitude of the oscillation is maximum when ω = ωd.
b) Driving frequency close to natural frequency:
- When ωd is very close to ω, \(m(ω^2-ω_d^2)\) would be much less than ωdb.
- For any reasonable value of b the amplitude of the oscillations is given as,
\(⇒ A=\frac{F_o}{ω_db}\)
- This makes it clear that the maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity.
- The phenomenon of increase in amplitude when the frequency of driving force is close to the natural frequency of the oscillator is called resonance.
EXPLANATION:
- We know that once the natural oscillations die out for the forced oscillation of the damped pendulum, the amplitude of the pendulum is given as,
\(⇒ A=\frac{F_o}{[m^2(ω^2-ω_d^2)^2+ω_d^2b^2]^{1/2}}\) ----(1)
Where ω = natural frequency, ωd = driving frequency, m = mass of the system, b = damping constant, and Fo = amplitude of the driving force
- By equation 1 it is clear that the amplitude will be maximum when,
⇒ ω = ωd
- Hence, option 1 is correct.
