A system that is oscillated at a frequency equivalent to its _____ by an external force will experience resonance.
A. highest frequency
B. natural frequency
C. forced frequency
D. highest wavelength
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ford, J.M. Tanner, in Physics for Students of Science and Engineering, 1985
Forced Oscillations: Resonance
Forced oscillations occur when an oscillating system is driven by a periodic force that is external to the oscillating system. In such a case, the oscillator is compelled to move at the frequency νD = ωD/2π of the driving force. The physically interesting aspect of a forced oscillator is its response—how much it moves—to the imposed driving force. Let us, therefore, examine qualitatively the response of an oscillator to a driving force.
Consider a boy seated in a playground swing. He and the swing constitute (very nearly) a simple pendulum, which has a natural frequency νN of oscillation. By a single push against the ground, the boy can start a gentle oscillation of the system at its natural frequency. The period TN of his oscillation is equal to 1/νN. Now suppose a girl stands behind the swing and periodically pushes on the oscillating boy but with a driving period TD = 2π/ωD that is much shorter, i.e., very different, from the natural period of the system. How does the swing respond? It accelerates every time she pushes, of course, moving at the driving frequency, but it does not move very far from its equilibrium position. The boy is rattled around near the vertical position of the swing. Technically, we could say that the amplitude of his oscillation is small. Suppose, then, that the girl, sensing the ineffectiveness of her efforts, begins to push on the boy only when he has reached the utmost extent of his backward swing. She could then observe two facts: (a) The boy still oscillates at the frequency at which she pushes, and (b) he smoothly (relatively speaking) moves through longer and longer arcs.
By injecting energy into the oscillator at an appropriate frequency, the girl is able to achieve a large response (amplitude) in the oscillating system. We may generalize the results of this experience. When a driving force is applied to an oscillating system at a frequency near the natural frequency of the system, the amplitude of the oscillation becomes large. This relatively large response of an oscillator to being driven at a frequency that is near its natural frequency is called resonance. An oscillator is, therefore, said to be resonant at its natural frequency.
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Resonance is a special case of forced vibration in which the natural frequency of vibration of the body is the same as the impressed frequency of external periodic force and the amplitude of forced vibration is maximum.
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