Question

In forced oscillation of a particle the amplitude is maximum for a frequency \[{\omega _1}\]of the force, while the energy is maximum for a frequency\[{{\omega }_{2}}\]of the force, then
(a) \[{{\omega }_{1}}\text{ }=\text{ }{{\omega }_{2}}\]
(b) \[{{\omega }_{1}}\text{ }{{\omega }_{2}}\]
(c) \[{{\omega }_{1}}\text{ }{{\omega }_{2}}\], when damping is small and\[{{\omega }_{1}}\text{ }{{\omega }_{2}}\]when damping is large
(d) \[{{\omega }_{1}}\text{ }{{\omega }_{2}}\]

Answer 1

Hint: Oscillation is the periodic fluctuation of a measure around a central value (often a point of equilibrium) or between two or more states, usually in time. Mechanical oscillation is exactly described by the term vibration. A swinging pendulum and alternating current are both instances of oscillation.

Complete answer:
The energy oscillates back and forth between kinetic and potential in undamped simple harmonic motion, moving entirely from one to the other as the system oscillates. The motion of an item on a frictionless surface linked to a spring begins with the entire amount of energy stored in the spring. The elastic potential energy is transformed to kinetic energy as the item moves, eventually becoming totally kinetic energy at the equilibrium point. The spring then converts it back into elastic potential energy, and when the kinetic energy is entirely converted, the velocity becomes zero, and so on.
It's worth noting that the maximum velocity is determined by three variables. The maximum velocity is proportional to the amplitude. As you may expect, the maximum velocity increases as the maximum displacement increases. Stiffer systems have a higher maximum velocity because they apply more force for the same displacement. The formula for \[{{v}_{max}}\] reflects this fact; it is proportional to the square root of the force constant k. Finally, because maximum velocity is inversely related to the square root of m, maximum velocity is less for objects with greater masses. Objects with enormous masses accelerate more slowly for a given force.
The frequency of force must be identical to the starting frequency for the amplitude of oscillation and energy to be maximal, which is only achievable in resonance. In a condition of resonance \[{{\omega }_{1}}\text{ }=\text{ }{{\omega }_{2}}\] is the correct answer

Hence option a is correct.

Note:
Forced oscillation occurs when a body oscillates as a result of an external periodic force. Because of the external energy provided to the system, the amplitude of the oscillation is damped but remains constant. When you push someone on a swing, for example, you must maintain pushing them at regular intervals to keep the swing from reducing.