Question

Explain the difference between Simple harmonic Motion, Damped harmonic
oscillation and coupled oscillation. Use the diagrams and give their equation of
motion for each.

Answer 1

Free Oscillation

The free oscillation possesses constant amplitude and period without any external force to set the oscillation. Ideally, free oscillation does not undergo damping. But in all-natural systems damping is observed unless and until any constant external force is supplied to overcome damping. In such a system, the amplitude, frequency, and energy all remain constant.

Damped Oscillation

The damping is a resistance offered to the oscillation. The oscillation that fades with time is called damped oscillation. Due to damping, the amplitude of oscillation reduces with time. Reduction in amplitude is a result of energy loss from the system in overcoming external forces like friction or air resistance and other resistive forces. Thus, with the decrease in amplitude, the energy of the system also keeps decreasing. There are two types of damping

Natural Damping

Artificial Damping

Forced Oscillation

When a body oscillates by being influenced by an external periodic force, it is called forced oscillation. Here, the amplitude of oscillation, experiences damping but remains constant due to the external energy supplied to the system.

For example, when you push someone on a swing, you have to keep periodically pushing them so that the swing doesn’t reduce.

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Answer 2

Answer:

Energy conserves in simple harmonic oscillation, dampens in damped oscillation and energy exchanges between the oscillators in coupled oscillations.

Explanation:

Simple harmonic Motion:

• Simple harmonic motion is defined as a motion where the restoring force is directly proportional to the displacement of the body from its mean position.

       $\vec F\propto - \vec x$

• where $\vec F$ is the force and $\vec x$ is the displacement from equilibrium position.
• The equation of motion for simple harmonic oscillation is given by

       $\dfrac{d^{2}x }{dt^{2} } +\omega^{2} x=0$

• where $\dfrac{d^{2}x }{dt^{2} }$ represents the acceleration and $\omega$ is the angular frequency.

Damped harmonic oscillation:

• Damped harmonic oscillation is the oscillation happening under the influence of a restoring force and friction.
• The amplitude of oscillation reduces with time, results in loss of energy that is referred to as damping.
• The equation of motion is given by

        $\dfrac{d^{2}x }{dt^{2} } +\dfrac{\mu}{m} \dfrac{dx }{dt }+\dfrac{k}{m}x=0$

• where $\mu$ is the coefficient of friction, m is the mass of the body and $k$ is the spring constant.

Coupled Oscillation:

• Coupled oscillation is the oscillation that occur when two or more oscillating systems are connected with each other.
• There will be an exchange of energy between the oscillating systems.
• The equation of motion is given by

       $M\dfrac{d^{2}x_1 }{dt^{2} } +(k+k_{12} )x_1+k_{12}x_2=0$

      $M\dfrac{d^{2}x_2 }{dt^{2} } +(k+k_{12} )x_2+k_{12}x_1=0$

• where $x_{1} ~\&~x_{2}$ are the respective displacements of the oscillators, M is the mass of the system and $k~\&~k_{12}$ is the spring constant of the oscillators and the third oscillator.

Energy conserves in simple harmonic oscillation, energy dampens in damped oscillation and energy exchanges between the oscillators in coupled oscillations.

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