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# Damped Harmonic Oscillations

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# Problem statement

Forces acting on an oscillation particle during damped harmonic oscillations.

# Solution

No body can oscillate with a constant amplitude in normal conditions owing to various factors like resistive forces acting on the body. When an object oscillates its natural frequency remains the same but amplitude keeps on decreases due to various resistive forces like damping force opposing the motion of the object. The given to the object at the beginning of the oscillation is dissipated in doing work against the resistive forces. As the energy decreases with time, the object comes to rest.

# Forces acting on an oscillating particle

## Restoring force

• This force tries to bring the object back to mean position. At the extreme position of oscillation the velocity is zero but due to inertia and restoring force( which is maximum at the extreme position) it comes back to its mean position. At the mean position velocity is maximum and restoring force is minimum.
• Restoring force is directly proportional to the negative of displacement of the particle from its mean position during oscillation.
``````Fr=-kx
``````
• Where k is the proportionality constant and x is the instantaneous displacement.
• Negative sign indicates that restoring force and displacement act in opposite directions.

## Damping force

• This force acts on a oscillating particle under normal conditions. It is a resistive force. Damping force is directly proportional to the negative of velocity. Negative sign indicates that damping force tries to reduce velocity.
``````Fd=-b dx/dt
``````
• Where b is the damping constant and dx/dt represents instantaneous velocity.
• According to the above relation damping force is maximum at the mean position as velocity is maximum there and minimum at the extreme position as velocity is minmum there.

## Force of inertia

• Every body set for oscillatory motion has inertia.
• Force is directly proportional to acceleration.
``````F=md^2x/dt^2
``````
• Where m= mass and d^2x/dt^2 represents instantaneous acceleration.
• At equilibrium, force of inertia is balanced by restoring and damping force.
``````md^2x/dt^2=-kx-bdx/dt
``````
Damped Harmonic Oscillations