# Free and Forced Oscillations

Physics

As

## example

### Find external force in forced oscillation

Example: A simple harmonic oscillator is of mass . It is oscillating with a frequency of Hz. If its amplitude of vibration is cm, what is the force acting on the particle at its extreme position?

Solution:

## example

### Frequency of oscillations in forced oscillations

Example: A kg block hangs without vibrating at the bottom end of a spring with a force constant of N/m. The top end of the spring is attached to the ceiling of an elevator car. The car is rising with an upward acceleration of m/s when the acceleration suddenly ceases at time and the car moves upward with constant speed ( m/s) What is the angular frequency of oscillation of the block after the acceleration ceases?

Solution:
Given :     Mass of the block
Spring constant
Let  angular frequency of the oscillation  be   .
Now using

## definition

### Write displacement as a function of time in forced oscillation

The object oscillates about the equilibrium position .  If we choose the origin of our coordinate system such that , then the displacement from the equilibrium position as a function of time is given by:

## example

### Examples of mechanical resonance

Various examples of mechanical resonance include:
1. Musical instruments (acoustic resonance).
2. Most clocks keep time by mechanical resonance in a balance wheel,   pendulum, or quartz crystal.
3. Tidal resonance of the Bay of Fundy.
4. Orbital resonance as in some moons of the solar system's gas giants.
5. The resonance of the basilar membrane in the ear.
6. Making a child's swing, swing higher by pushing it at each swing.
7. A wine glass breaking when someone sings a loud note at exactly the right   pitch.

## definition

### Oscillations when driving frequency is close to natural frequency

Amplitude of oscillations in shown in the attached plot and given by the formula:

where
Driving Force
Mass
Driving Frequency
Damping Frequency and Damping constant

When , amplitude of oscillations is maximum.
which is a very large value.

This is the phenomenon of resonance in forced oscillations.

## definition

### Explain maintained oscillation with example

Maintained Oscillation : In maintained oscillation, energy is supplied to the system from outside at the same rate at which the energy is lost by it, so that its amplitude remains constant & the system oscillates with its own natural frequency. Example: Swing, electric tuning fork, balance wheel of a watch, etc