Oscillations
Periodic and oscillatory motions

Periodic motion: A motion which repeats itself after a fixed interval of time

Examples:

Motion of the moon around the earth

Motion of the hands of a clock

Oscillatory motion: A body in oscillatory motion moves to and fro about its mean position in a fixed time interval.
Examples:

Motion of the pendulum of a wall clock

Motion of the liquid contained in a Utube when one of its limbs is compressed.

Period (T): It is the interval of time after which a motion is repeated. Its unit is seconds (s).

Frequency (ν): Number of repetitions that occur per unit time
Its unit is (second)^{1} or Hertz.

Displacement: Change in position
The figure shows a block attached to a spring.
Here, displacement is x.
An oscillating simple pendulum’s angular displacement isβ.

Displacement variable may take negative values.

Periodic functions can be expressed as a superposition of the sine and cosine functions.

Periodic motion: A motion which repeats itself after a fixed interval of time

Examples:

Motion of the moon around the earth

Motion of the hands of a clock

Oscillatory motion: A body in oscillatory motion moves to and fro about its mean position in a fixed time interval.
Examples:

Motion of the pendulum of a wall clock

Motion of the liquid contained in a Utube when one of its limbs is compressed.

Period (T): It is the interval of time after which a motion is repeated. Its unit is seconds (s).

Frequency (ν): Number of repetitions that occur per unit time
Its unit is (second)^{1} or Hertz.

Displacement: Change in position
The figure shows a block attached to a spring.
Here, displacement is x.
An oscillating simple pendulum’s angular displacement isβ.

Displacement variable may take negative values.

Periodic functions can be expressed as a superposition of the sine and cosine functions.

Oscillations of a block of mass, m fixed to a spring, which is in turn fixed to a rigid wall, are shown in the figure.

The block is pulled and released so that it executes to and fro motion (SHM).
Here,
m = Mass of the block
+A, −A = Maximum displacement
(x = 0) = Position of the centre of the block at the equilibrium of the spring

When the block is pushed to the right side, one spring is compressed while the other is elongated hence the block is subjected to a restoring force of F (x), which is proportional to the displacement, x (in the opposite direction).
As the block feels twice of restoring force because of two spring system,
∴ F (x) = −2kx …(i) Where k is the spring constant (depends on the property of the spring)
Using Newton’s law of motion, the force applied to pull the spring is
F = ma(which must be equal and opposite to the restoring force)
Since acceleration in SHM =−ω^{2}x
F =−mω^{2}x
On comparing it with equation (i)
2k = mω^{2}
Where ω is the angular speed of the spring
$\therefore \omega =\sqrt{\frac{2k}{m}}$

Time period (T) of the oscillator is,
$\therefore T=\frac{2\pi}{\omega}=2\pi \sqrt{\frac{m}{2k}}$

A simple pendulum is a heavy point mass suspended by a weightless, inextensible, flexible string attached to a rigid support from where it moves freely.

The periodic motion of a simple pendulum for small displacements is simple harmonic.
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