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.

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.
m − Mass of the bob
L − Length of the massless string

Given below is a free body diagram to show the forces acting on the bob.
θ − Angle made by the string with the vertical
T − Tension along the string
g − Acceleration due to gravity
Radial acceleration = ω^{2}L
Net radial force = T − mg cos θ
Tangential acceleration is provided by mg sin θ.
Torque, τ = −L (mg sin θ)

According to Newton’s law of rotational motion,
τ = Iα
Here,
I − Moment of inertia
α − Angular acceleration
∴ Iα = −mg sin θ L
If θ …
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