Oscillations
Periodic and oscillatory motions
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Periodic motion: A motion which repeats itself after a fixed interval of time
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Examples:
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Motion of the moon around the earth
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Motion of the hands of a clock
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Oscillatory motion: A body in oscillatory motion moves to and fro about its mean position in a fixed time interval.
Examples:
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Motion of the pendulum of a wall clock
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Motion of the liquid contained in a U-tube when one of its limbs is compressed.
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Period (T): It is the interval of time after which a motion is repeated. Its unit is seconds (s).
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Frequency (ν): Number of repetitions that occur per unit time
Its unit is (second)-1 or Hertz.
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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.
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Displacement variable may take negative values.
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Periodic functions can be expressed as a superposition of the sine and cosine functions.
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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.
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The periodic motion of a simple pendulum for small displacements is simple harmonic.
m − Mass of the bob
L − Length of the massless string
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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 = ω2L
Net radial force = T − mg cos θ
Tangential acceleration is provided by mg sin θ.
Torque, τ = −L (mg sin θ)
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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 is very small, then
Ignoring the higher powers of θ, we get
Angul…
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