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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 U-tube 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 U-tube 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 = ω2L

Net radial force = Tmg 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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