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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.

• 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 …(iWhere 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 =−ω2x

F =mω2x

On comparing it with equation (i)

2k = mω2

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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