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Monday, December 4, 2017

Simple Harmonic Motion

Periodic Motion: Any motion which repeats itself at regular intervals of time is called Periodic Motion or Harmonic Motion. Some of the examples of periodic motion are a rocking chair, a bouncing ball, a vibrating tuning fork, a swing in motion, the Earth in its orbit around the Sun, and a water wave.

•    The interval of time for a repetition, or cycle, of the motion is called a period.
•    The number of periods per unit time is called the frequency.

Thus, the period of the Earth’s orbit is one year, and its frequency is one orbit per year. A tuning fork might have a frequency of 1,000 cycles per second and a period of 1 millisecond (1 thousandth of a second).

The Simple harmonic motion is a special case of periodic motion. In the examples given above, the rocking chair, the tuning fork, the swing, and the water wave execute simple harmonic motion, but the bouncing ball and the Earth in its orbit do not.

  • Waves that can be represented by sine curves are periodic.
  • If the wave is propagated with a velocity v and has a wavelength λ, then the period T is equal to wavelength divided by velocity, or

T= λ/v.

The frequency f is the reciprocal of the period;

thus, f = 1/T = v/λ

Oscillatory motion
– The oscillatory motion is motion that is repetitive and that which cycles about a mean position. We can also call the oscillatory motion, the periodic motion.

Examples of oscillatory motion- Some of the examples of oscillatory motion are motion of a pendulums, vibrating strings, and elastic materials such as a spring. Physical phenomena such as light, sound, ocean waves, and molecular vibrations also exhibit oscillatory motion.

Time period and frequency
When an object performs oscillatory motion, the time taken for it to go from the mean position to the extreme positions and again return to the mean position (one oscillation) is called the time period. This is equal to the one oscillation. The number of oscillations an object performs in 1 second its frequency.

Simple Harmonic Motion

If a particle repeats its motion about a fixed point after a regular interval of time in such a way that at any moment the acceleration of the particle is directly proportional to its displacement from the fixed point at that moment and is always directed towards the fixed point then the motion of the particle is called Simple Harmonic Motion or S.H.M.

The fixed point around which particle oscillates is called equilibrium point.

Characteristics of S.H.M.

When a particle describes Simple Harmonic Motion (SHM) through its mean position –

•    No force acts on the particle.
•    Acceleration of the particle is zero.
•    Velocity is maximum.
•    Kinetic energy is maximum.
•    Potential energy is zero.

When a particle describes Simple Harmonic Motion (S.H.M.) at the extreme end then-

•    Acceleration of the particle is maximum.
•    Restoring force acting on the particle is maximum.
•    Velocity of particle is zero.
•    Kinetic Energy of particle is zero.
•    Potential Energy is maximum.

Simple Pendulum

If a point mass is suspended from a fixed support with the help of a mass less and inextensible string, the arrangement is called Simple Pendulum. Generally, a simple pendulum is made by hanging a small ball called bob from a fixed support with the help of a light string.

If the bob of a simple pendulum is slightly displaced from its mean position and then released, it starts oscillating in simple harmonic motion.

The time period of oscillation of a simple pendulum is given by equation T = 2π√l/g; where l is the effective length of the pendulum and g is the acceleration due to gravity.