# Summary

• The motions which repeat themselves are called periodic motion.
• The period T is the time required for one complete oscillation,or cycle. It is related to the frequency v by,
• The frequency v of periodic or oscillation motion is the number of oscillations per unit time. In the SI, it is measured in hertz:
• 1 hertz = 1 Hz = 1 oscillation per second = 1s-1
• In simple harmonic motion (SHM), the displacement x(t) of a particle from its equilibrium position is given by,
(displacement),
• in which A is the amplitude of the displacement, the quantity is the phase of the motion, and Î¦ is the phase constant. The angular frequency Ï‰ is related to the period and frequency of the motion by
(angular frequency),
• Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the latter motion occurs.
• The particle velocity and acceleration during SHM as functions of time are given by,
(Velocity),
• Thus we see that both velocity and acceleration of a body executing simple harmonic motion are periodic functions, having the velocity amplitude and acceleration amplitude , respectively.
• The force acting simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
• A particle executing simple harmonic motion has, at any time, kinectic energy and potential energy . If no friction is present the mechanical energy of the system, E = K+U always remains constant even though K and U change with time.
• A particle of mass m oscillating under the influence of a Hook's law restoring force given by F = -kx exhibits simple harmonic motion with
(angular frequency),
(period)
• Such a system is also called a linear oscillator.
• The motion of a simple pendulum swinging through small angles is approximately simple harmonic. The period of oscillation is given by,
• The mechanical energy in a real oscillating system decreases during oscillating because external forces, such as drag, inhibit the oscillations and transfer mechanical energy to thermal energy. The real oscillator and its motion are then said to be damped. If the damping force is given by , where v is the velocity of the oscillator and b is a damping constant, then the displacement of the oscillator is given by,
• Where Ï‰', the angular frequency of the damped oscillator, is given by
• If the damping constant is small then Ï‰' = Ï‰, where Ï‰ is the angular frequency of the undamped oscillator. The mechanical energy E of the damped oscillator is given by
• If an external force with angular frequency Ï‰d acts on an oscillating system with natural angular frequency Ï‰, the system oscillates with angular frequency Ï‰d. The amplitude of oscillations is the greatest when a condition called resonance.