A motion that repeats itself at regular intervals of time is called periodic motion.
If the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, giving rise to oscillations or vibrations.
The smallest interval of time after which the motion is repeated is called its period.
The reciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol The relation between v and T is
Any periodic function can be expressed as a superposition of sine and cosine functions of different time periods with suitable coefficients.
The quantity A is called the amplitude of the motion. It is a positive constant which represents the magnitude of the maximum displacement of the particle in either direction. The cosine function varies between the limits so the displacement varies between the limits
The time varying quantity, is called the phase of the motion. It describes the state of motion at a given time.
The constant is called the phase constant (or phase angle). The value of depends on the displacement and velocity of the particle at
The constant called the angular frequency of the motion, is related to the period
The angular frequency is,
The SI unit of angular frequency is radians per second.
Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion takes place.
Simple harmonic motion is the motion executed by a particle subject to a force, which is proportional to the displacement of the particle and is directed towards the mean position.
A particle executing simple harmonic motion has, at any time, kinetic energy and potential energy . If no friction is present the mechanical energy of the system, always remains constant even though K and U change with time.
A particle of mass m oscillating under the influence of a Hooke's law restoring force given by
exhibits simple harmonic motion with
Such a system is also called a linear oscillator.
The restoring torque is given by,
where the negative sign indicates that the torque acts to reduce and L is the length of the moment arm of the force sin about the pivot point. For rotational motion we have,
where I is the pendulum's rotational inertia about the pivot point and α is its angular acceleration about that point.
Period of oscillation =
The mechanical energy in a real oscillating system decreases during oscillations 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 the angular frequency of the damped oscillator, is given by
If the damping constant is small then where is the angular frequency of the un-damped oscillator. The mechanical energy of the damped oscillator is given by
If an external force with angular frequency acts on an oscillating system with natural angular frequency ω, the system oscillates with angular frequency.
The amplitude of oscillations is the greatest when
a condition called resonance
A body oscillates with SHM according to the equation (in SI units), s, calculate the (a) displacement, (b) speed and (c) acceleration of the body.
The angular frequency of the body and its time period s.
(a) displacement =
(b) the speed of the body
(c) the acceleration of the body =× displacement
Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case.
This function represents a simple harmonic motion having a period and a phase angle
The function is periodic having a period It also represents a harmonic motion with the point of equilibrium occurring at instead of zero.