Any motion, which represents itself in equal intervals of time is called periodic motion. Often, the displacement of a particle in periodic motion can be expressed in terms of sine and cosine functions. Since the term harmonic is applied to expressions containing these functions, periodic motion is also called harmonic motion.
If a particle in periodic motion moves back and forth (or to and fro) over the same path, then its motion is called oscillatory or vibratory. The examples of oscillatory or vibratory motion are:
– the motion of a pendulum
– the motion of a spring fixed at one end, which is stretched or compressed and then released
– the motion of a violin string
– the motion of atoms in molecules or in a solid lattice
– the motion of air molecules as a sound wave passes by
Characteristics of a Harmonic Motion
The basic quantities characterizing a periodic motion are the amplitude, period and frequency of vibrations.
The amplitude of oscillations is the maximum displacement of a vibrating body from the position of equilibrium.
Time Period (T)
The time period of oscillations is defined as the time between two successive identical positions passed by the body in the same direction.
The frequency of oscillations is the number of cycles of vibrations of a body completed in one second. The frequency is related to the time period as
The SI unit of frequency is s-1 or Hz (hertz)
In mechanical oscillations a body oscillates about its mean position which is also its equilibrium position. At the equilibrium position no net force (or torque) acts on the oscillating body. The displacement (linear or angular) of an oscillating particle is its distance (linear or angular) from the equilibrium position at any instant.
When a body oscillates along a straight line within two fixed limits; its displacement x changes periodically in both magnitude and direction; its velocity v and acceleration a also vary periodically in magnitude and direction, shown in figure. Since F = ma, therefore, the force acting on the body also varies in magnitude and direction with time.
In terms of energy, we can say that a particle executing harmonic motion moves back and forth about a point at which the potential energy is minimum (equilibrium position). The force acting on the body at any position is given by
When a body is displaced from its equilibrium it is acted upon by a restoring force (or torque) which always acts to accelerate the body in the direction of its equilibrium position as shown in the figure.
- A particle of mass m is executing oscillation about the origin on the x-axis. Its potential energy is U(x) = k |x|3, where k is a positive constant. If the amplitude of oscillation is a, then its time period T is
(a) proportional to 1÷√a (b) independent of a
(c) proportional to √a (d) proportional to a3/2
- A particle free to move along the x-axis has potential energy given by U(x) = k[1 – exp (–x2)] for –£ x £ + , where k is a positive constant of appropriate dimensions. Then
(a) at points away from the origin, the particle is in unstable equilibrium
(b) for any finite non-zero value of x, there is a force directed away from the origin
(c) if its total mechanical energy is it has its minimum kinetic energy at the origin
(d) for small displacements from x = 0, the motion is simple harmonic
SIMPLE HARMONIC MOTION
Let us consider an oscillatory particle along a straight line whose potential energy function varies as
where k is a constant
| The force acting on the particle is given by
Such an oscillatory motion in which restoring force acting on the particle is directly proportional to the displacement from the equilibrium position is called Simple Harmonic Motion.
The potential energy function of such a particle is represented by a symmetric curve as shown in the figure(2).
Note that the limits of oscillation are equally spaced about the equilibrium position. This is not true for the motion shown in figure(1), which is harmonic, but not simple harmonic.
Applying Newton’s Second Law in equation (4), we get
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