PERIODIC AND OSCILLATORY MOTION: -
Motion of a system at regular interval of time on a definite path about a definite point is
known as a periodic motion, e.g., uniform circular motion of a particle.
To and fro motion of a system on a linear path is called an oscillatory motion, e.g., motion
of the bob of a simple pendulum.
SIMPLE HARMONIC MOTION: -
DEFINATION: - “The periodic motion of a body about a fixed point, on a linear path, under the influence of the force acting towards the fixed point and proportional to the displacement of the body from the fixed point, is called a simple harmonic motion.”This is the simplest type of periodic motion which can be understood by considering the following example.
Suppose a body of mass m is suspended at the lower end of a massless elastic spring obeying Hooke’s law which is fixed to a rigid support in the vertical position. The spring elongates by length Δl and
attains equilibrium as shown in Fig. ( b ) Here two forces act on the body. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k Δl , upwards, where k = force constant of the spring.
For equilibrium, mg = k Δl … ( 1 )
The spring is constrained to move in the vertical direction only. Now, suppose the body is given some energy in its equilibrium condition and it undergoes displacement y in the upward direction as shown in Fig. ( c ).
Two forces act on the body in this displaced condition also. ( 1 ) Its weight, mg, downwards and
( 2 ) the restoring force developed in the spring, k ( Δl - y ), upwards. The resultant force acting on the body in this condition is given by F = - mg + k ( Δl - y ) … … ( 2 )
From equations ( 1 ) and ( 2 ),
F = - ky
DISPLACEMENT: - The distance of the body at any instant from the equilibrium point is known as its
displacement at that instant. The displacements along the positive Y-axis are taken as positive
and those on the negative Y-axis are taken as negative. In the equation, F = - ky, F is negative when y is positive and vice versa. Thus, the resultant force acting on the body is proportional to the displacement and is directed opposite to the displacement, i.e., towards the equilibrium point.
The body performing SHM is known as a simple harmonic oscillator ( SHO ).
COUPLED OSCILLATIONS: - The figure shows two pendulums connected by an elastic spring. Obviously, they cannot oscillate independently of each other. They are called coupled oscillators ( more
appropriately coupled pendula ) and their oscillations are known as coupled oscillations. The constituent particles of solids also undergo coupled oscillations. Oscillations of coupled oscillators are complex and not always simple harmonic, i.e., their displacements x1 and
x2 cannot be expressed in the form of sine or cosine functions. But by suitable transformation of the co-ordinate system, they can be expressed in the form of equations of SHM as under.
X1 = A sin ( ω1t + φ1 ) … … … ( 1 ) and
X2 = B sin ( ω2t + φ2 ) … … … ( 2 ),
where X1 = x1 + x2 and X2 = x1 - x2 .
ω1 and ω2 are normal frequencies and oscillations given by X1 and X2 with these frequencies are the normal modes of vibrations of the coupled oscillators. This oscillator has two normal modes as only two co-ordinates are present. With proper selection of initial Conditions, the coupled oscillator can be oscillated in any one of these two modes. If at t = 0, x1 = x2 , i.e., both the oscillators are given equal displacements in the same direction, then B = 0 from equation (2).
The coupled oscillator will oscillate with angular frequency ω1 = g / l
according to equation (1). As shown in the figure, both the oscillators undergo equal displacements in the same direction in the same time. Hence the length of the spring does not change. So in this mode, the oscillators oscillate independently of each other as if the spring is not present. Next, if at t = 0, x1 = - x2 , i.e., both the oscillators are given equal displacements in mutually opposite directions and released, then A = 0 from equation ( 1 ). The coupled oscillator will oscillate with angular frequency. Both these types of oscillations are the normal modes of oscillations of the given coupled oscillator. If the initial conditions were different from the above two conditions, then the oscillations of each oscillator would be complex. However, in such a situation, the displacements of both the oscillators can be represented as a linear combination of the above two equations as the function of time.
Motion of a system at regular interval of time on a definite path about a definite point is
known as a periodic motion, e.g., uniform circular motion of a particle.
To and fro motion of a system on a linear path is called an oscillatory motion, e.g., motion
of the bob of a simple pendulum.
SIMPLE HARMONIC MOTION: -DEFINATION: - “The periodic motion of a body about a fixed point, on a linear path, under the influence of the force acting towards the fixed point and proportional to the displacement of the body from the fixed point, is called a simple harmonic motion.”This is the simplest type of periodic motion which can be understood by considering the following example.
Suppose a body of mass m is suspended at the lower end of a massless elastic spring obeying Hooke’s law which is fixed to a rigid support in the vertical position. The spring elongates by length Δl and
attains equilibrium as shown in Fig. ( b ) Here two forces act on the body. ( 1 ) Its weight, mg, downwards and ( 2 ) the restoring force developed in the spring, k Δl , upwards, where k = force constant of the spring.
For equilibrium, mg = k Δl … ( 1 )
The spring is constrained to move in the vertical direction only. Now, suppose the body is given some energy in its equilibrium condition and it undergoes displacement y in the upward direction as shown in Fig. ( c ).
Two forces act on the body in this displaced condition also. ( 1 ) Its weight, mg, downwards and
( 2 ) the restoring force developed in the spring, k ( Δl - y ), upwards. The resultant force acting on the body in this condition is given by F = - mg + k ( Δl - y ) … … ( 2 )
From equations ( 1 ) and ( 2 ),
F = - ky
DISPLACEMENT: - The distance of the body at any instant from the equilibrium point is known as its
displacement at that instant. The displacements along the positive Y-axis are taken as positive
and those on the negative Y-axis are taken as negative. In the equation, F = - ky, F is negative when y is positive and vice versa. Thus, the resultant force acting on the body is proportional to the displacement and is directed opposite to the displacement, i.e., towards the equilibrium point.
The body performing SHM is known as a simple harmonic oscillator ( SHO ).
COUPLED OSCILLATIONS: - The figure shows two pendulums connected by an elastic spring. Obviously, they cannot oscillate independently of each other. They are called coupled oscillators ( moreappropriately coupled pendula ) and their oscillations are known as coupled oscillations. The constituent particles of solids also undergo coupled oscillations. Oscillations of coupled oscillators are complex and not always simple harmonic, i.e., their displacements x1 and
x2 cannot be expressed in the form of sine or cosine functions. But by suitable transformation of the co-ordinate system, they can be expressed in the form of equations of SHM as under.
X1 = A sin ( ω1t + φ1 ) … … … ( 1 ) and
X2 = B sin ( ω2t + φ2 ) … … … ( 2 ),
where X1 = x1 + x2 and X2 = x1 - x2 .
ω1 and ω2 are normal frequencies and oscillations given by X1 and X2 with these frequencies are the normal modes of vibrations of the coupled oscillators. This oscillator has two normal modes as only two co-ordinates are present. With proper selection of initial Conditions, the coupled oscillator can be oscillated in any one of these two modes. If at t = 0, x1 = x2 , i.e., both the oscillators are given equal displacements in the same direction, then B = 0 from equation (2). The coupled oscillator will oscillate with angular frequency ω1 = g / laccording to equation (1). As shown in the figure, both the oscillators undergo equal displacements in the same direction in the same time. Hence the length of the spring does not change. So in this mode, the oscillators oscillate independently of each other as if the spring is not present. Next, if at t = 0, x1 = - x2 , i.e., both the oscillators are given equal displacements in mutually opposite directions and released, then A = 0 from equation ( 1 ). The coupled oscillator will oscillate with angular frequency. Both these types of oscillations are the normal modes of oscillations of the given coupled oscillator. If the initial conditions were different from the above two conditions, then the oscillations of each oscillator would be complex. However, in such a situation, the displacements of both the oscillators can be represented as a linear combination of the above two equations as the function of time.
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