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Wave and Oscillation

Any motions that repeat themselves are called oscillation. But any motion or event that repeats itself at regular interval of time is said to be periodic motion. The motions of earth round the sun, the swinging motion of pendulum, the vibration of guitar string are some examples of periodic motion, and oscillatory motion.

                In above examples of mechanical oscillations, the body undergoes linear or angular displacement. There are some non mechanical oscillations also that involve the variation of quantities such as voltage, charge or even the electric and magnetic field. These are the examples of electromagnetic oscillations.

1.1 Simple Harmonic oscillator

When an object moves in an oscillatory motion such that its acceleration is always directed toward a certain fixed point and varies directly as its distance from that point, the object is said to execute simple harmonic motion (SHM). The fixed point of reference is referred as the mean position or center.

The force acting on the body for such motion to take place should be directed towards that fixed point and should be proportional to the displacement of the object from that fixed point. The force acted restores the object back to its equilibrium position. Therefore such force is often termed as restoring force.

Therefore for SHM to take place,

I.        The motion should be periodic.

II.       The object if displace from mean position, a restoring force tending to bring it to the mean position and directed towards the mean position must act on the object

III.    The restoring force should be directly proportional to displacement of the body form its mean position.

1.2 Differential Equation of simple Harmonic Oscillator

Consider a particle of mass m executes Simple Harmonic Motion. Let the displacement of the particle at any instant t be y. The restoring force F acting on particle by definition is

                              

or,                                                                                                                                    ……………..1.1

Where k is the constant of proportionality and is called force constant or stiffness factor. The negative sign in equation 1.1 indicates that restoring force and displacement are in opposite in direction which is necessary for repeated periodic motion.

Since , so equation 1.1 becomes,

m  + ky = 0

or,        +  = 0

Here,  and is often called the angular velocity of object executing SHM. So,

                                                                                         ……………..1.2

Equation 1.2 is the general differential equation of SHM.

Example 1.1

An oscillatory motion of a body is represented by  where y is displacement in time t, a is amplitude and  is angular frequency. Show that motion is simple harmonic.

Solution,

Here,

Differentiating,

Making second differentitation again,

 

This is the differential equation of SHM. So y= a  represent SHM.

Example 1.2

Show that the velocity of simple harmonic oscillator at any instance leads the displacement by a phase angle of . Hence the general expression for the velocity of a simple harmonic oscillator .

Solution,

           We know that the displacement of a simple harmonic oscillator at any instance of time t can be written as

The velocity is the rate of change of displacement so, velocity

          

             

Comparing the equations for displacement and velocity, the velocity of simple harmonic oscillator at any instant I leads the displacement by a phase difference of . However, the velocity varies simple harmonically with the same frequency.

But        

∴          

Also 

This velocity will be maximum if displacement is zero. Therefore,

V_mix =  at mean position

1.3 The Block-Spring System

Consider a block of mass m is oscillating at the end of a mass less spring as shown in figure 1.1 Let’s assume that the net force acting on the block  is that exerted by the spring , which is obtained from Hooke’s law.

or,         

this shows the acceleration is directly proportional to the displacement and is always in the opposite direction And this is the requirement for SHM. The acceleration is  therefore we can write,

             

                                                                                                                  ………………….1.3

Comparing this equation with differential equation of SHM, the block-spring system executes simple harmonic motion with an angular frequency ω given by

                                                                                                                                              ………………1.4

In this way the time period of a block-spring system is given by,

                                                                                                                              ………………….1.5

The period is independent of the amplitude which is requirement for SHM. From the equation 1.5 for a given spring constant (k), the period increases with the mass of the block. Therefore a massive block oscillates more slowly. For a given block, the period decreases as k increases indicating a stiffer produces a quiker oscillations.

Example 1.3

A 2kg mass hangs from a spring. A 300g body hung below the mass stretches the spring 2cm farther. If the 300g body is removed and the mass is set in oscillation, find the period of motion.

Solution,

When 300g body is hung, the spring stretches 2 cm further. So,

N/m

Again time period of oscillation,

1.4    Energy in Simple Harmonic Motion

The force exerted by an ideal spring is conservative. If there is no friction, the energy of simple harmonic motion like block-spring system is constant. The solution of differential equation of SHM is

                                                                                                       

The potential energy of the system is,

                                                                             ….…….1.6

Similarly kinetic energy of the system is,

                                         ………. 1.7

Therefore the total mechanical energy,

or,                                    [  

or,                                                                                                                 ………….1.8

This is the total energy of a block-spring system. It shows the total energy of any simple harmonic oscillator is constant. The total energy is also found to be proportional to the square of the amplitude. The variation of K and U as function of displacement x is shown in figure 1.2. When  , k is zero and the total energy is equal to the maximum U, E = U_max =  Similarly, when x =0, U is zero and the total energy is equal to maximum K.E=

So the total energy in simple harmonic motion is constant quantity.

1.5 Simple Pendulum

                A simple pendulum  is an idealized system in which a point mass is suspended at one end of a mass-less string. Consider a simple pendulum of length  l and a bob of mass m is suspended at the free end of the pendulum as shown in figure 1.3. The other end of the spring is suspended at the rigid support. The distance along the arc from the lowest point  where  is the radian angle made to the vertical. The weight of the mass is mg and acts vertically downward. This force due to gravity can be resolved into two rectangular components ;mg cos  is the force along the string and mg sin  perpendicular to the string. The tension on the string is T and is balanced by the components mg cos . The Newtonian force is balanced by the component of force due to gravity along the perpendicular direction.

                                                                                                                                     ……..1.9

The  negative sign is for way x is defined. Therefore this component of force due to gravity acts as restoring force. From Taylor’s series of expansion one can write,

Thus for small angles  where θ is in radians. From circular measurement

So  we can write

Now equation 1.9 becomes,

                                                                                             …..1.10

This equation is similar to the equation of simple harmonic motion

There[i]fore within  the small angle approximation, simple pendulum execute simple harmonic motion with an angular frequency

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