1 Periodic Motion:

A motion of the body which repeats itself after a regular interval of time is called periodic motion.

    e.g: The motion of all planets around Sun, motion of Moon around Earth, Motion of bob of pendulum, all are periodic motions.

2Oscillatory Motions:

If a body moves to and fro about a fixed point, then the motion of the body is called oscillatory motion.

    e.g: The motion of a bob of simple pendulum, motion of spring is oscillatory motion.

  Periodic Motion V/s Oscillatory Motion:

All oscillatory motions are periodic motions because they repeats their periods after a regular interval of time but all periodic motions are not oscillatory motions because it is not necessary to vibrate to and fro for periodic motions.

3 Periodic Functions:

Any functions which repeat itself at a regular interval of time is called periodic functions.

Consider a function  of time period  such that    

 

So  is periodic function over a time period.

e.g:-  and  are periodic over a time interval or period  

as  

 And  

Now suppose  

 

⟹

Similarly 

Hence   

and  are periodic functions.

Or we may write         

 

And  

4 Harmonic and Non Harmonic Periodic Functions:

A periodic function which can be represented by a sine or cosine curve is called harmonic function 

e.g:- 

A periodic function which cannot be represented by a sine or cosine is called non harmonic function

e.g:- etc.

    A non harmonic function may be constructed by two or more than two harmonic functions as 

  

    All harmonic functions are periodic but all periodic functions are not harmonic because all periodic functions cannot be represented by sine or cosine functions.

5  Simple Harmonic Motion:

If a particle moves to and fro about its mean position then the motion of particle is called simple harmonic motion.

The restoring force in the body is direction proportional to the displacement of the particle from mean position      

 i.e.  

Or       

Or 

Where  is called force constant.

Here  sign indicate that restoring force always act in opposite to displacement of the body.

Now from Newton’s second law

⟹ 

Or  

Hence in S.H.M., acceleration produced in the body is directly proportionality to displacement of the body.

 Example of S.H.M:

•                   Oscillations of a loaded spring
• Vibration of a tuning fork.
•                    Oscillation of a bob of simple pendulum.

6  Differential Equation for S.H.M.:

In S.H.M. the restoring force acting on the particle may be given as

 

Where  is spring constant and  is displacement

Now from Newton second law of motion.

⟹ Comparing eq. (i) and (ii) we get  

             

Or  

Or   

Putting  

we get        

Or 

The above eq. is called differential eq. of S.H.M.

The solution of above eq. is

 

Where  is amplitude,  is phase angle, is initial phase or epoch.

This can be proven as 

Then 

And 

⟹ 

Which is same as eq. (iii) hence solution of eq. (iii) is

 

7 Time Period of S.H.M.:

If we replace  by  in eq.

  

then we get  

 

Thus time period of S.H.M. is     

Or      

Or 

8 Some Important Terms Connected with S.H.M:

•                            Harmonic oscillator:

 A particle executing simple harmonic motion is called harmonic oscillator.

•                         Displacement:

The displacement of oscillating particle from its mean position at any instant of     time is called displacement of the particle. It is denoted by   or.

•                      Amplitude:

The maximum displacement of the particle from mean position is called amplitude. It is denoted by .

•                      Oscillation or Cycle:

One complete back and forth motion of particle starting and ending at same point is called one oscillation or one cycle one vibration.

•                         Time Period:

The time taken by the particle to complete one cycle is called time period. It is denoted by .

•                      Frequency:

The number of oscillations completed unit time is called frequency. It is denoted by 

Unit: 

(Vii) Angular Frequency:

A quantity obtained by multiplying frequency  by a factor  is called angular frequency. It is denoted by .   

Thus 

Unit:- 

9 Phase:

The phase of a particle at any instant gives the state of the particle w.r.t. its position and direction of motion at that instant. It is denoted by .

For a simple harmonic eq.   

The phase of the particle is     

 

The phase of the particle at  is called its initial phase or epoch.

 i.e. At  ,   

Thus  is initial phase or epoch.

10 Uniform Circular Motion and S.H.M.:

Suppose a particle  moving in a circle of radius. At any time suppose particle make angle  with  and  is foot of ⊥ drawn from  to. Now when particle move from  to  then  to  then  to  then   to . In all cases the  oscillate about mean position  on  axis. Thus uniform circular motion also exhibits S.H.M. or simple harmonic motion may be defined as the projection of uniform circular motion upon a diameter of the circle.

Displacement in Simple Harmonic Motion:

As from diagram 

In right angle  

∆ ONP     

Where  

Or    

Or    

⟹ displacement 

If point A lies below the x axis 

then   

⟹ 

Here  is the initial phase of S.H.M.

Q.  Show that  represents S.H.M.

Sol. As  

Differentiating w.r.t. time , we get  

 

Again differentiating w.r.t. time , 

we get  

Or    

Or   

Which is the eq. for S.H.M. Hence  represents S.H.M.

11 Velocity in S.H.M.:

As we know that rate of change of displacement of a particle represent velocity. So we can calculate velocity by differentiating displacement.

As displacement of the particle having S.H.M. can be given as  

 

Differentiating w.r.t. time we get  

     

Or 

Or 

Or     

Special Cases:

1.                            When particle is at mean position

 then     ⟹ 

2.                         When particle is at extreme position then

         ⟹ 

Hence velocity at extreme position is zero.

12 Acceleration of the Particle Having S.H.M.:

As we know the rate of change of velocity of a particle is called acceleration. So by differentiating velocity of the particle w.r.t. time we can obtain acceleration.

Here in S.H.M. the velocity of the particle can be given as 

 

Differentiating w.r.t. time we get   

    

Or 

Or      

 Here  sign indicate that acceleration produced in the particle will oppose the displacement of the particle.

Special Cases:

1.                            When particle is at mean position then

   ⟹  

Hence acceleration of the particle at mean position is zero.

2.                         When particle is at extreme position then

   ⟹ . 

3.                      Hence acceleration of the particle at extreme position is maximum.

 

13 Phase Relationship between Displacement, Velocity and Acceleration:

Graphical relationship between displacement, velocity and acceleration of particle having S.H.M. is                     

And 

The variation of  and  with respect to time may be given by

)

Now we can plot a graph between t and A for displacement, velocity and acceleration as below: 

Conclusion:

1.                            Displacement, velocity and acceleration all vary harmonically w.r.t. time.
2.                         The amplitude of velocity and acceleration are  and  times respectively for displacement.
3.                      Velocity leads displacements by phase angle .
4.                      Acceleration leads displacement by phase angle π. Also when displacement is maximum the acceleration is minimum and vice versa. But when displacement is zero then acceleration is also zero.

14 Energy in S.H.M.: Kinetic and Potential Energy:

There are two types of energy associated with S.H.M., Kinetic energy and potential energy both may be calculated as given below:

1.                            Kinetic Energy:

As we know the displacement of the particle having S.H.M. is 

 

So velocity    

Hence Kinetic energy

 

Or 

 

•                         Potential Energy:

As we know the displacement x of the particle from its mean position on is directly proportional to force acting on it as                         

  

Now small amount of work is to be done to move particle small distance  as    

So total work done      

 

This work is store in body in form of P.E. so  

  

Total Energy:

The sum of Kinetic energy and potential energy is called total energy of a harmonic oscillator as given below:

Or  

Hence total energy of a harmonic oscillator is independent of time or displacement, it is conserved.

15 Graphically Representation:

At mean position i.e.  

Total energy 

Kinetic energy 

Potential energy 

At extreme position i.e. 

Kinetic energy 

Potential energy 

Thus sum of K.E. and P.E. remains conserved. 

Example  A particle of mass 40 g executes a simple harmonic motion of amplitude 2.0 cm.  If the time period is 0.20 s, find the total mechanical energy of the system. 

Solution: The total mechanical energy of the system is

 

Example  A particle of mass 0.50 kg executes a simple harmonic motion under a force F = – (50 N/m)x.  If it crosses the centre of oscillation with a speed of  10 m/s,  find the amplitude of the motion. 

Solution:  The kinetic energy of the particle when it is at the centre of oscillation is

 

The potential energy is zero here.

At the maximum displacement x = A,

the speed is zero and hence the kinetic energy is zero.

The potential  , as there is no loss of energy,

 

The force on the particle is given

By

Thus, the spring constant is k = 50 N/m.

Equation (i) gives

or,  A = 1 m.

16 Oscillation due to a Spring:

1.                            Horizontal Oscillations of Body on Spring:

Suppose a mass less spring whose one end is fixed and other end is stretched to a distance  from equilibrium position then restoring force developed in spring is 

Or 

Or 

Or 

Thus spring will execute S.H.M. 

Now time period of oscillation is   

    

Or 

And frequency of oscillation

 

Thus smaller the time period larger will be the frequency of oscillation. Also for smaller body the time period will be small and frequency will be large.

•                         Vertical  Oscillation of a body on a Spring:

If a spring is suspended vertically and a body of mass  is attached to its lower end then spring get stretched to a distanced due to the weight of the body.

If the body is pulled vertically downward through a small distance  from its equilibrium position and then released. Then restoring force developed in the spring may be given by 

Here          

So 

If  is the amount of acceleration in the body, 

then        

Or 

Or   

Hence body execute S.H.M. and hence its time period may be given as  

 

Or  

Here clearly shown that acceleration due to gravity does not effects the time period of the oscillation of the particle.

•                      Oscillation of Loaded Spring Combination:
•             Spring Connected in parallel:

In fig. (a) and (b) two spring of spring constants  and  are connected in parallel. Let  is the extension produced in each spring then restoring force developed in the springs may be given as 

   and   

So the total restoring force in parallel combinations of spring is           

   

Now let   is the force constant in parallel combination 

then        

Or  

So the frequency of vibrations of the parallel combination is   

 

•             If Spring are connected in series:

In fig. (c) and (d) two springs of spring constants  and  are connected in series and suppose  and  are the extension produce in the springs.

Then restoring forces developed in spring is   

      and    

Or        and   

But in series combination      

So total extension

 

Or         

Or  

If  is force constant for series combination

 then    

⟹      

Or 

∴ The frequency of vibration for series combination is

  

Example  A particle of mass 200 g executes a simple harmonic motion. The restoring force is provided by a spring of spring constant 80 N/m.  Find the time period. 

Solution:  The time period is

 

17 Simple Pendulum:

A simple pendulum is consists of point mass suspended by a flexible, inelastic and weightless string from a rigid support of infinite mass. In practical, we can neither have a point mass nor a weightless string so no pendulum is an ideal simple pendulum.

Expression for Simple Pendulum:

Suppose at any time  the bob of simple pendulum lies at point. At  the weight of the bob acts in downward direction. Now resolving  into rectangular components we get 

Where T is the tension in the strings and  brings the bob toward mean position and act as restoring force as 

When θ is small then we can write  

Also 

Or 

Or 

Or 

Clearly 

Now time period of simple pendulum is 

            

Or 

    Thus time of a simple pendulum depends upon length, acceleration due to gravity  and independent of mass  of the bob.

Example  Calculate the time period of a simple pendulum of length one meter. The acceleration due to gravity at the place is  

Solution :  The time period is

 

Example  In a laboratory experiment with simple pendulum it was found that it took  36 s to complete 20  oscillations when the effective length was kept at  80 cm.  Calculate the acceleration due to gravity from these data.  

Solution:  The time period of a simple pendulum is given by

 

or 

In the experiment described in the question, the time period is

 

Thus, by (i),

18 Other Example of S.H.M.:

1.                    Oscillation of a liquid Column in a U-Tube:

Suppose a U-tube of cross section A contains a liquid of density ρ up to a height. Then mass of the liquid in U-tube is 

 

If liquid is depressed at one side then restoring force develop in liquid i.e.  

  Or 

i.e.  

Thus force on the liquid is proportional to displacement and acts in its opposite direction; hence the liquid in U-tube executes S.H.M. with force constant  

The time period of oscillation is 

If  is the length of the liquid column, then

   and  

         Oscillation of Body Dropped in a Tunnel Along the Diameter of Earth:

Suppose earth is a sphere of radius  and centre . again suppose that a pipe is drug along the diameter of the earth. Again suppose that a body of mass  is dropped in to the tunnel and is at point P which is at a depth d below the surface of earth.

If  is the acceleration due to gravity at  then           

  

If  is the displacement of the body from centre of the earth then   

  ⟹ 

Now force acting on the body is    

 

Or     

Or     

where   

Hence the displacement of the body is directly proportional to restoring force or applied force. Hence body will execute S.H.M. with time period of oscillation   

19 Free, Damped and Maintained Oscillation:

•                            Un-damped or Free Oscillation: 

If amplitude of oscillating particle does not changes with respect to time then the oscillation of particle is said to be free or un-damped oscillation. 

e.g.: The oscillation of a bob of simple pendulum.

(ii)  Damped Oscillation:

If the amplitude of the oscillating particle changes w.r.t. time then oscillation of the particle is said to be damped oscillation, this occurs due to lose of energy of the oscillating particle due to friction of viscous mediums such as air, water etc.  e.g: The oscillation of bob of pendulum in a fluid.

(iii)  Maintained Oscillations: 

If constant energy equal to lose of energy during oscillation of a body is supplied then damped oscillation becomes un-damped oscillation. Now the oscillation of the body under constant energy supply is called maintained oscillation.

20 Forced, Resonant and Coupled Oscillations:

When a body oscillate under the effect of an external force then the oscillation of the body is called forced oscillation.

e.g.:- When the free hand of the simple pendulum is held by the hand and the pendulum is made to oscillate by giving Jerks by hand, then the oscillation of the pendulum is called forced oscillation.

Resonant Oscillation:

If the frequency of driving force is equal to the natural frequency of the oscillator itself then the oscillation of the body is called resonant oscillation and the phenomenon is called resonance.e.g.:- Resonance causes disaster during earth quake i.e. when natural frequency of building become equal to earth quake then building damages.

Principal of Tuning of a radio receiver:

We know that there are many signals of many stations around radio but we obtain a particular radio signal by setting a certain frequency of radio. When external frequency is same as that of stetted frequency then we listen that station.

Coupled Oscillation:

A system of two or more oscillators linked together in such a way that there is a mutual exchange of energy between then, then the oscillation of the body is called coupled oscillations. 

E.g. Oscillations of loaded springs etc

                                                   

 

 

 

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