Simple Harmonic Motion (SHM)

Simple Harmonic Motion

It is the periodic (repeated) motion. in which a particle moves to and fro motion about a mean (equilibrium) position under a restoring force(acceleration) which is always directed toward mean position such that its magnitude is proportional to displacement (a∝ -y).

Or, F ∝ -y

F= -ky

 Types: (i) Linear SHM; a∝-y

           Or, a=-ky


(ii) Angular SHM

∝ = -Kθ


Equation of Simple Harmonic Motion in different conditions

  The general equation of SHM is α= Asin(wt+).

1) From position (x=0, t=0) to x=+A

X= Asinwt ( =0)

2) From right extreme (t=0, x=+A)

x= A sin (wt + π/2)

x= A coswt

3) From left extreme (t=0, x=-A)

 x= A sin(wt - \2)

x= -Acoswt

4) from half of maximum displacement to A

x= Asin(wt +/6) as,

= sin-1(x/a) where x = A/2

Time taken by the particle

1) from (x=0 to A/2)

x = A sinwt

or, A/2=Asinwt


 2)from (x=A to A/2)

x = A cosw t  



3)from one extreme to another extreme.(x=-Ato A)


4)From mean position to extreme position,

x = A sinwt

T= T/4 

Where, T isthe total time period from A to-A and from -A to A.

Velocity in SHM


So, at x=0


and at x = A

Also, v2=w² (A2-x²)



which shows graph between velocity and displacement on SHM is elliptical.

Acceleration on SHM

So, graph between a and x is st. line.

•At x=0, a=0 (min), at v=A; a=-w²A.

• For maximum amplitude a0 and maximum velocity V0 Amplitude and angular velocity are given as    so,

Spring mass system

•for both horizontal and vertical arrangements  



•If mass of sparing is considered ();  

Reduced Mass

It is the behaviour of combined two masses when they are attracted toward each other by a certain force.

**** Distance doesn't matters.

For two bodies joined as aside:

Time period is where μ=reduced mass=

Spring combination

Simple Pendulum

The time period of pendulum is  given by;

 for a very short pendulum; R=Radius of earth ;l<<<


For ;

For L=R, =59.8min.

Note time period is independent of mass of bob of pendulum.

Change on length

•If the length of pendulum increases, time period increases  as (Tα);

motion becomes slow and time with loose and vice versa.

(i)If initial time period is T1, and by some factor it changes to T₂ if (T2>T1)•Loss in time =*86400 secs.

(ii)if (T2>T1)gain in time =*86400 secs.

•If a simple pendulum is taken from mean sea level to height h, Loss in time = 13.5*h secs.(h in km)

•If a simple pendulum is taken from mean sea level to depth h, Loss in time =13.5*hsecs. (h in km)

•Due to change on temperature loss or gain=0.5αΔθ

Change in acceleration due to gravity

where is the effective acceleration due to gravity.

In a lift

i. Moving upward:

ii. Moving downward:

iii. Moving with constant velocity:

On inclined plane

where is angle made by inclined plane with horizontal

In a train

i. Accelerating with acceleration :

ii. Moving with constant speed in circular path:

Tension on string

we have,



= where A is the amplitude

Energy in simple harmonic motion



T.E.== constant

So, P.E α α α ==

K.E α (A2-x2) α = ==

Hence, K.E and P.E has double the frequency of SHM.


  1. A body executes SHM under action of force F1 with frequency f₁. If force changed to so it executes SHM with frequency  f2 when both force simultaneously acts. what is frequency of oscillation?    


F=mrw2 = α f2

so,and ,  

 for, (say)                                               

or, ,f is required frequency.

  1. A load of mass m falls from height h onto the scale pan hung from a spring of spring constant k and mass of scale pan is zero and mass m doesn’t bounce relative to pan, then amplitude of vibration is


Initial Energy= Final energy


Taking +ve as > mg