# 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

Or,

(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

t=T/12

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

x = A cosw t

Or,A/2=Acoswt

t=T/6

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

t=T/2

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

V=

So, at x=0

So,

and at x = A

Also, v2=w² (A2-x²)

Or,

Or,

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

F=-kx=ma

Or,

•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<<<

so,

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,

so

=

= where A is the amplitude

Energy in simple harmonic motion

P.E.====

K.E.===

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.

Examples

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?

solution:

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

Solution:

Initial Energy= Final energy

so,

Taking +ve as > mg