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**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,$dt_{2}d_{2}y +Ky=0$

(ii) Angular SHM

∝ = -Kθ

$dt_{2}d_{2}θ +Kθ=0$

**Equation of Simple Harmonic Motion in different conditions**

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

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

X= Asinwt ($ϕ$ =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 - $π$\2)

x= -Acoswt

4) from half of maximum displacement to A

x= Asin(wt +$π$/6) as,

$ϕ$ = 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= $dtdx =Awcos(wt+ϕ)=wA_{2}−x_{2} $

So, $V_{max}=Aw$at x=0

So, $V=AwA_{2}1−x_{2} =V_{max}∗A_{2}1−x_{2} $

and $V_{min}=0$at x = A

Also, v^{2}=w² (A^{2}-x²)

Or, $w_{2}v_{2} +x_{2}=A_{2}$

Or, $A_{2}w_{2}v_{2} +A_{2}x_{2} =1$

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

**Acceleration on SHM**

•$a=dtdv =−Aw_{2}sin(wt+ϕ)=−w_{2}x$

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

•At x=0, a=0 (min), at v=A; a=-w²A.

• For maximum amplitude a_{0} and maximum velocity V_{0} Amplitude and angular velocity are given as $A=a_{0}v_{0} $ $w=v_{0}d_{0} $ so,$T=a_{0}2πv_{0} $

**Spring mass system**

•for both horizontal and vertical arrangements

F=-kx=ma

$a=m−kx =−w_{2}x$

Or, $w=mk $

$T=2πkm $

•If mass of sparing is considered ($m_{s}$); $T=2πk3m+ms $

**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 $T=2πkμ $ where μ=reduced mass=$M_{a}+M_{b}M_{a}.M_{b} $

**Spring combination**

**Simple Pendulum**

The time period of pendulum is given by;

$T=2π1+LR R g $

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

so,$T=2πgL $

For $L→∞$; $T=2πgR $

For L=R, $T=2π2gR $ =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$l $);

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 =$T_{1}T_{2}−T_{1} $*86400 secs.

(ii)if (T2>T1)gain in time =$T_{1}T_{2}−T_{1} $*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**

$T=2πg_{r}l $ where $g_{r}$ is the effective acceleration due to gravity.

**In a lift**

i. Moving upward: $g_{r}=g+a_{0}$

ii. Moving downward: $g_{r}=g−a_{0}$

iii. Moving with constant velocity: $g_{r}=g$

**On inclined plane**

$g=gcosθ$ where $θ$is angle made by inclined plane with horizontal

**In a train**

i. Accelerating with acceleration $a_{0}$: $g_{r}=g_{2}+a_{0} $

ii. Moving with constant speed in circular path: $g_{r}=g_{2}+(rv_{2} )_{2} $

**Tension on string**

$T_{max}=rmv_{2} +mg$

we have, $V_{max}=Aw=Alg $

so $T_{max}=rm(Alg )_{2} +mg$

= $mg+mg(l_{2}A_{2} )$

= $mg(1+l_{2}A_{2} )$ where A is the amplitude

**Energy in simple harmonic motion**

P.E.=$21 kx$*=*$21 $$(mxw_{2})∗x$=$21 mx_{2}w_{2}$=$21 mA_{2}w_{2}sin_{2}θ$

K.E.=$21 mv_{2}$=$21 A_{2}w_{2}cos_{2}θ$=$21 mw_{2}(A_{2}−x_{2})$

T.E.=$21 mA_{2}w_{2}$= constant

So, P.E α $x_{2}$α $sin_{2}θ$ α $21−cos2θ $ =$2(1−cos2wt) $=$2(1−cos2πft) $

K.E α (A^{2}-x^{2}) α $cos_{2}θ$= $2(1+cos2θ $=$2(1+cos2wt) $=$2(1+cos2πft) $

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

**Examples**

- 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=mrw^{2} =$mr(2π∗f)_{2}$ α f^{2}

so,$F_{1}αf_{1}$and , $F_{2}αf_{2}$

for, $F_{1}+F_{2}$$∝$$(f_{1}+f_{2})$$∝f_{2}$ (say)

or, $f_{1}+f_{2}=f_{2}$,f is required frequency.

- 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

$mgh=−21 kx_{2}+mgx$

$kx_{2}−2mgkx+2mgh=0$

so, $x=2k2mg±4m_{2}g_{2}+4∗2mgh∗k $

Taking +ve as $4m_{2}g_{2}+..........$> mg

$x=kmg (1+1+mg2hk )$